Optimal. Leaf size=115 \[ \frac {\sqrt {\pi } g F^{a f} (d+e x)^2 e^{-\frac {1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} n \sqrt {\log (F)}} \]
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Rubi [A] time = 0.21, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {12, 2276, 2234, 2204} \[ \frac {\sqrt {\pi } g F^{a f} (d+e x)^2 e^{-\frac {1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-2/n} \text {Erfi}\left (\frac {b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} n \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
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Rule 12
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) \, dx &=\frac {\operatorname {Subst}\left (\int F^{f \left (a+b \log ^2\left (c x^n\right )\right )} g x \, dx,x,d+e x\right )}{e}\\ &=\frac {g \operatorname {Subst}\left (\int F^{f \left (a+b \log ^2\left (c x^n\right )\right )} x \, dx,x,d+e x\right )}{e}\\ &=\frac {\left (g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int e^{\frac {2 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}{b f n^2 \log (F)}} F^{a f} g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int e^{\frac {\left (\frac {2}{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}{b f n^2 \log (F)}} F^{a f} g \sqrt {\pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {1+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} n \sqrt {\log (F)}}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 115, normalized size = 1.00 \[ \frac {\sqrt {\pi } g F^{a f} (d+e x)^2 e^{-\frac {1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} n \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 112, normalized size = 0.97 \[ -\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \relax (F)} g \operatorname {erf}\left (\frac {{\left (b f n^{2} \log \left (e x + d\right ) \log \relax (F) + b f n \log \relax (F) \log \relax (c) + 1\right )} \sqrt {-b f n^{2} \log \relax (F)}}{b f n^{2} \log \relax (F)}\right ) e^{\left (\frac {a b f^{2} n^{2} \log \relax (F)^{2} - 2 \, b f n \log \relax (F) \log \relax (c) - 1}{b f n^{2} \log \relax (F)}\right )}}{2 \, e n} \]
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 \[ \int {\left (e g x + d g\right )} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \left (e g x +d g \right ) F^{\left (b \ln \left (c \left (e x +d \right )^{n}\right )^{2}+a \right ) f}\, dx \]
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 \[ \int {\left (e g x + d g\right )} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}\,{d x} \]
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 \[ \int {\mathrm {e}}^{f\,\ln \relax (F)\,\left (b\,{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2+a\right )}\,\left (d\,g+e\,g\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 110.48, size = 838, normalized size = 7.29 \[ \begin {cases} - \frac {F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b d^{2} f g n^{2} \log {\relax (F )} \log {\left (d + e x \right )}}{2 e} - \frac {F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b d^{2} f g n^{2} \log {\relax (F )}}{2 e} - \frac {F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b d^{2} f g n \log {\relax (F )} \log {\relax (c )}}{2 e} - F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b d f g n^{2} x \log {\relax (F )} \log {\left (d + e x \right )} + \frac {F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b d f g n^{2} x \log {\relax (F )}}{2} - F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b d f g n x \log {\relax (F )} \log {\relax (c )} - \frac {F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b e f g n^{2} x^{2} \log {\relax (F )} \log {\left (d + e x \right )}}{2} + \frac {F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b e f g n^{2} x^{2} \log {\relax (F )}}{4} - \frac {F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b e f g n x^{2} \log {\relax (F )} \log {\relax (c )}}{2} + \frac {F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} d^{2} g}{2 e} + F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} d g x + \frac {F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} e g x^{2}}{2} & \text {for}\: e \neq 0 \\F^{f \left (a + b \log {\left (c d^{n} \right )}^{2}\right )} d g x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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