Optimal. Leaf size=131 \[ \frac {105 \sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{16 b^{9/2} \log ^{\frac {9}{2}}(F)}-\frac {105 \sqrt {x} F^{a+b x}}{8 b^4 \log ^4(F)}+\frac {35 x^{3/2} F^{a+b x}}{4 b^3 \log ^3(F)}-\frac {7 x^{5/2} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac {x^{7/2} F^{a+b x}}{b \log (F)} \]
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Rubi [A] time = 0.15, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2176, 2180, 2204} \[ \frac {105 \sqrt {\pi } F^a \text {Erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{16 b^{9/2} \log ^{\frac {9}{2}}(F)}-\frac {7 x^{5/2} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac {35 x^{3/2} F^{a+b x}}{4 b^3 \log ^3(F)}-\frac {105 \sqrt {x} F^{a+b x}}{8 b^4 \log ^4(F)}+\frac {x^{7/2} F^{a+b x}}{b \log (F)} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2180
Rule 2204
Rubi steps
\begin {align*} \int F^{a+b x} x^{7/2} \, dx &=\frac {F^{a+b x} x^{7/2}}{b \log (F)}-\frac {7 \int F^{a+b x} x^{5/2} \, dx}{2 b \log (F)}\\ &=-\frac {7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{7/2}}{b \log (F)}+\frac {35 \int F^{a+b x} x^{3/2} \, dx}{4 b^2 \log ^2(F)}\\ &=\frac {35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac {7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{7/2}}{b \log (F)}-\frac {105 \int F^{a+b x} \sqrt {x} \, dx}{8 b^3 \log ^3(F)}\\ &=-\frac {105 F^{a+b x} \sqrt {x}}{8 b^4 \log ^4(F)}+\frac {35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac {7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{7/2}}{b \log (F)}+\frac {105 \int \frac {F^{a+b x}}{\sqrt {x}} \, dx}{16 b^4 \log ^4(F)}\\ &=-\frac {105 F^{a+b x} \sqrt {x}}{8 b^4 \log ^4(F)}+\frac {35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac {7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{7/2}}{b \log (F)}+\frac {105 \operatorname {Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt {x}\right )}{8 b^4 \log ^4(F)}\\ &=\frac {105 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{16 b^{9/2} \log ^{\frac {9}{2}}(F)}-\frac {105 F^{a+b x} \sqrt {x}}{8 b^4 \log ^4(F)}+\frac {35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac {7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{7/2}}{b \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 0.27 \[ \frac {F^a \sqrt {-b x \log (F)} \Gamma \left (\frac {9}{2},-b x \log (F)\right )}{b^5 \sqrt {x} \log ^5(F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 89, normalized size = 0.68 \[ -\frac {105 \, \sqrt {\pi } \sqrt {-b \log \relax (F)} F^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (F)} \sqrt {x}\right ) - 2 \, {\left (8 \, b^{4} x^{3} \log \relax (F)^{4} - 28 \, b^{3} x^{2} \log \relax (F)^{3} + 70 \, b^{2} x \log \relax (F)^{2} - 105 \, b \log \relax (F)\right )} F^{b x + a} \sqrt {x}}{16 \, b^{5} \log \relax (F)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 94, normalized size = 0.72 \[ -\frac {105 \, \sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} \sqrt {x}\right )}{16 \, \sqrt {-b \log \relax (F)} b^{4} \log \relax (F)^{4}} + \frac {{\left (8 \, b^{3} x^{\frac {7}{2}} \log \relax (F)^{3} - 28 \, b^{2} x^{\frac {5}{2}} \log \relax (F)^{2} + 70 \, b x^{\frac {3}{2}} \log \relax (F) - 105 \, \sqrt {x}\right )} e^{\left (b x \log \relax (F) + a \log \relax (F)\right )}}{8 \, b^{4} \log \relax (F)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 99, normalized size = 0.76 \[ -\frac {\left (-\frac {\left (-b \right )^{\frac {9}{2}} \left (-72 b^{3} x^{3} \ln \relax (F )^{3}+252 b^{2} x^{2} \ln \relax (F )^{2}-630 b x \ln \relax (F )+945\right ) \sqrt {x}\, {\mathrm e}^{b x \ln \relax (F )} \sqrt {\ln \relax (F )}}{72 b^{4}}+\frac {105 \left (-b \right )^{\frac {9}{2}} \sqrt {\pi }\, \erfi \left (\sqrt {b}\, \sqrt {x}\, \sqrt {\ln \relax (F )}\right )}{16 b^{\frac {9}{2}}}\right ) F^{a}}{\left (-b \right )^{\frac {7}{2}} b \ln \relax (F )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 24, normalized size = 0.18 \[ -\frac {F^{a} x^{\frac {9}{2}} \Gamma \left (\frac {9}{2}, -b x \log \relax (F)\right )}{\left (-b x \log \relax (F)\right )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.43, size = 82, normalized size = 0.63 \[ \frac {F^a\,x^{7/2}\,\left (\frac {105\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,x\,\ln \relax (F)}\right )}{16}+F^{b\,x}\,\left (\frac {105\,\sqrt {-b\,x\,\ln \relax (F)}}{8}+\frac {35\,{\left (-b\,x\,\ln \relax (F)\right )}^{3/2}}{4}+\frac {7\,{\left (-b\,x\,\ln \relax (F)\right )}^{5/2}}{2}+{\left (-b\,x\,\ln \relax (F)\right )}^{7/2}\right )\right )}{b\,\ln \relax (F)\,{\left (-b\,x\,\ln \relax (F)\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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