Optimal. Leaf size=67 \[ \frac {F^{a f} \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {f} \sqrt {\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2308, 2235}
\begin {gather*} \frac {\sqrt {\pi } F^{a f} \text {Erfi}\left (\sqrt {b} \sqrt {f} \sqrt {\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2308
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} \, dx &=\frac {\text {Subst}\left (\int \frac {F^{f \left (a+b \log ^2\left (c x^n\right )\right )}}{g x} \, dx,x,d+e x\right )}{e}\\ &=\frac {\text {Subst}\left (\int \frac {F^{f \left (a+b \log ^2\left (c x^n\right )\right )}}{x} \, dx,x,d+e x\right )}{e g}\\ &=\frac {\text {Subst}\left (\int e^{a f \log (F)+b f x^2 \log (F)} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e g n}\\ &=\frac {F^{a f} \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {f} \sqrt {\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 67, normalized size = 1.00 \begin {gather*} \frac {F^{a f} \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {f} \sqrt {\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.34, size = 382, normalized size = 5.70 \[\frac {\sqrt {\pi }\, F^{f \left (-i b \ln \left (c \right ) \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )+b \,\pi ^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )-b \,\pi ^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )+b \,\pi ^{2} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )+i b \ln \left (c \right ) \pi \,\mathrm {csgn}\left (i c \right )-i b \ln \left (c \right ) \pi \,\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i b \ln \left (c \right ) \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )-b \,\pi ^{2}+b \ln \left (c \right )^{2}+a \right )} F^{-\frac {f b \left (i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )-i \pi \,\mathrm {csgn}\left (i c \right )+i \pi \,\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )-i \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )-2 \ln \left (c \right )\right )^{2}}{4}} \erf \left (\sqrt {-\ln \left (F \right ) b f}\, \ln \left (\left (e x +d \right )^{n}\right )-\frac {f b \left (2 \ln \left (c \right )-i \pi \,\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \left (-\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (-\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )\right )\right ) \ln \left (F \right )}{2 \sqrt {-\ln \left (F \right ) b f}}\right )}{2 g e n \sqrt {-\ln \left (F \right ) b f}}\]
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 = 0.36, size = 57, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} F^{a f} \operatorname {erf}\left (\frac {\sqrt {-b f n^{2} \log \left (F\right )} {\left (n \log \left (x e + d\right ) + \log \left (c\right )\right )}}{n}\right ) e^{\left (-1\right )}}{2 \, g n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {F^{a f} F^{b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}}{d + e x}\, dx}{g} \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 [B]
time = 3.74, size = 49, normalized size = 0.73 \begin {gather*} \frac {F^{a\,f}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,f\,\ln \left (F\right )\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\sqrt {b\,f\,\ln \left (F\right )}}\right )}{2\,e\,g\,n\,\sqrt {b\,f\,\ln \left (F\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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