Optimal. Leaf size=77 \[ \frac {a (c+d x)^2}{2 d}+\frac {b (c+d x) \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac {b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)} \]
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Rubi [A] time = 0.08, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2183, 2176, 2194} \[ \frac {a (c+d x)^2}{2 d}+\frac {b (c+d x) \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac {b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2183
Rule 2194
Rubi steps
\begin {align*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x) \, dx &=\int \left (a (c+d x)+b \left (F^{e g+f g x}\right )^n (c+d x)\right ) \, dx\\ &=\frac {a (c+d x)^2}{2 d}+b \int \left (F^{e g+f g x}\right )^n (c+d x) \, dx\\ &=\frac {a (c+d x)^2}{2 d}+\frac {b \left (F^{e g+f g x}\right )^n (c+d x)}{f g n \log (F)}-\frac {(b d) \int \left (F^{e g+f g x}\right )^n \, dx}{f g n \log (F)}\\ &=\frac {a (c+d x)^2}{2 d}-\frac {b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {b \left (F^{e g+f g x}\right )^n (c+d x)}{f g n \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 73, normalized size = 0.95 \[ \frac {1}{2} a x (2 c+d x)+\frac {b (c+d x) \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}-\frac {b d \left (F^{g (e+f x)}\right )^n}{f^2 g^2 n^2 \log ^2(F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 87, normalized size = 1.13 \[ \frac {{\left (a d f^{2} g^{2} n^{2} x^{2} + 2 \, a c f^{2} g^{2} n^{2} x\right )} \log \relax (F)^{2} - 2 \, {\left (b d - {\left (b d f g n x + b c f g n\right )} \log \relax (F)\right )} F^{f g n x + e g n}}{2 \, f^{2} g^{2} n^{2} \log \relax (F)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.37, size = 1117, normalized size = 14.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 105, normalized size = 1.36 \[ \frac {a d \,x^{2}}{2}+a c x +\frac {b d x \,{\mathrm e}^{n \ln \left ({\mathrm e}^{\left (f x +e \right ) g \ln \relax (F )}\right )}}{f g n \ln \relax (F )}+\frac {b c \,{\mathrm e}^{n \ln \left ({\mathrm e}^{\left (f x +e \right ) g \ln \relax (F )}\right )}}{f g n \ln \relax (F )}-\frac {b d \,{\mathrm e}^{n \ln \left ({\mathrm e}^{\left (f x +e \right ) g \ln \relax (F )}\right )}}{f^{2} g^{2} n^{2} \ln \relax (F )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 88, normalized size = 1.14 \[ \frac {1}{2} \, a d x^{2} + a c x + \frac {{\left (F^{f g x + e g}\right )}^{n} b c}{f g n \log \relax (F)} + \frac {{\left ({\left (F^{e g}\right )}^{n} f g n x \log \relax (F) - {\left (F^{e g}\right )}^{n}\right )} {\left (F^{f g x}\right )}^{n} b d}{f^{2} g^{2} n^{2} \log \relax (F)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.60, size = 72, normalized size = 0.94 \[ a\,c\,x-\left (\frac {b\,\left (d-c\,f\,g\,n\,\ln \relax (F)\right )}{f^2\,g^2\,n^2\,{\ln \relax (F)}^2}-\frac {b\,d\,x}{f\,g\,n\,\ln \relax (F)}\right )\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n+\frac {a\,d\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 94, normalized size = 1.22 \[ a c x + \frac {a d x^{2}}{2} + \begin {cases} \frac {\left (b c f g n \log {\relax (F )} + b d f g n x \log {\relax (F )} - b d\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{2} g^{2} n^{2} \log {\relax (F )}^{2}} & \text {for}\: f^{2} g^{2} n^{2} \log {\relax (F )}^{2} \neq 0 \\b c x + \frac {b d x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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