Optimal. Leaf size=118 \[ -\frac{\sqrt{\pi } F^{a f} e^{-\frac{1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{2/n} \text{Erfi}\left (\frac{1-b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e \sqrt{f} g^3 n \sqrt{\log (F)} (d+e x)^2} \]
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Rubi [A] time = 0.230671, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {12, 2276, 2234, 2204} \[ -\frac{\sqrt{\pi } F^{a f} e^{-\frac{1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{2/n} \text{Erfi}\left (\frac{1-b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e \sqrt{f} g^3 n \sqrt{\log (F)} (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2276
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int \frac{F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )}}{(d g+e g x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{F^{f \left (a+b \log ^2\left (c x^n\right )\right )}}{g^3 x^3} \, dx,x,d+e x\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{F^{f \left (a+b \log ^2\left (c x^n\right )\right )}}{x^3} \, dx,x,d+e x\right )}{e g^3}\\ &=\frac{\left (c (d+e x)^n\right )^{2/n} \operatorname{Subst}\left (\int e^{-\frac{2 x}{n}+a f \log (F)+b f x^2 \log (F)} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e g^3 n (d+e x)^2}\\ &=\frac{\left (e^{-\frac{1}{b f n^2 \log (F)}} F^{a f} \left (c (d+e x)^n\right )^{2/n}\right ) \operatorname{Subst}\left (\int e^{\frac{\left (-\frac{2}{n}+2 b f x \log (F)\right )^2}{4 b f \log (F)}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e g^3 n (d+e x)^2}\\ &=-\frac{e^{-\frac{1}{b f n^2 \log (F)}} F^{a f} \sqrt{\pi } \left (c (d+e x)^n\right )^{2/n} \text{erfi}\left (\frac{1-b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e \sqrt{f} g^3 n (d+e x)^2 \sqrt{\log (F)}}\\ \end{align*}
Mathematica [A] time = 0.207316, size = 117, normalized size = 0.99 \[ \frac{\sqrt{\pi } F^{a f} e^{-\frac{1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{2/n} \text{Erfi}\left (\frac{b f n \log (F) \log \left (c (d+e x)^n\right )-1}{\sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e \sqrt{f} g^3 n \sqrt{\log (F)} (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.614, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{f \left ( a+b \left ( \ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2} \right ) }}{ \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 )^{2} + a\right )} 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 = 1.05993, size = 296, normalized size = 2.51 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b f n^{2} \log \left (F\right )} \operatorname{erf}\left (\frac{{\left (b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + b f n \log \left (F\right ) \log \left (c\right ) - 1\right )} \sqrt{-b f n^{2} \log \left (F\right )}}{b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac{a b f^{2} n^{2} \log \left (F\right )^{2} + 2 \, b f n \log \left (F\right ) \log \left (c\right ) - 1}{b f n^{2} \log \left (F\right )}\right )}}{2 \, 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 )^{2} + a\right )} 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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