Optimal. Leaf size=137 \[ \frac {\sqrt {\pi } F^{a f} (d g+e g x)^{m+1} e^{-\frac {(m+1)^2}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+m+1}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}} \]
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Rubi [A] time = 0.39, antiderivative size = 136, normalized size of antiderivative = 0.99, 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 = {2276, 2234, 2204} \[ \frac {\sqrt {\pi } F^{a f} (g (d+e x))^{m+1} e^{-\frac {(m+1)^2}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-\frac {m+1}{n}} \text {Erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+m+1}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} g n \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2276
Rubi steps
\begin {align*} \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (d g+e g x)^m \, dx &=\frac {\operatorname {Subst}\left (\int F^{f \left (a+b \log ^2\left (c x^n\right )\right )} (g x)^m \, dx,x,d+e x\right )}{e}\\ &=\frac {\left ((g (d+e x))^{1+m} \left (c (d+e x)^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int e^{\frac {(1+m) 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 n}\\ &=\frac {\left (e^{-\frac {(1+m)^2}{4 b f n^2 \log (F)}} F^{a f} (g (d+e x))^{1+m} \left (c (d+e x)^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int e^{\frac {\left (\frac {1+m}{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 n}\\ &=\frac {e^{-\frac {(1+m)^2}{4 b f n^2 \log (F)}} F^{a f} \sqrt {\pi } (g (d+e x))^{1+m} \left (c (d+e x)^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {1+m+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} g n \sqrt {\log (F)}}\\ \end {align*}
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Mathematica [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (d g+e g x)^m \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.42, size = 143, normalized size = 1.04 \[ -\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \relax (F)} \operatorname {erf}\left (\frac {{\left (2 \, b f n^{2} \log \left (e x + d\right ) \log \relax (F) + 2 \, b f n \log \relax (F) \log \relax (c) + m + 1\right )} \sqrt {-b f n^{2} \log \relax (F)}}{2 \, b f n^{2} \log \relax (F)}\right ) e^{\left (\frac {4 \, a b f^{2} n^{2} \log \relax (F)^{2} + 4 \, b f m n^{2} \log \relax (F) \log \relax (g) - 4 \, {\left (b f m + b f\right )} n \log \relax (F) \log \relax (c) - m^{2} - 2 \, m - 1}{4 \, b f n^{2} \log \relax (F)}\right )}}{2 \, e n} \]
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 \[ \int {\left (e g x + d g\right )}^{m} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 84.22, size = 0, normalized size = 0.00 \[ \int F^{\left (b \ln \left (c \left (e x +d \right )^{n}\right )^{2}+a \right ) f} \left (e g x +d g \right )^{m}\, dx \]
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 \[ \int {\left (e g x + d g\right )}^{m} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}\,{d x} \]
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 \[ \int {\mathrm {e}}^{f\,\ln \relax (F)\,\left (b\,{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2+a\right )}\,{\left (d\,g+e\,g\,x\right )}^m \,d x \]
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 \[ \int F^{f \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}^{2}\right )} \left (g \left (d + e x\right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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