Integrand size = 11, antiderivative size = 29 \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^3} \, dx=\frac {1}{2} a^2 \Gamma \left (-\frac {1}{2},a x\right )-\frac {\Gamma \left (\frac {3}{2},a x\right )}{2 x^2} \] Output:
1/2*a^2*(-2*Pi^(1/2)*erfc((a*x)^(1/2))+2/(a*x)^(1/2)*exp(-a*x))-1/2*((a*x) ^(1/2)*exp(-a*x)+1/2*Pi^(1/2)*erfc((a*x)^(1/2)))/x^2
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^3} \, dx=\frac {1}{2} a^2 \Gamma \left (-\frac {1}{2},a x\right )-\frac {\Gamma \left (\frac {3}{2},a x\right )}{2 x^2} \] Input:
Integrate[Gamma[3/2, a*x]/x^3,x]
Output:
(a^2*Gamma[-1/2, a*x])/2 - Gamma[3/2, a*x]/(2*x^2)
Time = 0.29 (sec) , antiderivative size = 29, 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 = {7116}
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.
\(\displaystyle \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 7116 |
\(\displaystyle \frac {1}{2} a^2 \Gamma \left (-\frac {1}{2},a x\right )-\frac {\Gamma \left (\frac {3}{2},a x\right )}{2 x^2}\) |
Input:
Int[Gamma[3/2, a*x]/x^3,x]
Output:
(a^2*Gamma[-1/2, a*x])/2 - Gamma[3/2, a*x]/(2*x^2)
Int[Gamma[n_, (b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*(Gamma[n, b*x]/(d*(m + 1))), x] - Simp[(d*x)^m*(Gamma[m + n + 1, b*x]/(b *(m + 1)*(b*x)^m)), x] /; FreeQ[{b, d, m, n}, x] && NeQ[m, -1]
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.41
method | result | size |
derivativedivides | \(-\frac {4 \sqrt {\pi }\, a^{4} \operatorname {erfc}\left (\sqrt {x a}\right ) x^{2}-4 a^{2} {\mathrm e}^{-x a} \left (x a \right )^{\frac {3}{2}}+\sqrt {\pi }\, a^{2} \operatorname {erfc}\left (\sqrt {x a}\right )+2 \sqrt {x a}\, a^{2} {\mathrm e}^{-x a}}{4 x^{2} a^{2}}\) | \(70\) |
default | \(a^{2} \left (\sqrt {\pi }\, \left (-\frac {\operatorname {erfc}\left (\sqrt {x a}\right )}{4 x^{2} a^{2}}-\frac {-\frac {{\mathrm e}^{-x a}}{3 \left (x a \right )^{\frac {3}{2}}}+\frac {2 \,{\mathrm e}^{-x a}}{3 \sqrt {x a}}+\frac {2 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{3}}{2 \sqrt {\pi }}\right )-\frac {2 \,{\mathrm e}^{-x a}}{3 \left (x a \right )^{\frac {3}{2}}}+\frac {4 \,{\mathrm e}^{-x a}}{3 \sqrt {x a}}+\frac {4 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{3}\right )\) | \(105\) |
parts | \(2 a^{2} \left (-\frac {{\mathrm e}^{-x a}}{3 \left (x a \right )^{\frac {3}{2}}}+\frac {2 \,{\mathrm e}^{-x a}}{3 \sqrt {x a}}+\frac {2 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{3}\right )+\sqrt {\pi }\, a^{2} \left (-\frac {\operatorname {erfc}\left (\sqrt {x a}\right )}{4 x^{2} a^{2}}-\frac {-\frac {{\mathrm e}^{-x a}}{3 \left (x a \right )^{\frac {3}{2}}}+\frac {2 \,{\mathrm e}^{-x a}}{3 \sqrt {x a}}+\frac {2 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{3}}{2 \sqrt {\pi }}\right )\) | \(110\) |
Input:
int(((x*a)^(1/2)*exp(-x*a)+1/2*Pi^(1/2)*erfc((x*a)^(1/2)))/x^3,x,method=_R ETURNVERBOSE)
Output:
-1/4*(4*Pi^(1/2)*a^4*erfc((x*a)^(1/2))*x^2-4*a^2*exp(-x*a)*(x*a)^(3/2)+Pi^ (1/2)*a^2*erfc((x*a)^(1/2))+2*(x*a)^(1/2)*a^2*exp(-x*a))/x^2/a^2
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^3} \, dx=\frac {\sqrt {\pi } {\left (4 \, a^{2} x^{2} + 1\right )} \operatorname {erf}\left (\sqrt {a x}\right ) + 2 \, {\left (2 \, a x - 1\right )} \sqrt {a x} e^{\left (-a x\right )} - \sqrt {\pi }}{4 \, x^{2}} \] Input:
integrate(((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2)))/x^3,x, al gorithm="fricas")
Output:
1/4*(sqrt(pi)*(4*a^2*x^2 + 1)*erf(sqrt(a*x)) + 2*(2*a*x - 1)*sqrt(a*x)*e^( -a*x) - sqrt(pi))/x^2
Time = 0.43 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.28 \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^3} \, dx=- \sqrt {\pi } a^{2} \operatorname {erfc}{\left (\sqrt {a x} \right )} + \frac {a \sqrt {a x} e^{- a x}}{x} - \frac {\sqrt {a x} e^{- a x}}{2 x^{2}} - \frac {\sqrt {\pi } \operatorname {erfc}{\left (\sqrt {a x} \right )}}{4 x^{2}} \] Input:
integrate(((a*x)**(1/2)*exp(-a*x)+1/2*pi**(1/2)*erfc((a*x)**(1/2)))/x**3,x )
Output:
-sqrt(pi)*a**2*erfc(sqrt(a*x)) + a*sqrt(a*x)*exp(-a*x)/x - sqrt(a*x)*exp(- a*x)/(2*x**2) - sqrt(pi)*erfc(sqrt(a*x))/(4*x**2)
\[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^3} \, dx=\int { \frac {\sqrt {\pi } \operatorname {erfc}\left (\sqrt {a x}\right ) + 2 \, \sqrt {a x} e^{\left (-a x\right )}}{2 \, x^{3}} \,d x } \] Input:
integrate(((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2)))/x^3,x, al gorithm="maxima")
Output:
1/2*integrate((sqrt(pi)*erfc(sqrt(a*x)) + 2*sqrt(a*x)*e^(-a*x))/x^3, x)
\[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^3} \, dx=\int { \frac {\sqrt {\pi } \operatorname {erfc}\left (\sqrt {a x}\right ) + 2 \, \sqrt {a x} e^{\left (-a x\right )}}{2 \, x^{3}} \,d x } \] Input:
integrate(((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2)))/x^3,x, al gorithm="giac")
Output:
integrate(1/2*(sqrt(pi)*erfc(sqrt(a*x)) + 2*sqrt(a*x)*e^(-a*x))/x^3, x)
Timed out. \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^3} \, dx=\int \frac {\frac {\sqrt {\Pi }\,\mathrm {erfc}\left (\sqrt {a\,x}\right )}{2}+{\mathrm {e}}^{-a\,x}\,\sqrt {a\,x}}{x^3} \,d x \] Input:
int(((Pi^(1/2)*erfc((a*x)^(1/2)))/2 + exp(-a*x)*(a*x)^(1/2))/x^3,x)
Output:
int(((Pi^(1/2)*erfc((a*x)^(1/2)))/2 + exp(-a*x)*(a*x)^(1/2))/x^3, x)
\[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^3} \, dx=\frac {\sqrt {x}\, \sqrt {\pi }\, e^{a x} \mathrm {erf}\left (\sqrt {x}\, \sqrt {a}\right )-2 \sqrt {x}\, e^{a x} \sqrt {a}\, \left (\int \frac {\sqrt {x}}{e^{a x} x^{2}}d x \right ) a \,x^{2}-2 \sqrt {a}\, x -\sqrt {x}\, \sqrt {\pi }\, e^{a x}}{4 \sqrt {x}\, e^{a x} x^{2}} \] Input:
int(((a*x)^(1/2)*exp(-a*x)+1/2*Pi^(1/2)*erfc((a*x)^(1/2)))/x^3,x)
Output:
(sqrt(x)*sqrt(pi)*e**(a*x)*erf(sqrt(x)*sqrt(a)) - 2*sqrt(x)*e**(a*x)*sqrt( a)*int(sqrt(x)/(e**(a*x)*x**2),x)*a*x**2 - 2*sqrt(a)*x - sqrt(x)*sqrt(pi)* e**(a*x))/(4*sqrt(x)*e**(a*x)*x**2)