Optimal. Leaf size=103 \[ \frac{3 a^2 b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}+a^3 x+\frac{3 a b^2 \left (F^{g (e+f x)}\right )^{2 n}}{2 f g n \log (F)}+\frac{b^3 \left (F^{g (e+f x)}\right )^{3 n}}{3 f g n \log (F)} \]
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Rubi [A] time = 0.0493079, antiderivative size = 103, normalized size of antiderivative = 1., 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 2282
Rule 266
Rule 43
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*}
Mathematica [A] time = 0.0487694, size = 74, normalized size = 0.72 \[ \frac{b \left (F^{g (e+f x)}\right )^n \left (18 a^2+9 a b \left (F^{g (e+f x)}\right )^n+2 b^2 \left (F^{g (e+f x)}\right )^{2 n}\right )}{6 f g n \log (F)}+a^3 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 124, normalized size = 1.2 \begin{align*}{\frac{{b}^{3} \left ( \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{3}}{3\,ngf\ln \left ( F \right ) }}+{\frac{3\,a{b}^{2} \left ( \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{2}}{2\,ngf\ln \left ( F \right ) }}+3\,{\frac{{a}^{2}b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n}}{ngf\ln \left ( F \right ) }}+{\frac{{a}^{3}\ln \left ( \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) }{ngf\ln \left ( F \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22977, size = 155, normalized size = 1.5 \begin{align*} 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 \left (F\right )} + \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 \left (F\right )} + \frac{{\left (F^{f g x}\right )}^{3 \, n}{\left (F^{e g}\right )}^{3 \, n} b^{3}}{3 \, f g n \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51905, size = 189, normalized size = 1.83 \begin{align*} \frac{6 \, a^{3} f g n x \log \left (F\right ) + 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 \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.291525, size = 153, normalized size = 1.49 \begin{align*} 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{\left (F \right )}^{2} + 9 a b^{2} f^{2} g^{2} n^{2} \left (F^{g \left (e + f x\right )}\right )^{2 n} \log{\left (F \right )}^{2} + 2 b^{3} f^{2} g^{2} n^{2} \left (F^{g \left (e + f x\right )}\right )^{3 n} \log{\left (F \right )}^{2}}{6 f^{3} g^{3} n^{3} \log{\left (F \right )}^{3}} & \text{for}\: 6 f^{3} g^{3} n^{3} \log{\left (F \right )}^{3} \neq 0 \\x \left (3 a^{2} b + 3 a b^{2} + b^{3}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.37772, size = 1393, normalized size = 13.52 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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