Integral - Combinations of logarithms, exponentials, and powers $\int_0^\infty x^{\nu-1}\ e^{-\mu x}\ \ln(x+a)\,dx$

Question

$$\int_0^\infty x^{\nu-1}\ e^{-\mu x}\ \ln(x)\ dx = \frac1{\mu^\nu}\Gamma(\nu)\left[\psi(\nu)-\ln(\mu)\right] \quad\qquad [\Re\,\mu \gt 0, \quad \Re\,\nu\gt 0]$$

Hello, I found above equation on Table of Integrals, Series, and Products by I.S. GradshteynI.M. Ryzhik. I want extend the problem from above integral, by changing $\ln(x)$ to $(\ln(x+a))$, which $a$ is a variable. Anybody can help me to solve it ?

Answer 1

$$\int_0^{\infty } x^{v-1} \exp (-\mu x) \log (x+a) \, dx=\\\mathcal{L}_x\left[x^{v-1} \log (x+a)\right](\mu )=\\\mathcal{M}_x[\exp (-\mu x) \log (x+a)](v)$$

Integral is equivalent to Laplace Transform or Mellin Transform.

With Maple help:

$\int_0^{\infty } x^{v-1} \exp (-\mu x) \log (x+a) \, dx={\frac {\mu\,{\mbox{$_2$F$_2$}(1,1;\,2,2-v;\,\mu\,a)}\Gamma \left( v \right) a}{{\mu}^{v} \left( v-1 \right) }}-{\frac {{a}^{v}\pi\,\Gamma \left( v,-\mu\,a \right) }{\sin \left( \pi\,v \right) \left( -\mu\,a \right) ^{v}}}+{\frac {{a}^{v}\pi\,\Gamma \left( v \right) v}{\sin \left( \pi\,v \right) \left( -\mu\,a \right) ^{v} \left( v-1 \right) }}-{\frac {{a}^{v}\pi\,\Gamma \left( v \right) }{\sin \left( \pi\,v \right) \left( -\mu\,a \right) ^{v} \left( v-1 \right) }}+{ \frac {\Gamma \left( v \right) \Psi \left( v \right) v}{{\mu}^{v} \left( v-1 \right) }}-{\frac {\Gamma \left( v \right) \Psi \left( v \right) }{{\mu}^{v} \left( v-1 \right) }}-{\frac {\Gamma \left( v \right) v\ln \left( \mu \right) }{{\mu}^{v} \left( v-1 \right) }}+{ \frac {\Gamma \left( v \right) \ln \left( \mu \right) }{{\mu}^{v} \left( v-1 \right) }} $

where:$\, _2F_2(1,1;2,2-v;a \mu )$ is the generalized hypergeometric function

where: $\Psi \left( v \right)$ is PolyGamma function

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Edited 11.04.2018:

On Maple 2018:

 `assuming`([inttrans:-laplace(x^(v-1)*ln(x+a), x, mu)], [a > 0, mu > 0, v > 
 0, v in integer, x > 0])# Can't find.

 `assuming`([inttrans:-mellin(exp(-mu*x)*ln(x+a), x, v)], [a in real, mu > 0])# Can find.

 #(Pi*GAMMA(v)*mu^(-v+1)*csc(Pi*v)*GAMMA(-v+1)*mu^(v-1)*(1/a)^(-v)*(- 
 a*mu)^(-v)*mu-Pi*mu^(-v+1)*csc(Pi*v)*GAMMA(-v+1)*mu^(v-1)*(1/a)^(-v)*(- 
 a*mu)^(-v)*GAMMA(v, -a*mu)*mu+GAMMA(v-1)*mu^(-v+1)*GAMMA(- 
 v+1)*hypergeom([1, 1], [2, 2-v], a*mu)*a*mu-Pi^2*mu^(- 
 v+1)*cot(Pi*v)*csc(Pi*v)+ln(a)*mu^(-v)*GAMMA(v)*GAMMA(-v+1)*mu+Pi*mu^(- 
 v+1)*Psi(-v+1)*csc(Pi*v)-Pi*mu^(-v+1)*ln(mu)*csc(Pi*v)+Pi*mu^(- 
 v+1)*ln(1/a)*csc(Pi*v))/(GAMMA(-v+1)*mu)

 `assuming`([int(x^(v-1)*exp(-mu*x)*ln(x+a), x = 0 .. infinity)], [a in real, mu > 0, x > 0, v > 0])# Can find.

 #ln(a)*mu^(-v)*GAMMA(v)+mu^(-v+1)*a*(-(-Psi(-v+1)+Pi*cot(Pi*v)+ln(mu)- 
 ln(1/a))*Pi*csc(Pi*v)/(GAMMA(-v+1)*mu*a)+GAMMA(v-1)*hypergeom([1, 1], [2, 
 2-v], a*mu)+Pi*mu^(v-1)*(1/a)^(-v)*(-a*mu)^(-v)*csc(Pi*v)*GAMMA(v)/a- 
 Pi*mu^(v-1)*(1/a)^(-v)*(-a*mu)^(-v)*GAMMA(v, -a*mu)*csc(Pi*v)/a)