Optimal. Leaf size=133 \[ \frac {e^{-\frac {3 (3+4 a b f n \log (F))}{4 b^2 f n^2 \log (F)}} g^2 \sqrt {\pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\frac {3}{n}+2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )}{2 b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} n \sqrt {\log (F)}} \]
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Rubi [A]
time = 0.16, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2314, 2308,
2266, 2235} \begin {gather*} \frac {\sqrt {\pi } g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \exp \left (-\frac {3 (4 a b f n \log (F)+3)}{4 b^2 f n^2 \log (F)}\right ) \text {Erfi}\left (\frac {2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac {3}{n}}{2 b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} n \sqrt {\log (F)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 2308
Rule 2314
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)^2 \, dx &=\frac {\text {Subst}\left (\int F^{f \left (a+b \log \left (c x^n\right )\right )^2} g^2 x^2 \, dx,x,d+e x\right )}{e}\\ &=\frac {g^2 \text {Subst}\left (\int F^{f \left (a+b \log \left (c x^n\right )\right )^2} x^2 \, dx,x,d+e x\right )}{e}\\ &=\frac {g^2 \text {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^2 \, dx,x,d+e x\right )}{e}\\ &=\frac {g^2 \text {Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} x^2 \left (c x^n\right )^{2 a b f \log (F)} \, dx,x,d+e x\right )}{e}\\ &=\frac {\left (g^2 (d+e x)^{-2 a b f n \log (F)} \left (c (d+e x)^n\right )^{2 a b f \log (F)}\right ) \text {Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} x^{2+2 a b f n \log (F)} \, dx,x,d+e x\right )}{e}\\ &=\frac {\left (g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac {3+2 a b f n \log (F)}{n}}\right ) \text {Subst}\left (\int \exp \left (a^2 f \log (F)+b^2 f x^2 \log (F)+\frac {x (3+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 {(3+2 a b f n \log (F))^2}{4 b^2 f n^2 \log (F)}\right ) g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac {3+2 a b f n \log (F)}{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {\left (2 b^2 f x \log (F)+\frac {3+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 {\exp \left (-\frac {3 (3+4 a b f n \log (F))}{4 b^2 f n^2 \log (F)}\right ) g^2 \sqrt {\pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\frac {3}{n}+2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )}{2 b \sqrt {f} \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} n \sqrt {\log (F)}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 129, normalized size = 0.97 \begin {gather*} \frac {e^{-\frac {3 \left (3+4 b f n \log (F) \left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right )\right )}{4 b^2 f n^2 \log (F)}} g^2 \sqrt {\pi } \text {erfi}\left (\frac {3+2 b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 b \sqrt {f} n \sqrt {\log (F)}}\right )}{2 b e \sqrt {f} n \sqrt {\log (F)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int F^{f \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}} \left (e g x +d g \right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 134, normalized size = 1.01 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {-b^{2} f n^{2} \log \left (F\right )} g^{2} \operatorname {erf}\left (\frac {{\left (2 \, b^{2} f n^{2} \log \left (x e + d\right ) \log \left (F\right ) + 2 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 2 \, a b f n \log \left (F\right ) + 3\right )} \sqrt {-b^{2} f n^{2} \log \left (F\right )}}{2 \, b^{2} f n^{2} \log \left (F\right )}\right ) e^{\left (-\frac {3 \, {\left (4 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 4 \, a b f n \log \left (F\right ) + 3\right )}}{4 \, b^{2} f n^{2} \log \left (F\right )} - 1\right )}}{2 \, b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {e}}^{f\,\ln \left (F\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}\,{\left (d\,g+e\,g\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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