Optimal. Leaf size=67 \[ \frac {\sqrt {\pi } F^{a f} \text {erfi}\left (\sqrt {b} \sqrt {f} \sqrt {\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}} \]
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Rubi [A] time = 0.14, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {12, 2276, 2204} \[ \frac {\sqrt {\pi } F^{a f} \text {Erfi}\left (\sqrt {b} \sqrt {f} \sqrt {\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2204
Rule 2276
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 {\operatorname {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 {\operatorname {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 {\operatorname {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 \[ \frac {\sqrt {\pi } F^{a f} \text {erfi}\left (\sqrt {b} \sqrt {f} \sqrt {\log (F)} \log \left (c (d+e x)^n\right )\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 57, normalized size = 0.85 \[ -\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \relax (F)} F^{a f} \operatorname {erf}\left (\frac {\sqrt {-b f n^{2} \log \relax (F)} {\left (n \log \left (e x + d\right ) + \log \relax (c)\right )}}{n}\right )}{2 \, e g n} \]
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 \[ \int \frac {F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}}{e g x + d g}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{\left (b \ln \left (c \left (e x +d \right )^{n}\right )^{2}+a \right ) f}}{e g x +d g}\, dx \]
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 \[ \int \frac {F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}}{e g x + d g}\,{d x} \]
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 \[ \frac {F^{a\,f}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,f\,\ln \relax (F)\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\sqrt {b\,f\,\ln \relax (F)}}\right )}{2\,e\,g\,n\,\sqrt {b\,f\,\ln \relax (F)}} \]
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 \[ \frac {\int \frac {F^{a f} F^{b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}}{d + e x}\, dx}{g} \]
Verification of antiderivative is not currently implemented for this CAS.
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