Optimal. Leaf size=126 \[ -\frac{\sqrt{\pi } \left (c (d+e x)^n\right )^{2/n} e^{-\frac{1-2 a b f n \log (F)}{b^2 f n^2 \log (F)}} \text{Erfi}\left (\frac{-a b f \log (F)+b^2 (-f) \log (F) \log \left (c (d+e x)^n\right )+\frac{1}{n}}{b \sqrt{f} \sqrt{\log (F)}}\right )}{2 b e \sqrt{f} g^3 n \sqrt{\log (F)} (d+e x)^2} \]
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Rubi [A] time = 0.405278, antiderivative size = 126, 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} \[ -\frac{\sqrt{\pi } \left (c (d+e x)^n\right )^{2/n} e^{-\frac{1-2 a b f n \log (F)}{b^2 f n^2 \log (F)}} \text{Erfi}\left (\frac{-a b f \log (F)+b^2 (-f) \log (F) \log \left (c (d+e x)^n\right )+\frac{1}{n}}{b \sqrt{f} \sqrt{\log (F)}}\right )}{2 b e \sqrt{f} g^3 n \sqrt{\log (F)} (d+e x)^2} \]
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)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{F^{f \left (a+b \log \left (c x^n\right )\right )^2}}{g^3 x^3} \, 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^3} \, dx,x,d+e x\right )}{e g^3}\\ &=\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^3} \, dx,x,d+e x\right )}{e g^3}\\ &=\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^3} \, dx,x,d+e x\right )}{e g^3}\\ &=\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^{-3+2 a b f n \log (F)} \, dx,x,d+e x\right )}{e g^3}\\ &=\frac{\left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac{-2+2 a b f n \log (F)}{n}} \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 g^3 n (d+e x)^2}\\ &=\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 ) \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 g^3 n (d+e x)^2}\\ &=-\frac{e^{-\frac{1-2 a b f n \log (F)}{b^2 f n^2 \log (F)}} \sqrt{\pi } \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} g^3 n (d+e x)^2 \sqrt{\log (F)}}\\ \end{align*}
Mathematica [A] time = 0.244847, size = 121, normalized size = 0.96 \[ \frac{\sqrt{\pi } \left (c (d+e x)^n\right )^{2/n} e^{\frac{2 a b f n \log (F)-1}{b^2 f n^2 \log (F)}} \text{Erfi}\left (\frac{b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )-1}{b \sqrt{f} n \sqrt{\log (F)}}\right )}{2 b e \sqrt{f} g^3 n \sqrt{\log (F)} (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.665, 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}}}{ \left ( egx+dg \right ) ^{3}}}\, 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}}{{\left (e g x + d g\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.0907, size = 335, normalized size = 2.66 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b^{2} f n^{2} \log \left (F\right )} \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 g^{3} 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}}{{\left (e g x + d g\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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