Optimal. Leaf size=67 \[ a^2 x+\frac {2 a b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}+\frac {b^2 \left (F^{g (e+f x)}\right )^{2 n}}{2 f g n \log (F)} \]
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Rubi [A] time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2282, 266, 43} \[ a^2 x+\frac {2 a b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}+\frac {b^2 \left (F^{g (e+f x)}\right )^{2 n}}{2 f g n \log (F)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2282
Rubi steps
\begin {align*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^n\right )^2}{x} \, dx,x,F^{g (e+f x)}\right )}{f g \log (F)}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^2}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}\\ &=a^2 x+\frac {2 a b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}+\frac {b^2 \left (F^{g (e+f x)}\right )^{2 n}}{2 f g n \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 0.78 \[ a^2 x+\frac {b \left (F^{g (e+f x)}\right )^n \left (4 a+b \left (F^{g (e+f x)}\right )^n\right )}{2 f g n \log (F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 61, normalized size = 0.91 \[ \frac {2 \, a^{2} f g n x \log \relax (F) + 4 \, F^{f g n x + e g n} a b + F^{2 \, f g n x + 2 \, e g n} b^{2}}{2 \, f g n \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 77, normalized size = 1.15 \[ \frac {4 \, F^{f g n x} F^{g n e} a b + F^{2 \, f g n x} F^{2 \, g n e} b^{2} + 2 \, a^{2} \log \left ({\left | F \right |}^{f g n x} {\left | F \right |}^{g n e}\right )}{2 \, f g n \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 90, normalized size = 1.34 \[ \frac {a^{2} \ln \left (\left (F^{\left (f x +e \right ) g}\right )^{n}\right )}{f g n \ln \relax (F )}+\frac {2 a b \left (F^{\left (f x +e \right ) g}\right )^{n}}{f g n \ln \relax (F )}+\frac {b^{2} \left (F^{\left (f x +e \right ) g}\right )^{2 n}}{2 f g n \ln \relax (F )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 75, normalized size = 1.12 \[ a^{2} x + \frac {2 \, {\left (F^{f g x}\right )}^{n} {\left (F^{e g}\right )}^{n} a b}{f g n \log \relax (F)} + \frac {{\left (F^{f g x}\right )}^{2 \, n} {\left (F^{e g}\right )}^{2 \, n} b^{2}}{2 \, f g n \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.72, size = 75, normalized size = 1.12 \[ \frac {\frac {b^2\,{\left (F^{e\,g+f\,g\,x}\right )}^{2\,n}}{2}+2\,a\,b\,{\left (F^{e\,g+f\,g\,x}\right )}^n}{f\,g\,n\,\ln \relax (F)}+\frac {a^2\,\ln \left (F^{g\,\left (e+f\,x\right )}\right )}{f\,g\,\ln \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 94, normalized size = 1.40 \[ a^{2} x + \begin {cases} \frac {4 a b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log {\relax (F )} + b^{2} f g n \left (F^{g \left (e + f x\right )}\right )^{2 n} \log {\relax (F )}}{2 f^{2} g^{2} n^{2} \log {\relax (F )}^{2}} & \text {for}\: 2 f^{2} g^{2} n^{2} \log {\relax (F )}^{2} \neq 0 \\x \left (2 a b + b^{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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