How to evaluate $\int\frac{3x^{2}+7}{\sqrt{x^{4}+5x^{2}+9}}dx$?

Question

Q) How to evaluate $$\int\frac{3x^{2}+7}{\sqrt{x^{4}+5x^{2}+9}}dx$$?

Ans): First of all let me show my approach. I wrote the above integral as $$\int\frac{3x^{2}}{\sqrt{x^{4}+5x^{2}+9}}dx+\int\frac{7}{\sqrt{x^{4}+5x^{2}+9}}dx$$. Now $x^{4}+5x^{2}+9$ can be factorized as $x^{4}+5x^{2}+9=(x^{2})^{2}+2(\frac{5}{2})(x^{2})+(\frac{5}{2})^{2}+9-(\frac{5}{2})^{2}=(x^{2}+\frac{5}{2})^{2}+\frac{11}{4}$.

Therefore let's integrate $$\int\frac{3x^{2}}{\sqrt{x^{4}+5x^{2}+9}}dx$$ first.

$$\int\frac{3x^{2}}{\sqrt{x^{4}+5x^{2}+9}}dx=3\int\frac{x^{2}}{\sqrt{x^{4}+5x^{2}+9}}dx=3\int\frac{(x)(x)}{\sqrt{x^{4}+5x^{2}+9}}dx=3\int\frac{(x)(x)}{(x^{2}+\frac{5}{2})^{2}+\frac{11}{4}}dx=$$

$$\frac{3}{2}\int\frac{(x)(2x)}{\sqrt{(x^{2}+\frac{5}{2})^{2}+\frac{11}{4}}}dx$$ Now let's substitute $(x^{2}+\frac{5}{2})=t$. Therefore $2xdx=dt$. Therefore the integral $$\frac{3}{2}\int\frac{(x)(2x)}{\sqrt{(x^{2}+\frac{5}{2})^{2}+\frac{11}{4}}}dx$$ can be evaluated by applying Integration By Parts. Here we have to consider $u=x$ and $v=\frac{2x}{\sqrt{(x^{2}+\frac{5}{2})^{2}+\frac{11}{4}}}$.

Therefore the result of the integration of the $1st$ part will be $$\frac{3}{2}[x\int\frac{2x}{\sqrt{(x^{2}+\frac{5}{2})^{2}+\frac{11}{4}}}dx-\int\frac{d}{dx}(x)\int\frac{2x}{\sqrt{(x^{2}+\frac{5}{2})^{2}+\frac{11}{4}}}dx]=\frac{3}{2}[x\int\frac{1}{\sqrt{t^{2}+\frac{11}{4}}}dt-\int\int\frac{1}{\sqrt{t^{2}+\frac{11}{4}}}dt]=\frac{3}{2}[x\log|t+\sqrt{t^{2}+\frac{11}{4}}|-\int\log|t+\sqrt{t^{2}+\frac{11}{4}}|dt]=\frac{3}{2}[x\log|(x^{2}+\frac{5}{2})+\sqrt{(x^{2}+\frac{5}{2})^{2}+\frac{11}{4}}|-\int\log|t+\sqrt{t^{2}+\frac{11}{4}}|dt]$$. And after this step I can't proceed further in integrating the $1st$ part, i.e. $$\int\frac{3x^{2}}{\sqrt{x^{4}+5x^{2}+9}}dx$$.
But still once confirm whether my procedure is correct or not.

Now also we can write the $2nd$ part as $$\int\frac{7}{\sqrt{(x^{2}+\frac{5}{2})^{2}+\frac{11}{4}}}dx$$. But I don't know how to integrate the $2nd$ part. Please help me out with the evaluation of the above integral.

Answer 1

Just for your curiosity.

As said, this would involve elliptic integrals.

To give you a taste, consider the case of $$I=\int \frac {x^2+a}{\sqrt {(x^2+b)\,(x^2+c)}}\,dx$$ Skipping the intermediate steps and simplifications $$\color{blue}{I=\frac{i}{\sqrt{c}}\Bigg((c-a)\, F\left(i \sinh ^{-1}\left(\frac{x}{\sqrt{b}}\right)\bigg|\frac{b}{c}\right)-c \,E\left(i \sinh ^{-1}\left(\frac{x}{\sqrt{b}}\right)\bigg|\frac{b}{c}\right) \Bigg)}$$ where appear the elliptic integrals of the first and second kinds.

The problem is still more nasty if, as in your case, $b$ and $c$ are complex conjugate.

For your specific case where $a=\frac 7 3$, $b=\frac{5+i \sqrt{11}}{2}$ and $c=\frac{5-i \sqrt{11}}{2}$ and bounds $(0,1)$, the result will be $0.809427$ which, more or less, corresponds to one third of $2.42$.