Optimal. Leaf size=151 \[ \frac {135135 \sqrt {\pi } \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{128 b^{15/2}}-\frac {135135 \sqrt {x} e^{-b x}}{64 b^7}-\frac {45045 x^{3/2} e^{-b x}}{32 b^6}-\frac {9009 x^{5/2} e^{-b x}}{16 b^5}-\frac {1287 x^{7/2} e^{-b x}}{8 b^4}-\frac {143 x^{9/2} e^{-b x}}{4 b^3}-\frac {13 x^{11/2} e^{-b x}}{2 b^2}-\frac {x^{13/2} e^{-b x}}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2176, 2180, 2205} \[ \frac {135135 \sqrt {\pi } \text {Erf}\left (\sqrt {b} \sqrt {x}\right )}{128 b^{15/2}}-\frac {13 x^{11/2} e^{-b x}}{2 b^2}-\frac {143 x^{9/2} e^{-b x}}{4 b^3}-\frac {1287 x^{7/2} e^{-b x}}{8 b^4}-\frac {9009 x^{5/2} e^{-b x}}{16 b^5}-\frac {45045 x^{3/2} e^{-b x}}{32 b^6}-\frac {135135 \sqrt {x} e^{-b x}}{64 b^7}-\frac {x^{13/2} e^{-b x}}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2176
Rule 2180
Rule 2205
Rubi steps
\begin {align*} \int e^{-b x} x^{13/2} \, dx &=-\frac {e^{-b x} x^{13/2}}{b}+\frac {13 \int e^{-b x} x^{11/2} \, dx}{2 b}\\ &=-\frac {13 e^{-b x} x^{11/2}}{2 b^2}-\frac {e^{-b x} x^{13/2}}{b}+\frac {143 \int e^{-b x} x^{9/2} \, dx}{4 b^2}\\ &=-\frac {143 e^{-b x} x^{9/2}}{4 b^3}-\frac {13 e^{-b x} x^{11/2}}{2 b^2}-\frac {e^{-b x} x^{13/2}}{b}+\frac {1287 \int e^{-b x} x^{7/2} \, dx}{8 b^3}\\ &=-\frac {1287 e^{-b x} x^{7/2}}{8 b^4}-\frac {143 e^{-b x} x^{9/2}}{4 b^3}-\frac {13 e^{-b x} x^{11/2}}{2 b^2}-\frac {e^{-b x} x^{13/2}}{b}+\frac {9009 \int e^{-b x} x^{5/2} \, dx}{16 b^4}\\ &=-\frac {9009 e^{-b x} x^{5/2}}{16 b^5}-\frac {1287 e^{-b x} x^{7/2}}{8 b^4}-\frac {143 e^{-b x} x^{9/2}}{4 b^3}-\frac {13 e^{-b x} x^{11/2}}{2 b^2}-\frac {e^{-b x} x^{13/2}}{b}+\frac {45045 \int e^{-b x} x^{3/2} \, dx}{32 b^5}\\ &=-\frac {45045 e^{-b x} x^{3/2}}{32 b^6}-\frac {9009 e^{-b x} x^{5/2}}{16 b^5}-\frac {1287 e^{-b x} x^{7/2}}{8 b^4}-\frac {143 e^{-b x} x^{9/2}}{4 b^3}-\frac {13 e^{-b x} x^{11/2}}{2 b^2}-\frac {e^{-b x} x^{13/2}}{b}+\frac {135135 \int e^{-b x} \sqrt {x} \, dx}{64 b^6}\\ &=-\frac {135135 e^{-b x} \sqrt {x}}{64 b^7}-\frac {45045 e^{-b x} x^{3/2}}{32 b^6}-\frac {9009 e^{-b x} x^{5/2}}{16 b^5}-\frac {1287 e^{-b x} x^{7/2}}{8 b^4}-\frac {143 e^{-b x} x^{9/2}}{4 b^3}-\frac {13 e^{-b x} x^{11/2}}{2 b^2}-\frac {e^{-b x} x^{13/2}}{b}+\frac {135135 \int \frac {e^{-b x}}{\sqrt {x}} \, dx}{128 b^7}\\ &=-\frac {135135 e^{-b x} \sqrt {x}}{64 b^7}-\frac {45045 e^{-b x} x^{3/2}}{32 b^6}-\frac {9009 e^{-b x} x^{5/2}}{16 b^5}-\frac {1287 e^{-b x} x^{7/2}}{8 b^4}-\frac {143 e^{-b x} x^{9/2}}{4 b^3}-\frac {13 e^{-b x} x^{11/2}}{2 b^2}-\frac {e^{-b x} x^{13/2}}{b}+\frac {135135 \operatorname {Subst}\left (\int e^{-b x^2} \, dx,x,\sqrt {x}\right )}{64 b^7}\\ &=-\frac {135135 e^{-b x} \sqrt {x}}{64 b^7}-\frac {45045 e^{-b x} x^{3/2}}{32 b^6}-\frac {9009 e^{-b x} x^{5/2}}{16 b^5}-\frac {1287 e^{-b x} x^{7/2}}{8 b^4}-\frac {143 e^{-b x} x^{9/2}}{4 b^3}-\frac {13 e^{-b x} x^{11/2}}{2 b^2}-\frac {e^{-b x} x^{13/2}}{b}+\frac {135135 \sqrt {\pi } \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{128 b^{15/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 24, normalized size = 0.16 \[ -\frac {\sqrt {b x} \Gamma \left (\frac {15}{2},b x\right )}{b^8 \sqrt {x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 82, normalized size = 0.54 \[ -\frac {2 \, {\left (64 \, b^{7} x^{6} + 416 \, b^{6} x^{5} + 2288 \, b^{5} x^{4} + 10296 \, b^{4} x^{3} + 36036 \, b^{3} x^{2} + 90090 \, b^{2} x + 135135 \, b\right )} \sqrt {x} e^{\left (-b x\right )} - 135135 \, \sqrt {\pi } \sqrt {b} \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right )}{128 \, b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.36, size = 80, normalized size = 0.53 \[ -\frac {{\left (64 \, b^{6} x^{\frac {13}{2}} + 416 \, b^{5} x^{\frac {11}{2}} + 2288 \, b^{4} x^{\frac {9}{2}} + 10296 \, b^{3} x^{\frac {7}{2}} + 36036 \, b^{2} x^{\frac {5}{2}} + 90090 \, b x^{\frac {3}{2}} + 135135 \, \sqrt {x}\right )} e^{\left (-b x\right )}}{64 \, b^{7}} - \frac {135135 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} \sqrt {x}\right )}{128 \, b^{\frac {15}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.14, size = 145, normalized size = 0.96 \[ -\frac {x^{\frac {13}{2}} {\mathrm e}^{-b x}}{b}+\frac {-\frac {13 x^{\frac {11}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {13 \left (-\frac {11 x^{\frac {9}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {11 \left (-\frac {9 x^{\frac {7}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {9 \left (-\frac {7 x^{\frac {5}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {7 \left (-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \erf \left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{2 b}\right )}{2 b}\right )}{2 b}\right )}{b}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.84, size = 79, normalized size = 0.52 \[ -\frac {{\left (64 \, b^{6} x^{\frac {13}{2}} + 416 \, b^{5} x^{\frac {11}{2}} + 2288 \, b^{4} x^{\frac {9}{2}} + 10296 \, b^{3} x^{\frac {7}{2}} + 36036 \, b^{2} x^{\frac {5}{2}} + 90090 \, b x^{\frac {3}{2}} + 135135 \, \sqrt {x}\right )} e^{\left (-b x\right )}}{64 \, b^{7}} + \frac {135135 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right )}{128 \, b^{\frac {15}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.43, size = 89, normalized size = 0.59 \[ -\frac {135135\,x^{13/2}\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {b\,x}\right )}{128\,b\,{\left (b\,x\right )}^{13/2}}-\frac {x^{13/2}\,{\mathrm {e}}^{-b\,x}\,\left (\frac {135135\,\sqrt {b\,x}}{64}+\frac {45045\,{\left (b\,x\right )}^{3/2}}{32}+\frac {9009\,{\left (b\,x\right )}^{5/2}}{16}+\frac {1287\,{\left (b\,x\right )}^{7/2}}{8}+\frac {143\,{\left (b\,x\right )}^{9/2}}{4}+\frac {13\,{\left (b\,x\right )}^{11/2}}{2}+{\left (b\,x\right )}^{13/2}\right )}{b\,{\left (b\,x\right )}^{13/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________