Optimal. Leaf size=46 \[ -\frac{\sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{2 \sqrt{b} d \sqrt{\log (F)}} \]
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Rubi [A] time = 0.054265, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2211, 2204} \[ -\frac{\sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{2 \sqrt{b} d \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
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Rule 2211
Rule 2204
Rubi steps
\begin{align*} \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^2} \, dx &=-\frac{\operatorname{Subst}\left (\int F^{a+b x^2} \, dx,x,\frac{1}{c+d x}\right )}{d}\\ &=-\frac{F^a \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{2 \sqrt{b} d \sqrt{\log (F)}}\\ \end{align*}
Mathematica [A] time = 0.0093136, size = 46, normalized size = 1. \[ -\frac{\sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{2 \sqrt{b} d \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 35, normalized size = 0.8 \begin{align*} -{\frac{{F}^{a}\sqrt{\pi }}{2\,d}{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-b\ln \left ( F \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53979, size = 116, normalized size = 2.52 \begin{align*} \frac{\sqrt{\pi } F^{a} \sqrt{-\frac{b \log \left (F\right )}{d^{2}}} \operatorname{erf}\left (\frac{d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{d x + c}\right )}{2 \, b \log \left (F\right )} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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