Optimal. Leaf size=70 \[ \frac{\sqrt{\pi } \text{Erfi}\left (a \sqrt{f} \sqrt{\log (F)}+b \sqrt{f} \sqrt{\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 b e \sqrt{f} g n \sqrt{\log (F)}} \]
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Rubi [A] time = 0.267745, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {12, 2278, 2274, 15, 2276, 2234, 2204} \[ \frac{\sqrt{\pi } \text{Erfi}\left (a \sqrt{f} \sqrt{\log (F)}+b \sqrt{f} \sqrt{\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 b e \sqrt{f} g n \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2278
Rule 2274
Rule 15
Rule 2276
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int \frac{F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{d g+e g x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{F^{f \left (a+b \log \left (c x^n\right )\right )^2}}{g x} \, dx,x,d+e x\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{F^{f \left (a+b \log \left (c x^n\right )\right )^2}}{x} \, dx,x,d+e x\right )}{e g}\\ &=\frac{\operatorname{Subst}\left (\int \frac{F^{a^2 f+2 a b f \log \left (c x^n\right )+b^2 f \log ^2\left (c x^n\right )}}{x} \, dx,x,d+e x\right )}{e g}\\ &=\frac{\operatorname{Subst}\left (\int \frac{F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} \left (c x^n\right )^{2 a b f \log (F)}}{x} \, dx,x,d+e x\right )}{e g}\\ &=\frac{\left ((d+e x)^{-2 a b f n \log (F)} \left (c (d+e x)^n\right )^{2 a b f \log (F)}\right ) \operatorname{Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} x^{-1+2 a b f n \log (F)} \, dx,x,d+e x\right )}{e g}\\ &=\frac{\operatorname{Subst}\left (\int \exp \left (a^2 f \log (F)+2 a b f x \log (F)+b^2 f x^2 \log (F)\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e g n}\\ &=\frac{\operatorname{Subst}\left (\int \exp \left (\frac{\left (2 a b f \log (F)+2 b^2 f x \log (F)\right )^2}{4 b^2 f \log (F)}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e g n}\\ &=\frac{\sqrt{\pi } \text{erfi}\left (a \sqrt{f} \sqrt{\log (F)}+b \sqrt{f} \sqrt{\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 b e \sqrt{f} g n \sqrt{\log (F)}}\\ \end{align*}
Mathematica [A] time = 0.0460468, size = 59, normalized size = 0.84 \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{f} \sqrt{\log (F)} \left (a+b \log \left (c (d+e x)^n\right )\right )\right )}{2 b e \sqrt{f} g n \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{f \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}}{egx+dg}}\, dx \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^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f}}{e g x + d g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.00244, size = 159, normalized size = 2.27 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b^{2} f n^{2} \log \left (F\right )} \operatorname{erf}\left (\frac{\sqrt{-b^{2} f n^{2} \log \left (F\right )}{\left (b n \log \left (e x + d\right ) + b \log \left (c\right ) + a\right )}}{b n}\right )}{2 \, b e g n} \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^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f}}{e g x + d g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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