Optimal. Leaf size=103 \[ a^3 x+\frac {3 a^2 b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}+\frac {3 a b^2 \left (F^{g (e+f x)}\right )^{2 n}}{2 f g n \log (F)}+\frac {b^3 \left (F^{g (e+f x)}\right )^{3 n}}{3 f g n \log (F)} \]
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Rubi [A] time = 0.05, antiderivative size = 103, 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} \[ \frac {3 a^2 b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}+a^3 x+\frac {3 a b^2 \left (F^{g (e+f x)}\right )^{2 n}}{2 f g n \log (F)}+\frac {b^3 \left (F^{g (e+f x)}\right )^{3 n}}{3 f g n \log (F)} \]
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 )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^n\right )^3}{x} \, dx,x,F^{g (e+f x)}\right )}{f g \log (F)}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^3}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}\\ &=\frac {\operatorname {Subst}\left (\int \left (3 a^2 b+\frac {a^3}{x}+3 a b^2 x+b^3 x^2\right ) \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}\\ &=a^3 x+\frac {3 a^2 b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}+\frac {3 a b^2 \left (F^{g (e+f x)}\right )^{2 n}}{2 f g n \log (F)}+\frac {b^3 \left (F^{g (e+f x)}\right )^{3 n}}{3 f g n \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 74, normalized size = 0.72 \[ a^3 x+\frac {b \left (F^{g (e+f x)}\right )^n \left (18 a^2+9 a b \left (F^{g (e+f x)}\right )^n+2 b^2 \left (F^{g (e+f x)}\right )^{2 n}\right )}{6 f g n \log (F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 84, normalized size = 0.82 \[ \frac {6 \, a^{3} f g n x \log \relax (F) + 18 \, F^{f g n x + e g n} a^{2} b + 9 \, F^{2 \, f g n x + 2 \, e g n} a b^{2} + 2 \, F^{3 \, f g n x + 3 \, e g n} b^{3}}{6 \, f g n \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 102, normalized size = 0.99 \[ \frac {18 \, F^{f g n x} F^{g n e} a^{2} b + 9 \, F^{2 \, f g n x} F^{2 \, g n e} a b^{2} + 2 \, F^{3 \, f g n x} F^{3 \, g n e} b^{3} + 6 \, a^{3} \log \left ({\left | F \right |}^{f g n x} {\left | F \right |}^{g n e}\right )}{6 \, f g n \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 124, normalized size = 1.20 \[ \frac {a^{3} \ln \left (\left (F^{\left (f x +e \right ) g}\right )^{n}\right )}{f g n \ln \relax (F )}+\frac {3 a^{2} b \left (F^{\left (f x +e \right ) g}\right )^{n}}{f g n \ln \relax (F )}+\frac {3 a \,b^{2} \left (F^{\left (f x +e \right ) g}\right )^{2 n}}{2 f g n \ln \relax (F )}+\frac {b^{3} \left (F^{\left (f x +e \right ) g}\right )^{3 n}}{3 f g n \ln \relax (F )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.56, size = 115, normalized size = 1.12 \[ a^{3} x + \frac {3 \, {\left (F^{f g x}\right )}^{n} {\left (F^{e g}\right )}^{n} a^{2} b}{f g n \log \relax (F)} + \frac {3 \, {\left (F^{f g x}\right )}^{2 \, n} {\left (F^{e g}\right )}^{2 \, n} a b^{2}}{2 \, f g n \log \relax (F)} + \frac {{\left (F^{f g x}\right )}^{3 \, n} {\left (F^{e g}\right )}^{3 \, n} b^{3}}{3 \, f g n \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.78, size = 124, normalized size = 1.20 \[ \frac {a^3\,\ln \left (F^{f\,g\,x}\right )}{f\,g\,\ln \relax (F)}+\frac {b^3\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{3\,n}}{3\,f\,g\,n\,\ln \relax (F)}+\frac {3\,a^2\,b\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n}{f\,g\,n\,\ln \relax (F)}+\frac {3\,a\,b^2\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}}{2\,f\,g\,n\,\ln \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 153, normalized size = 1.49 \[ a^{3} x + \begin {cases} \frac {18 a^{2} b f^{2} g^{2} n^{2} \left (F^{g \left (e + f x\right )}\right )^{n} \log {\relax (F )}^{2} + 9 a b^{2} f^{2} g^{2} n^{2} \left (F^{g \left (e + f x\right )}\right )^{2 n} \log {\relax (F )}^{2} + 2 b^{3} f^{2} g^{2} n^{2} \left (F^{g \left (e + f x\right )}\right )^{3 n} \log {\relax (F )}^{2}}{6 f^{3} g^{3} n^{3} \log {\relax (F )}^{3}} & \text {for}\: 6 f^{3} g^{3} n^{3} \log {\relax (F )}^{3} \neq 0 \\x \left (3 a^{2} b + 3 a b^{2} + b^{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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