Optimal. Leaf size=108 \[ -\frac{15 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{8 b^{7/2} \log ^{\frac{7}{2}}(F)}-\frac{5 x^{3/2} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac{15 \sqrt{x} F^{a+b x}}{4 b^3 \log ^3(F)}+\frac{x^{5/2} F^{a+b x}}{b \log (F)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0938183, antiderivative size = 108, normalized size of antiderivative = 1., 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.
[In]
[Out]
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*}
Mathematica [A] time = 0.0064971, size = 36, normalized size = 0.33 \[ \frac{\sqrt{x} F^a \text{Gamma}\left (\frac{7}{2},-b x \log (F)\right )}{b^3 \log ^3(F) \sqrt{-b x \log (F)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 87, normalized size = 0.8 \begin{align*} -{\frac{{F}^{a}}{b} \left ({\frac{ \left ( 28\,{b}^{2}{x}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}-70\,b\ln \left ( F \right ) x+105 \right ){{\rm e}^{b\ln \left ( F \right ) x}}}{28\,{b}^{3}}\sqrt{x} \left ( -b \right ) ^{{\frac{7}{2}}}\sqrt{\ln \left ( F \right ) }}-{\frac{15\,\sqrt{\pi }}{8} \left ( -b \right ) ^{{\frac{7}{2}}}{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ){b}^{-{\frac{7}{2}}}} \right ) \left ( -b \right ) ^{-{\frac{5}{2}}} \left ( \ln \left ( F \right ) \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.22414, size = 32, normalized size = 0.3 \begin{align*} -\frac{F^{a} x^{\frac{7}{2}} \Gamma \left (\frac{7}{2}, -b x \log \left (F\right )\right )}{\left (-b x \log \left (F\right )\right )^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.51819, size = 219, normalized size = 2.03 \begin{align*} \frac{15 \, \sqrt{\pi } \sqrt{-b \log \left (F\right )} F^{a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right ) + 2 \,{\left (4 \, b^{3} x^{2} \log \left (F\right )^{3} - 10 \, b^{2} x \log \left (F\right )^{2} + 15 \, b \log \left (F\right )\right )} F^{b x + a} \sqrt{x}}{8 \, b^{4} \log \left (F\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.25743, size = 111, normalized size = 1.03 \begin{align*} \frac{15 \, \sqrt{\pi } F^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{8 \, \sqrt{-b \log \left (F\right )} b^{3} \log \left (F\right )^{3}} + \frac{{\left (4 \, b^{2} x^{\frac{5}{2}} \log \left (F\right )^{2} - 10 \, b x^{\frac{3}{2}} \log \left (F\right ) + 15 \, \sqrt{x}\right )} e^{\left (b x \log \left (F\right ) + a \log \left (F\right )\right )}}{4 \, b^{3} \log \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]