Optimal. Leaf size=131 \[ \frac{105 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{16 b^{9/2} \log ^{\frac{9}{2}}(F)}-\frac{7 x^{5/2} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac{35 x^{3/2} F^{a+b x}}{4 b^3 \log ^3(F)}-\frac{105 \sqrt{x} F^{a+b x}}{8 b^4 \log ^4(F)}+\frac{x^{7/2} F^{a+b x}}{b \log (F)} \]
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Rubi [A] time = 0.147164, antiderivative size = 131, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0062582, size = 36, normalized size = 0.27 \[ \frac{F^a \sqrt{-b x \log (F)} \text{Gamma}\left (\frac{9}{2},-b x \log (F)\right )}{b^5 \sqrt{x} \log ^5(F)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 99, normalized size = 0.8 \begin{align*} -{\frac{{F}^{a}}{b} \left ( -{\frac{ \left ( -72\,{b}^{3}{x}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}+252\,{b}^{2}{x}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}-630\,b\ln \left ( F \right ) x+945 \right ){{\rm e}^{b\ln \left ( F \right ) x}}}{72\,{b}^{4}}\sqrt{x} \left ( -b \right ) ^{{\frac{9}{2}}}\sqrt{\ln \left ( F \right ) }}+{\frac{105\,\sqrt{\pi }}{16} \left ( -b \right ) ^{{\frac{9}{2}}}{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ){b}^{-{\frac{9}{2}}}} \right ) \left ( -b \right ) ^{-{\frac{7}{2}}} \left ( \ln \left ( F \right ) \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22927, size = 32, normalized size = 0.24 \begin{align*} -\frac{F^{a} x^{\frac{9}{2}} \Gamma \left (\frac{9}{2}, -b x \log \left (F\right )\right )}{\left (-b x \log \left (F\right )\right )^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53279, size = 254, normalized size = 1.94 \begin{align*} -\frac{105 \, \sqrt{\pi } \sqrt{-b \log \left (F\right )} F^{a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right ) - 2 \,{\left (8 \, b^{4} x^{3} \log \left (F\right )^{4} - 28 \, b^{3} x^{2} \log \left (F\right )^{3} + 70 \, b^{2} x \log \left (F\right )^{2} - 105 \, b \log \left (F\right )\right )} F^{b x + a} \sqrt{x}}{16 \, b^{5} \log \left (F\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2203, size = 127, normalized size = 0.97 \begin{align*} -\frac{105 \, \sqrt{\pi } F^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{16 \, \sqrt{-b \log \left (F\right )} b^{4} \log \left (F\right )^{4}} + \frac{{\left (8 \, b^{3} x^{\frac{7}{2}} \log \left (F\right )^{3} - 28 \, b^{2} x^{\frac{5}{2}} \log \left (F\right )^{2} + 70 \, b x^{\frac{3}{2}} \log \left (F\right ) - 105 \, \sqrt{x}\right )} e^{\left (b x \log \left (F\right ) + a \log \left (F\right )\right )}}{8 \, b^{4} \log \left (F\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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