Integrand size = 11, antiderivative size = 42 \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x} \, dx=-\frac {4}{9} (a x)^{3/2} \, _2F_2\left (\frac {3}{2},\frac {3}{2};\frac {5}{2},\frac {5}{2};-a x\right )+\frac {1}{2} \sqrt {\pi } \log (x) \] Output:
-4/9*(a*x)^(3/2)*hypergeom([3/2, 3/2],[5/2, 5/2],-a*x)+1/2*Pi^(1/2)*ln(x)
Time = 0.23 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.86 \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x} \, dx=-\frac {4}{9} (a x)^{3/2} \, _2F_2\left (\frac {3}{2},\frac {3}{2};\frac {5}{2},\frac {5}{2};-a x\right )+\frac {1}{2} \left (-2 e^{-a x} \sqrt {a x}+\sqrt {\pi } \text {erf}\left (\sqrt {a x}\right )+2 \Gamma \left (\frac {3}{2},a x\right )\right ) \log (a x) \] Input:
Integrate[Gamma[3/2, a*x]/x,x]
Output:
(-4*(a*x)^(3/2)*HypergeometricPFQ[{3/2, 3/2}, {5/2, 5/2}, -(a*x)])/9 + ((( -2*Sqrt[a*x])/E^(a*x) + Sqrt[Pi]*Erf[Sqrt[a*x]] + 2*Gamma[3/2, a*x])*Log[a *x])/2
Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {7115}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x} \, dx\) |
\(\Big \downarrow \) 7115 |
\(\displaystyle \frac {1}{2} \sqrt {\pi } \log (x)-\frac {4}{9} (a x)^{3/2} \, _2F_2\left (\frac {3}{2},\frac {3}{2};\frac {5}{2},\frac {5}{2};-a x\right )\) |
Input:
Int[Gamma[3/2, a*x]/x,x]
Output:
(-4*(a*x)^(3/2)*HypergeometricPFQ[{3/2, 3/2}, {5/2, 5/2}, -(a*x)])/9 + (Sq rt[Pi]*Log[x])/2
Int[Gamma[n_, (b_.)*(x_)]/(x_), x_Symbol] :> Simp[Gamma[n]*Log[x], x] - Sim p[((b*x)^n/n^2)*HypergeometricPFQ[{n, n}, {1 + n, 1 + n}, (-b)*x], x] /; Fr eeQ[{b, n}, x] && !IntegerQ[n]
\[\int \frac {\sqrt {x a}\, {\mathrm e}^{-x a}+\frac {\sqrt {\pi }\, \operatorname {erfc}\left (\sqrt {x a}\right )}{2}}{x}d x\]
Input:
int(((x*a)^(1/2)*exp(-x*a)+1/2*Pi^(1/2)*erfc((x*a)^(1/2)))/x,x)
Output:
int(((x*a)^(1/2)*exp(-x*a)+1/2*Pi^(1/2)*erfc((x*a)^(1/2)))/x,x)
\[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x} \, dx=\int { \frac {\sqrt {\pi } \operatorname {erfc}\left (\sqrt {a x}\right ) + 2 \, \sqrt {a x} e^{\left (-a x\right )}}{2 \, x} \,d x } \] Input:
integrate(((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2)))/x,x, algo rithm="fricas")
Output:
integral(-1/2*(sqrt(pi)*erf(sqrt(a*x)) - 2*sqrt(a*x)*e^(-a*x) - sqrt(pi))/ x, x)
\[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x} \, dx=\frac {\int \frac {\sqrt {\pi } \operatorname {erfc}{\left (\sqrt {a x} \right )}}{x}\, dx + \int \frac {2 \sqrt {a x} e^{- a x}}{x}\, dx}{2} \] Input:
integrate(((a*x)**(1/2)*exp(-a*x)+1/2*pi**(1/2)*erfc((a*x)**(1/2)))/x,x)
Output:
(Integral(sqrt(pi)*erfc(sqrt(a*x))/x, x) + Integral(2*sqrt(a*x)*exp(-a*x)/ x, x))/2
\[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x} \, dx=\int { \frac {\sqrt {\pi } \operatorname {erfc}\left (\sqrt {a x}\right ) + 2 \, \sqrt {a x} e^{\left (-a x\right )}}{2 \, x} \,d x } \] Input:
integrate(((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2)))/x,x, algo rithm="maxima")
Output:
1/2*integrate((sqrt(pi)*erfc(sqrt(a*x)) + 2*sqrt(a*x)*e^(-a*x))/x, x)
\[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x} \, dx=\int { \frac {\sqrt {\pi } \operatorname {erfc}\left (\sqrt {a x}\right ) + 2 \, \sqrt {a x} e^{\left (-a x\right )}}{2 \, x} \,d x } \] Input:
integrate(((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2)))/x,x, algo rithm="giac")
Output:
integrate(1/2*(sqrt(pi)*erfc(sqrt(a*x)) + 2*sqrt(a*x)*e^(-a*x))/x, x)
Timed out. \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x} \, dx=\int \frac {\frac {\sqrt {\Pi }\,\mathrm {erfc}\left (\sqrt {a\,x}\right )}{2}+{\mathrm {e}}^{-a\,x}\,\sqrt {a\,x}}{x} \,d x \] Input:
int(((Pi^(1/2)*erfc((a*x)^(1/2)))/2 + exp(-a*x)*(a*x)^(1/2))/x,x)
Output:
int(((Pi^(1/2)*erfc((a*x)^(1/2)))/2 + exp(-a*x)*(a*x)^(1/2))/x, x)
\[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x} \, dx=\frac {\sqrt {\pi }\, \left (2 \,\mathrm {erf}\left (\sqrt {x}\, \sqrt {a}\right )-\left (\int \frac {\mathrm {erf}\left (\sqrt {x}\, \sqrt {a}\right )}{x}d x \right )+\mathrm {log}\left (x \right )\right )}{2} \] Input:
int(((a*x)^(1/2)*exp(-a*x)+1/2*Pi^(1/2)*erfc((a*x)^(1/2)))/x,x)
Output:
(sqrt(pi)*(2*erf(sqrt(x)*sqrt(a)) - int(erf(sqrt(x)*sqrt(a))/x,x) + log(x) ))/2