Optimal. Leaf size=118 \[ \frac{\sqrt{\pi } F^{a f} (d+e x) e^{-\frac{1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e \sqrt{f} n \sqrt{\log (F)}} \]
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Rubi [A] time = 0.0938932, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2275, 2234, 2204} \[ \frac{\sqrt{\pi } F^{a f} (d+e x) e^{-\frac{1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e \sqrt{f} n \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
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Rule 2275
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int F^{f \left (a+b \log ^2\left (c x^n\right )\right )} \, dx,x,d+e x\right )}{e}\\ &=\frac{\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{\frac{x}{n}+a f \log (F)+b f x^2 \log (F)} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n}\\ &=\frac{\left (e^{-\frac{1}{4 b f n^2 \log (F)}} F^{a f} (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{\frac{\left (\frac{1}{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 n}\\ &=\frac{e^{-\frac{1}{4 b f n^2 \log (F)}} F^{a f} \sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{erfi}\left (\frac{1+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e \sqrt{f} n \sqrt{\log (F)}}\\ \end{align*}
Mathematica [A] time = 0.0229734, size = 118, normalized size = 1. \[ \frac{\sqrt{\pi } F^{a f} (d+e x) e^{-\frac{1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e \sqrt{f} n \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.013, size = 0, normalized size = 0. \begin{align*} \int{F}^{f \left ( a+b \left ( \ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2} \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 F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.986247, size = 309, normalized size = 2.62 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b f n^{2} \log \left (F\right )} \operatorname{erf}\left (\frac{{\left (2 \, b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + 2 \, b f n \log \left (F\right ) \log \left (c\right ) + 1\right )} \sqrt{-b f n^{2} \log \left (F\right )}}{2 \, b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac{4 \, a b f^{2} n^{2} \log \left (F\right )^{2} - 4 \, b f n \log \left (F\right ) \log \left (c\right ) - 1}{4 \, b f n^{2} \log \left (F\right )}\right )}}{2 \, e n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 105.909, size = 532, normalized size = 4.51 \begin{align*} \begin{cases} - \frac{2 F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} b d f n^{2} \log{\left (F \right )} \log{\left (d + e x \right )}}{e} - \frac{2 F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} b d f n^{2} \log{\left (F \right )}}{e} - \frac{2 F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} b d f n \log{\left (F \right )} \log{\left (c \right )}}{e} - 2 F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} b f n^{2} x \log{\left (F \right )} \log{\left (d + e x \right )} + 2 F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} b f n^{2} x \log{\left (F \right )} - 2 F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} b f n x \log{\left (F \right )} \log{\left (c \right )} + \frac{F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} d}{e} + F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} x & \text{for}\: e \neq 0 \\F^{f \left (a + b \log{\left (c d^{n} \right )}^{2}\right )} x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36132, size = 136, normalized size = 1.15 \begin{align*} -\frac{\sqrt{\pi } F^{a f} \operatorname{erf}\left (-\sqrt{-b f \log \left (F\right )} n \log \left (x e + d\right ) - \sqrt{-b f \log \left (F\right )} \log \left (c\right ) - \frac{\sqrt{-b f \log \left (F\right )}}{2 \, b f n \log \left (F\right )}\right ) e^{\left (-\frac{1}{4 \, b f n^{2} \log \left (F\right )} - 1\right )}}{2 \, \sqrt{-b f \log \left (F\right )} c^{\left (\frac{1}{n}\right )} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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