Integrand size = 11, antiderivative size = 29 \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^4} \, dx=\frac {1}{3} a^3 \Gamma \left (-\frac {3}{2},a x\right )-\frac {\Gamma \left (\frac {3}{2},a x\right )}{3 x^3} \] Output:
1/3*a^3*(4/3*Pi^(1/2)*erfc((a*x)^(1/2))-4/3/(a*x)^(1/2)*exp(-a*x)+2/3/(a*x )^(3/2)*exp(-a*x))-1/3*((a*x)^(1/2)*exp(-a*x)+1/2*Pi^(1/2)*erfc((a*x)^(1/2 )))/x^3
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^4} \, dx=\frac {1}{3} a^3 \Gamma \left (-\frac {3}{2},a x\right )-\frac {\Gamma \left (\frac {3}{2},a x\right )}{3 x^3} \] Input:
Integrate[Gamma[3/2, a*x]/x^4,x]
Output:
(a^3*Gamma[-3/2, a*x])/3 - Gamma[3/2, a*x]/(3*x^3)
Time = 0.30 (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.
\(\displaystyle \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 7116 |
\(\displaystyle \frac {1}{3} a^3 \Gamma \left (-\frac {3}{2},a x\right )-\frac {\Gamma \left (\frac {3}{2},a x\right )}{3 x^3}\) |
Input:
Int[Gamma[3/2, a*x]/x^4,x]
Output:
(a^3*Gamma[-3/2, a*x])/3 - Gamma[3/2, a*x]/(3*x^3)
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 = 0.37 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.97
method | result | size |
derivativedivides | \(\frac {8 \sqrt {\pi }\, a^{6} \operatorname {erfc}\left (\sqrt {x a}\right ) x^{3}-8 a^{3} {\mathrm e}^{-x a} \left (x a \right )^{\frac {5}{2}}+4 a^{3} {\mathrm e}^{-x a} \left (x a \right )^{\frac {3}{2}}-3 \sqrt {\pi }\, a^{3} \operatorname {erfc}\left (\sqrt {x a}\right )-6 a^{3} {\mathrm e}^{-x a} \sqrt {x a}}{18 x^{3} a^{3}}\) | \(86\) |
default | \(a^{3} \left (\sqrt {\pi }\, \left (-\frac {\operatorname {erfc}\left (\sqrt {x a}\right )}{6 x^{3} a^{3}}-\frac {-\frac {{\mathrm e}^{-x a}}{5 \left (x a \right )^{\frac {5}{2}}}+\frac {2 \,{\mathrm e}^{-x a}}{15 \left (x a \right )^{\frac {3}{2}}}-\frac {4 \,{\mathrm e}^{-x a}}{15 \sqrt {x a}}-\frac {4 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{15}}{3 \sqrt {\pi }}\right )-\frac {2 \,{\mathrm e}^{-x a}}{5 \left (x a \right )^{\frac {5}{2}}}+\frac {4 \,{\mathrm e}^{-x a}}{15 \left (x a \right )^{\frac {3}{2}}}-\frac {8 \,{\mathrm e}^{-x a}}{15 \sqrt {x a}}-\frac {8 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{15}\right )\) | \(131\) |
parts | \(2 a^{3} \left (-\frac {{\mathrm e}^{-x a}}{5 \left (x a \right )^{\frac {5}{2}}}+\frac {2 \,{\mathrm e}^{-x a}}{15 \left (x a \right )^{\frac {3}{2}}}-\frac {4 \,{\mathrm e}^{-x a}}{15 \sqrt {x a}}-\frac {4 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{15}\right )+\sqrt {\pi }\, a^{3} \left (-\frac {\operatorname {erfc}\left (\sqrt {x a}\right )}{6 x^{3} a^{3}}-\frac {-\frac {{\mathrm e}^{-x a}}{5 \left (x a \right )^{\frac {5}{2}}}+\frac {2 \,{\mathrm e}^{-x a}}{15 \left (x a \right )^{\frac {3}{2}}}-\frac {4 \,{\mathrm e}^{-x a}}{15 \sqrt {x a}}-\frac {4 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {x a}\right )}{15}}{3 \sqrt {\pi }}\right )\) | \(136\) |
Input:
int(((x*a)^(1/2)*exp(-x*a)+1/2*Pi^(1/2)*erfc((x*a)^(1/2)))/x^4,x,method=_R ETURNVERBOSE)
Output:
1/18*(8*Pi^(1/2)*a^6*erfc((x*a)^(1/2))*x^3-8*a^3*exp(-x*a)*(x*a)^(5/2)+4*a ^3*exp(-x*a)*(x*a)^(3/2)-3*Pi^(1/2)*a^3*erfc((x*a)^(1/2))-6*a^3*exp(-x*a)* (x*a)^(1/2))/x^3/a^3
Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97 \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^4} \, dx=-\frac {\sqrt {\pi } {\left (8 \, a^{3} x^{3} - 3\right )} \operatorname {erf}\left (\sqrt {a x}\right ) + 2 \, {\left (4 \, a^{2} x^{2} - 2 \, a x + 3\right )} \sqrt {a x} e^{\left (-a x\right )} + 3 \, \sqrt {\pi }}{18 \, x^{3}} \] Input:
integrate(((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2)))/x^4,x, al gorithm="fricas")
Output:
-1/18*(sqrt(pi)*(8*a^3*x^3 - 3)*erf(sqrt(a*x)) + 2*(4*a^2*x^2 - 2*a*x + 3) *sqrt(a*x)*e^(-a*x) + 3*sqrt(pi))/x^3
Time = 0.75 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.28 \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^4} \, dx=\frac {4 \sqrt {\pi } a^{3} \operatorname {erfc}{\left (\sqrt {a x} \right )}}{9} - \frac {4 a^{2} \sqrt {a x} e^{- a x}}{9 x} + \frac {2 a \sqrt {a x} e^{- a x}}{9 x^{2}} - \frac {\sqrt {a x} e^{- a x}}{3 x^{3}} - \frac {\sqrt {\pi } \operatorname {erfc}{\left (\sqrt {a x} \right )}}{6 x^{3}} \] Input:
integrate(((a*x)**(1/2)*exp(-a*x)+1/2*pi**(1/2)*erfc((a*x)**(1/2)))/x**4,x )
Output:
4*sqrt(pi)*a**3*erfc(sqrt(a*x))/9 - 4*a**2*sqrt(a*x)*exp(-a*x)/(9*x) + 2*a *sqrt(a*x)*exp(-a*x)/(9*x**2) - sqrt(a*x)*exp(-a*x)/(3*x**3) - sqrt(pi)*er fc(sqrt(a*x))/(6*x**3)
\[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^4} \, dx=\int { \frac {\sqrt {\pi } \operatorname {erfc}\left (\sqrt {a x}\right ) + 2 \, \sqrt {a x} e^{\left (-a x\right )}}{2 \, x^{4}} \,d x } \] Input:
integrate(((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2)))/x^4,x, al gorithm="maxima")
Output:
1/2*integrate((sqrt(pi)*erfc(sqrt(a*x)) + 2*sqrt(a*x)*e^(-a*x))/x^4, x)
\[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^4} \, dx=\int { \frac {\sqrt {\pi } \operatorname {erfc}\left (\sqrt {a x}\right ) + 2 \, \sqrt {a x} e^{\left (-a x\right )}}{2 \, x^{4}} \,d x } \] Input:
integrate(((a*x)^(1/2)*exp(-a*x)+1/2*pi^(1/2)*erfc((a*x)^(1/2)))/x^4,x, al gorithm="giac")
Output:
integrate(1/2*(sqrt(pi)*erfc(sqrt(a*x)) + 2*sqrt(a*x)*e^(-a*x))/x^4, x)
Timed out. \[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^4} \, dx=\int \frac {\frac {\sqrt {\Pi }\,\mathrm {erfc}\left (\sqrt {a\,x}\right )}{2}+{\mathrm {e}}^{-a\,x}\,\sqrt {a\,x}}{x^4} \,d x \] Input:
int(((Pi^(1/2)*erfc((a*x)^(1/2)))/2 + exp(-a*x)*(a*x)^(1/2))/x^4,x)
Output:
int(((Pi^(1/2)*erfc((a*x)^(1/2)))/2 + exp(-a*x)*(a*x)^(1/2))/x^4, x)
\[ \int \frac {\Gamma \left (\frac {3}{2},a x\right )}{x^4} \, dx=\frac {3 \sqrt {x}\, \sqrt {\pi }\, e^{a x} \mathrm {erf}\left (\sqrt {x}\, \sqrt {a}\right )+4 \sqrt {x}\, e^{a x} \sqrt {a}\, \left (\int \frac {\sqrt {x}}{e^{a x} x^{2}}d x \right ) a^{2} x^{3}+4 \sqrt {a}\, a \,x^{2}-6 \sqrt {a}\, x -3 \sqrt {x}\, \sqrt {\pi }\, e^{a x}}{18 \sqrt {x}\, e^{a x} x^{3}} \] Input:
int(((a*x)^(1/2)*exp(-a*x)+1/2*Pi^(1/2)*erfc((a*x)^(1/2)))/x^4,x)
Output:
(3*sqrt(x)*sqrt(pi)*e**(a*x)*erf(sqrt(x)*sqrt(a)) + 4*sqrt(x)*e**(a*x)*sqr t(a)*int(sqrt(x)/(e**(a*x)*x**2),x)*a**2*x**3 + 4*sqrt(a)*a*x**2 - 6*sqrt( a)*x - 3*sqrt(x)*sqrt(pi)*e**(a*x))/(18*sqrt(x)*e**(a*x)*x**3)