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 = 0.03, 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 = {2320, 272, 45}
\begin {gather*} 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 {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 2320
Rubi steps
\begin {align*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 \, dx &=\frac {\text {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 {\text {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 {\text {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 = 0.09, size = 65, normalized size = 0.97 \begin {gather*} \frac {b \left (F^{g (e+f x)}\right )^n \left (4 a+b \left (F^{g (e+f x)}\right )^n\right )+2 a^2 \log \left (\left (F^{g (e+f x)}\right )^n\right )}{2 f g n \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 65, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {b^{2} \left (F^{g \left (f x +e \right )}\right )^{2 n}}{2}+2 a b \left (F^{g \left (f x +e \right )}\right )^{n}+a^{2} \ln \left (\left (F^{g \left (f x +e \right )}\right )^{n}\right )}{g f \ln \left (F \right ) n}\) | \(65\) |
default | \(\frac {\frac {b^{2} \left (F^{g \left (f x +e \right )}\right )^{2 n}}{2}+2 a b \left (F^{g \left (f x +e \right )}\right )^{n}+a^{2} \ln \left (\left (F^{g \left (f x +e \right )}\right )^{n}\right )}{g f \ln \left (F \right ) n}\) | \(65\) |
norman | \(a^{2} x +\frac {b^{2} {\mathrm e}^{2 n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{2 n g f \ln \left (F \right )}+\frac {2 a b \,{\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{n g f \ln \left (F \right )}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 64, normalized size = 0.96 \begin {gather*} a^{2} x + \frac {2 \, F^{{\left (f x + e\right )} g n} a b}{f g n \log \left (F\right )} + \frac {F^{2 \, {\left (f x + e\right )} g n} b^{2}}{2 \, f g n \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 63, normalized size = 0.94 \begin {gather*} \frac {2 \, a^{2} f g n x \log \left (F\right ) + 4 \, F^{f g n x + g n e} a b + F^{2 \, f g n x + 2 \, g n e} b^{2}}{2 \, f g n \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 92, normalized size = 1.37 \begin {gather*} a^{2} x + \begin {cases} \frac {4 a b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log {\left (F \right )} + b^{2} f g n \left (F^{g \left (e + f x\right )}\right )^{2 n} \log {\left (F \right )}}{2 f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} & \text {for}\: f^{2} g^{2} n^{2} \log {\left (F \right )}^{2} \neq 0 \\x \left (2 a b + b^{2}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.01, size = 77, normalized size = 1.15 \begin {gather*} \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 \left (F\right )} \end {gather*}
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 \begin {gather*} \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 \left (F\right )}+\frac {a^2\,\ln \left (F^{g\,\left (e+f\,x\right )}\right )}{f\,g\,\ln \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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