Optimal. Leaf size=496 \[ \frac{18 a^2 b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac{9 a^2 b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{3 a^2 b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{18 a^2 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}+\frac{a^3 (c+d x)^4}{4 d}+\frac{9 a b^2 d^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac{9 a b^2 d (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)}+\frac{3 a b^2 (c+d x)^3 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac{9 a b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)}+\frac{2 b^3 d^2 (c+d x) \left (F^{e g+f g x}\right )^{3 n}}{9 f^3 g^3 n^3 \log ^3(F)}-\frac{b^3 d (c+d x)^2 \left (F^{e g+f g x}\right )^{3 n}}{3 f^2 g^2 n^2 \log ^2(F)}+\frac{b^3 (c+d x)^3 \left (F^{e g+f g x}\right )^{3 n}}{3 f g n \log (F)}-\frac{2 b^3 d^3 \left (F^{e g+f g x}\right )^{3 n}}{27 f^4 g^4 n^4 \log ^4(F)} \]
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Rubi [A] time = 0.705136, antiderivative size = 496, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2183, 2176, 2194} \[ \frac{18 a^2 b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac{9 a^2 b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{3 a^2 b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{18 a^2 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}+\frac{a^3 (c+d x)^4}{4 d}+\frac{9 a b^2 d^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac{9 a b^2 d (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)}+\frac{3 a b^2 (c+d x)^3 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac{9 a b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)}+\frac{2 b^3 d^2 (c+d x) \left (F^{e g+f g x}\right )^{3 n}}{9 f^3 g^3 n^3 \log ^3(F)}-\frac{b^3 d (c+d x)^2 \left (F^{e g+f g x}\right )^{3 n}}{3 f^2 g^2 n^2 \log ^2(F)}+\frac{b^3 (c+d x)^3 \left (F^{e g+f g x}\right )^{3 n}}{3 f g n \log (F)}-\frac{2 b^3 d^3 \left (F^{e g+f g x}\right )^{3 n}}{27 f^4 g^4 n^4 \log ^4(F)} \]
Antiderivative was successfully verified.
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Rule 2183
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^3 \, dx &=\int \left (a^3 (c+d x)^3+3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^3+3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3+b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3\right ) \, dx\\ &=\frac{a^3 (c+d x)^4}{4 d}+\left (3 a^2 b\right ) \int \left (F^{e g+f g x}\right )^n (c+d x)^3 \, dx+\left (3 a b^2\right ) \int \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3 \, dx+b^3 \int \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3 \, dx\\ &=\frac{a^3 (c+d x)^4}{4 d}+\frac{3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}+\frac{b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3}{3 f g n \log (F)}-\frac{\left (9 a^2 b d\right ) \int \left (F^{e g+f g x}\right )^n (c+d x)^2 \, dx}{f g n \log (F)}-\frac{\left (9 a b^2 d\right ) \int \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2 \, dx}{2 f g n \log (F)}-\frac{\left (b^3 d\right ) \int \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2 \, dx}{f g n \log (F)}\\ &=\frac{a^3 (c+d x)^4}{4 d}-\frac{9 a^2 b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}-\frac{9 a b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{4 f^2 g^2 n^2 \log ^2(F)}-\frac{b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f^2 g^2 n^2 \log ^2(F)}+\frac{3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}+\frac{b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3}{3 f g n \log (F)}+\frac{\left (18 a^2 b d^2\right ) \int \left (F^{e g+f g x}\right )^n (c+d x) \, dx}{f^2 g^2 n^2 \log ^2(F)}+\frac{\left (9 a b^2 d^2\right ) \int \left (F^{e g+f g x}\right )^{2 n} (c+d x) \, dx}{2 f^2 g^2 n^2 \log ^2(F)}+\frac{\left (2 b^3 d^2\right ) \int \left (F^{e g+f g x}\right )^{3 n} (c+d x) \, dx}{3 f^2 g^2 n^2 \log ^2(F)}\\ &=\frac{a^3 (c+d x)^4}{4 d}+\frac{18 a^2 b d^2 \left (F^{e g+f g x}\right )^n (c+d x)}{f^3 g^3 n^3 \log ^3(F)}+\frac{9 a b^2 d^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{4 f^3 g^3 n^3 \log ^3(F)}+\frac{2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{9 f^3 g^3 n^3 \log ^3(F)}-\frac{9 a^2 b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}-\frac{9 a b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{4 f^2 g^2 n^2 \log ^2(F)}-\frac{b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f^2 g^2 n^2 \log ^2(F)}+\frac{3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}+\frac{b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3}{3 f g n \log (F)}-\frac{\left (18 a^2 b d^3\right ) \int \left (F^{e g+f g x}\right )^n \, dx}{f^3 g^3 n^3 \log ^3(F)}-\frac{\left (9 a b^2 d^3\right ) \int \left (F^{e g+f g x}\right )^{2 n} \, dx}{4 f^3 g^3 n^3 \log ^3(F)}-\frac{\left (2 b^3 d^3\right ) \int \left (F^{e g+f g x}\right )^{3 n} \, dx}{9 f^3 g^3 n^3 \log ^3(F)}\\ &=\frac{a^3 (c+d x)^4}{4 d}-\frac{18 a^2 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}-\frac{9 a b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)}-\frac{2 b^3 d^3 \left (F^{e g+f g x}\right )^{3 n}}{27 f^4 g^4 n^4 \log ^4(F)}+\frac{18 a^2 b d^2 \left (F^{e g+f g x}\right )^n (c+d x)}{f^3 g^3 n^3 \log ^3(F)}+\frac{9 a b^2 d^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{4 f^3 g^3 n^3 \log ^3(F)}+\frac{2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{9 f^3 g^3 n^3 \log ^3(F)}-\frac{9 a^2 b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}-\frac{9 a b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{4 f^2 g^2 n^2 \log ^2(F)}-\frac{b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f^2 g^2 n^2 \log ^2(F)}+\frac{3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}+\frac{b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3}{3 f g n \log (F)}\\ \end{align*}
Mathematica [A] time = 0.721437, size = 341, normalized size = 0.69 \[ \frac{3 a^2 b \left (F^{g (e+f x)}\right )^n \left (6 d^2 f g n \log (F) (c+d x)-3 d f^2 g^2 n^2 \log ^2(F) (c+d x)^2+f^3 g^3 n^3 \log ^3(F) (c+d x)^3-6 d^3\right )}{f^4 g^4 n^4 \log ^4(F)}+\frac{3}{2} a^3 c^2 d x^2+a^3 c^3 x+a^3 c d^2 x^3+\frac{1}{4} a^3 d^3 x^4+\frac{3 a b^2 \left (F^{g (e+f x)}\right )^{2 n} \left (6 d^2 f g n \log (F) (c+d x)-6 d f^2 g^2 n^2 \log ^2(F) (c+d x)^2+4 f^3 g^3 n^3 \log ^3(F) (c+d x)^3-3 d^3\right )}{8 f^4 g^4 n^4 \log ^4(F)}+\frac{b^3 \left (F^{g (e+f x)}\right )^{3 n} \left (6 d^2 f g n \log (F) (c+d x)-9 d f^2 g^2 n^2 \log ^2(F) (c+d x)^2+9 f^3 g^3 n^3 \log ^3(F) (c+d x)^3-2 d^3\right )}{27 f^4 g^4 n^4 \log ^4(F)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{3} \left ( dx+c \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65667, size = 1463, normalized size = 2.95 \begin{align*} \frac{54 \,{\left (a^{3} d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, a^{3} c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, a^{3} c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, a^{3} c^{3} f^{4} g^{4} n^{4} x\right )} \log \left (F\right )^{4} - 8 \,{\left (2 \, b^{3} d^{3} - 9 \,{\left (b^{3} d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, b^{3} c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, b^{3} c^{2} d f^{3} g^{3} n^{3} x + b^{3} c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 9 \,{\left (b^{3} d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{3} c d^{2} f^{2} g^{2} n^{2} x + b^{3} c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \,{\left (b^{3} d^{3} f g n x + b^{3} c d^{2} f g n\right )} \log \left (F\right )\right )} F^{3 \, f g n x + 3 \, e g n} - 81 \,{\left (3 \, a b^{2} d^{3} - 4 \,{\left (a b^{2} d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, a b^{2} c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, a b^{2} c^{2} d f^{3} g^{3} n^{3} x + a b^{2} c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 6 \,{\left (a b^{2} d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, a b^{2} c d^{2} f^{2} g^{2} n^{2} x + a b^{2} c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \,{\left (a b^{2} d^{3} f g n x + a b^{2} c d^{2} f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} - 648 \,{\left (6 \, a^{2} b d^{3} -{\left (a^{2} b d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, a^{2} b c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, a^{2} b c^{2} d f^{3} g^{3} n^{3} x + a^{2} b c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 3 \,{\left (a^{2} b d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, a^{2} b c d^{2} f^{2} g^{2} n^{2} x + a^{2} b c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \,{\left (a^{2} b d^{3} f g n x + a^{2} b c d^{2} f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{216 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.985326, size = 1074, normalized size = 2.17 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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