Optimal. Leaf size=108 \[ -\frac {15 \sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{8 b^{7/2} \log ^{\frac {7}{2}}(F)}+\frac {15 \sqrt {x} F^{a+b x}}{4 b^3 \log ^3(F)}-\frac {5 x^{3/2} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac {x^{5/2} F^{a+b x}}{b \log (F)} \]
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Rubi [A] time = 0.09, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2176, 2180, 2204} \[ -\frac {15 \sqrt {\pi } F^a \text {Erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{8 b^{7/2} \log ^{\frac {7}{2}}(F)}-\frac {5 x^{3/2} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac {15 \sqrt {x} F^{a+b x}}{4 b^3 \log ^3(F)}+\frac {x^{5/2} F^{a+b x}}{b \log (F)} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2180
Rule 2204
Rubi steps
\begin {align*} \int F^{a+b x} x^{5/2} \, dx &=\frac {F^{a+b x} x^{5/2}}{b \log (F)}-\frac {5 \int F^{a+b x} x^{3/2} \, dx}{2 b \log (F)}\\ &=-\frac {5 F^{a+b x} x^{3/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{5/2}}{b \log (F)}+\frac {15 \int F^{a+b x} \sqrt {x} \, dx}{4 b^2 \log ^2(F)}\\ &=\frac {15 F^{a+b x} \sqrt {x}}{4 b^3 \log ^3(F)}-\frac {5 F^{a+b x} x^{3/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{5/2}}{b \log (F)}-\frac {15 \int \frac {F^{a+b x}}{\sqrt {x}} \, dx}{8 b^3 \log ^3(F)}\\ &=\frac {15 F^{a+b x} \sqrt {x}}{4 b^3 \log ^3(F)}-\frac {5 F^{a+b x} x^{3/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{5/2}}{b \log (F)}-\frac {15 \operatorname {Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 b^3 \log ^3(F)}\\ &=-\frac {15 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{8 b^{7/2} \log ^{\frac {7}{2}}(F)}+\frac {15 F^{a+b x} \sqrt {x}}{4 b^3 \log ^3(F)}-\frac {5 F^{a+b x} x^{3/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{5/2}}{b \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 0.33 \[ \frac {\sqrt {x} F^a \Gamma \left (\frac {7}{2},-b x \log (F)\right )}{b^3 \log ^3(F) \sqrt {-b x \log (F)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 77, normalized size = 0.71 \[ \frac {15 \, \sqrt {\pi } \sqrt {-b \log \relax (F)} F^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (F)} \sqrt {x}\right ) + 2 \, {\left (4 \, b^{3} x^{2} \log \relax (F)^{3} - 10 \, b^{2} x \log \relax (F)^{2} + 15 \, b \log \relax (F)\right )} F^{b x + a} \sqrt {x}}{8 \, b^{4} \log \relax (F)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 82, normalized size = 0.76 \[ \frac {15 \, \sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} \sqrt {x}\right )}{8 \, \sqrt {-b \log \relax (F)} b^{3} \log \relax (F)^{3}} + \frac {{\left (4 \, b^{2} x^{\frac {5}{2}} \log \relax (F)^{2} - 10 \, b x^{\frac {3}{2}} \log \relax (F) + 15 \, \sqrt {x}\right )} e^{\left (b x \log \relax (F) + a \log \relax (F)\right )}}{4 \, b^{3} \log \relax (F)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 87, normalized size = 0.81 \[ -\frac {\left (\frac {\left (-b \right )^{\frac {7}{2}} \left (28 b^{2} x^{2} \ln \relax (F )^{2}-70 b x \ln \relax (F )+105\right ) \sqrt {x}\, {\mathrm e}^{b x \ln \relax (F )} \sqrt {\ln \relax (F )}}{28 b^{3}}-\frac {15 \left (-b \right )^{\frac {7}{2}} \sqrt {\pi }\, \erfi \left (\sqrt {b}\, \sqrt {x}\, \sqrt {\ln \relax (F )}\right )}{8 b^{\frac {7}{2}}}\right ) F^{a}}{\left (-b \right )^{\frac {5}{2}} b \ln \relax (F )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 24, normalized size = 0.22 \[ -\frac {F^{a} x^{\frac {7}{2}} \Gamma \left (\frac {7}{2}, -b x \log \relax (F)\right )}{\left (-b x \log \relax (F)\right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.46, size = 72, normalized size = 0.67 \[ \frac {F^a\,x^{5/2}\,\left (F^{b\,x}\,\left (\frac {15\,\sqrt {-b\,x\,\ln \relax (F)}}{4}+\frac {5\,{\left (-b\,x\,\ln \relax (F)\right )}^{3/2}}{2}+{\left (-b\,x\,\ln \relax (F)\right )}^{5/2}\right )+\frac {15\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,x\,\ln \relax (F)}\right )}{8}\right )}{b\,\ln \relax (F)\,{\left (-b\,x\,\ln \relax (F)\right )}^{5/2}} \]
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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