Optimal. Leaf size=122 \[ \frac{\sqrt{\pi } g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} e^{-\frac{2 a b f n \log (F)+1}{b^2 f n^2 \log (F)}} \text{Erfi}\left (\frac{a b f \log (F)+b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac{1}{n}}{b \sqrt{f} \sqrt{\log (F)}}\right )}{2 b e \sqrt{f} n \sqrt{\log (F)}} \]
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Rubi [A] time = 0.369419, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {12, 2278, 2274, 15, 2276, 2234, 2204} \[ \frac{\sqrt{\pi } g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} e^{-\frac{2 a b f n \log (F)+1}{b^2 f n^2 \log (F)}} \text{Erfi}\left (\frac{a b f \log (F)+b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac{1}{n}}{b \sqrt{f} \sqrt{\log (F)}}\right )}{2 b e \sqrt{f} n \sqrt{\log (F)}} \]
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 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 F^{f \left (a+b \log \left (c x^n\right )\right )^2} g x \, dx,x,d+e x\right )}{e}\\ &=\frac{g \operatorname{Subst}\left (\int F^{f \left (a+b \log \left (c x^n\right )\right )^2} x \, dx,x,d+e x\right )}{e}\\ &=\frac{g \operatorname{Subst}\left (\int 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}\\ &=\frac{g \operatorname{Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} x \left (c x^n\right )^{2 a b f \log (F)} \, dx,x,d+e x\right )}{e}\\ &=\frac{\left (g (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}\\ &=\frac{\left (g (d+e x)^2 \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac{2+2 a b f n \log (F)}{n}}\right ) \operatorname{Subst}\left (\int \exp \left (a^2 f \log (F)+b^2 f x^2 \log (F)+\frac{x (2+2 a b f n \log (F))}{n}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n}\\ &=\frac{\left (\exp \left (a^2 f \log (F)-\frac{(2+2 a b f n \log (F))^2}{4 b^2 f n^2 \log (F)}\right ) g (d+e x)^2 \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac{2+2 a b f n \log (F)}{n}}\right ) \operatorname{Subst}\left (\int \exp \left (\frac{\left (2 b^2 f x \log (F)+\frac{2+2 a b f n \log (F)}{n}\right )^2}{4 b^2 f \log (F)}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n}\\ &=\frac{e^{-\frac{1+2 a b f n \log (F)}{b^2 f n^2 \log (F)}} g \sqrt{\pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{erfi}\left (\frac{\frac{1}{n}+a b f \log (F)+b^2 f \log (F) \log \left (c (d+e x)^n\right )}{b \sqrt{f} \sqrt{\log (F)}}\right )}{2 b e \sqrt{f} n \sqrt{\log (F)}}\\ \end{align*}
Mathematica [A] time = 0.703555, size = 120, normalized size = 0.98 \[ \frac{\sqrt{\pi } g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} e^{-\frac{2 a b f n \log (F)+1}{b^2 f n^2 \log (F)}} \text{Erfi}\left (\frac{b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )+1}{b \sqrt{f} n \sqrt{\log (F)}}\right )}{2 b e \sqrt{f} n \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.353, size = 0, normalized size = 0. \begin{align*} \int{F}^{f \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}} \left ( egx+dg \right ) \, 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{\left (e g x + d g\right )} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07586, size = 333, normalized size = 2.73 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b^{2} f n^{2} \log \left (F\right )} g \operatorname{erf}\left (\frac{{\left (b^{2} f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + b^{2} f n \log \left (F\right ) \log \left (c\right ) + a b f n \log \left (F\right ) + 1\right )} \sqrt{-b^{2} f n^{2} \log \left (F\right )}}{b^{2} f n^{2} \log \left (F\right )}\right ) e^{\left (-\frac{2 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 2 \, a b f n \log \left (F\right ) + 1}{b^{2} f n^{2} \log \left (F\right )}\right )}}{2 \, b e 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{\left (e g x + d g\right )} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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