I wonder if there exists an explicit solution to the integral $$\int_{1}^{x} \frac{d{u}}{\sqrt{(1-a/\sqrt{u}-b~u)^{3}}}\tag{1}$$ Any help or advice is welcome.
Appendix
The full expression with integral $(1)$ reads \begin{equation} f(x)=(1-a/\sqrt{x}-b~x)^{1/2}\bigg\{1+p_1/4~(1-a-b)^{1/2}\int_{1}^{x}(1-a/\sqrt{u}-b~u)^{-3/2}d{u}\bigg\}\tag{2} \end{equation} where $x\in [1,\infty)$, $p_1\ge 0, $ $a\in[0,1]$ and $b\le 1-a$. The expression $(2)$ defines a real-valued non-negative function $f(x)$.
It there any possibility to get that integral without buying Maple?
I do not know why such a thing would be useful.
Here is a shorter one, with an answer only one page long.
We are interested in the indefinte integral $$ \int \frac{d{x}}{\sqrt{(1-a/\sqrt{x}-bx)^{3}}} \tag1$$ substitute $u=\sqrt{x}$ to get $$ 2i\int\frac{u^{5/2}\,du}{(bu^3-u+a)^{3/2}} \tag2$$ Factor the cubic polynomial (over the complex numbers); then it suffices to evaluate $$ \int\frac{u^{5/2}}{\big((u-a1)(u-a2)(u-a3)\big)^{3/2}} \tag3$$ According to Maple, this is
-2/u^(1/2)*(2*u^2*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3* (u-a1)/a1/(u-a3))^(1/2)*EllipticE(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/ (-a3+a1))^(1/2))*a1^2*a2^3*a3-u^2*a1^2*a3^4+u^2*a1^3*a3^3+u^2*a2^3*a3^3+6*u*((- a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/ 2)*EllipticF(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^2 *a2^2*a3^3-4*u*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u- a1)/a1/(u-a3))^(1/2)*EllipticF(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(- a3+a1))^(1/2))*a1*a2^3*a3^3+2*u^2*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u -a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticE(((-a3+a1)*u/a1/(u-a3))^(1/2), (a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^3*a2^2*a3-2*u^2*((-a3+a1)*u/a1/(u-a3))^(1/2 )*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticE(((-a3+a1)*u /a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^3*a2*a3^2-4*u*((-a3+a1)*u /a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)* EllipticE(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^2*a2 ^2*a3^3+4*u*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1) /a1/(u-a3))^(1/2)*EllipticE(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+ a1))^(1/2))*a1*a2^3*a3^3+u^2*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3)) ^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticF(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*( -a3+a2)/a2/(-a3+a1))^(1/2))*a1^3*a2^2*a3+u^2*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*( u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticF(((-a3+a1)*u/a1/(u- a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^3*a2*a3^2-u^2*((-a3+a1)*u/a1/(u- a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticF((( -a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^2*a2^3*a3-3*u^2* ((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^ (1/2)*EllipticF(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))* a1^2*a2^2*a3^2+2*u^2*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*( a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticF(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/ a2/(-a3+a1))^(1/2))*a1*a2^3*a3^2-2*u*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2 /(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticF(((-a3+a1)*u/a1/(u-a3))^(1/ 2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^3*a2^2*a3^2-2*u*((-a3+a1)*u/a1/(u-a3))^( 1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticF(((-a3+a1 )*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^3*a2*a3^3-4*u*((-a3+a1 )*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)* EllipticE(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^3*a2 ^2*a3^2+4*u*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1) /a1/(u-a3))^(1/2)*EllipticE(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+ a1))^(1/2))*a1^3*a2*a3^3-4*u*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3)) ^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticE(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*( -a3+a2)/a2/(-a3+a1))^(1/2))*a1^2*a2^3*a3^2+2*u^2*((-a3+a1)*u/a1/(u-a3))^(1/2)*( a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticE(((-a3+a1)*u/a1 /(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^2*a2^2*a3^2-2*u^2*((-a3+a1)* u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)* EllipticE(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1*a2^3 *a3^2+4*u*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/ a1/(u-a3))^(1/2)*EllipticE(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1 ))^(1/2))*a1^3*a2^3*a3+2*u*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^( 1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticF(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(- a3+a2)/a2/(-a3+a1))^(1/2))*a1^2*a2^3*a3^2+((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u- a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticF(((-a3+a1)*u/a1/(u-a3 ))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^3*a2^2*a3^3+((-a3+a1)*u/a1/(u-a3)) ^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticF(((-a3+ a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^3*a2*a3^4-((-a3+a1)* u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)* EllipticF(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^2*a2 ^3*a3^3-3*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/ a1/(u-a3))^(1/2)*EllipticF(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1 ))^(1/2))*a1^2*a2^2*a3^4+2*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^( 1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticF(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(- a3+a2)/a2/(-a3+a1))^(1/2))*a1*a2^3*a3^4-2*u^2*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3* (u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticE(((-a3+a1)*u/a1/(u -a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^3*a2^3-2*((-a3+a1)*u/a1/(u-a3)) ^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticE(((-a3+ a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^3*a2^3*a3^2+2*((-a3+ a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)* EllipticE(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^3*a2 ^2*a3^3-2*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/ a1/(u-a3))^(1/2)*EllipticE(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1 ))^(1/2))*a1^3*a2*a3^4+2*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/ 2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticE(((-a3+a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+ a2)/a2/(-a3+a1))^(1/2))*a1^2*a2^3*a3^3+2*((-a3+a1)*u/a1/(u-a3))^(1/2)*(a3*(u-a2 )/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticE(((-a3+a1)*u/a1/(u-a3)) ^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1^2*a2^2*a3^4-2*((-a3+a1)*u/a1/(u-a3)) ^(1/2)*(a3*(u-a2)/a2/(u-a3))^(1/2)*(a3*(u-a1)/a1/(u-a3))^(1/2)*EllipticE(((-a3+ a1)*u/a1/(u-a3))^(1/2),(a1*(-a3+a2)/a2/(-a3+a1))^(1/2))*a1*a2^3*a3^4-2*u^2*a1^2 *a2^2*a3^2+2*u^2*a1*a2^2*a3^3+3*u*a1^3*a2^2*a3^2+2*u^2*a1^2*a2*a3^3+3*u*a1^2*a2 ^3*a3^2-2*u^2*a1^3*a2*a3^2-2*u^2*a1*a2^3*a3^2-2*u*a1^3*a2^3*a3-u*a1^3*a2*a3^3-u *a1*a2^3*a3^3-4*u*a1^2*a2^2*a3^3+u*a1^2*a2*a3^4+u^2*a1^3*a2^2*a3+u^2*a1^2*a2^3* a3+u*a1*a2^2*a3^4-u^2*a2^2*a3^4)/((u-a1)*(u-a2)*(u-a3))^(1/2)/(-a3+a1)^2/(-a3+ a2)^2/a3/(-a2+a1)^2Writing the three roots
a1,a2,a3in terms of radicals using Cardano's formula produces a multi-page answer as I mentioned in a comment.