Optimal. Leaf size=322 \[ \frac{a^2 (c+d x)^4}{4 d}+\frac{12 a b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac{6 a b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{2 a b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{12 a b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}+\frac{3 b^2 d^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac{3 b^2 d (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)}+\frac{b^2 (c+d x)^3 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac{3 b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)} \]
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Rubi [A] time = 0.486487, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2183, 2176, 2194} \[ \frac{a^2 (c+d x)^4}{4 d}+\frac{12 a b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac{6 a b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{2 a b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{12 a b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}+\frac{3 b^2 d^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac{3 b^2 d (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)}+\frac{b^2 (c+d x)^3 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac{3 b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)} \]
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 )^2 (c+d x)^3 \, dx &=\int \left (a^2 (c+d x)^3+2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3+b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3\right ) \, dx\\ &=\frac{a^2 (c+d x)^4}{4 d}+(2 a b) \int \left (F^{e g+f g x}\right )^n (c+d x)^3 \, dx+b^2 \int \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3 \, dx\\ &=\frac{a^2 (c+d x)^4}{4 d}+\frac{2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac{b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}-\frac{(6 a b d) \int \left (F^{e g+f g x}\right )^n (c+d x)^2 \, dx}{f g n \log (F)}-\frac{\left (3 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{a^2 (c+d x)^4}{4 d}-\frac{6 a 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{3 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{2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac{b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}+\frac{\left (12 a 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 (3 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{a^2 (c+d x)^4}{4 d}+\frac{12 a 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{3 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{6 a 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{3 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{2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac{b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}-\frac{\left (12 a 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 (3 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{a^2 (c+d x)^4}{4 d}-\frac{12 a b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}-\frac{3 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{12 a 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{3 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{6 a 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{3 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{2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac{b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}\\ \end{align*}
Mathematica [A] time = 0.545496, size = 239, normalized size = 0.74 \[ \frac{3}{2} a^2 c^2 d x^2+a^2 c^3 x+a^2 c d^2 x^3+\frac{1}{4} a^2 d^3 x^4+\frac{2 a b \left (F^{g (e+f x)}\right )^n \left (6 d^2 f g n \log (F) (c+d x)-3 d f^2 g^2 n^2 \log ^2(F) (c+d x)^2+f^3 g^3 n^3 \log ^3(F) (c+d x)^3-6 d^3\right )}{f^4 g^4 n^4 \log ^4(F)}+\frac{b^2 \left (F^{g (e+f x)}\right )^{2 n} \left (6 d^2 f g n \log (F) (c+d x)-6 d f^2 g^2 n^2 \log ^2(F) (c+d x)^2+4 f^3 g^3 n^3 \log ^3(F) (c+d x)^3-3 d^3\right )}{8 f^4 g^4 n^4 \log ^4(F)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.022, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{2} \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 = 1.56931, size = 999, normalized size = 3.1 \begin{align*} \frac{2 \,{\left (a^{2} d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, a^{2} c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, a^{2} c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, a^{2} c^{3} f^{4} g^{4} n^{4} x\right )} \log \left (F\right )^{4} -{\left (3 \, b^{2} d^{3} - 4 \,{\left (b^{2} d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, b^{2} c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, b^{2} c^{2} d f^{3} g^{3} n^{3} x + b^{2} c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 6 \,{\left (b^{2} d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c d^{2} f^{2} g^{2} n^{2} x + b^{2} c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \,{\left (b^{2} d^{3} f g n x + b^{2} c d^{2} f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} - 16 \,{\left (6 \, a b d^{3} -{\left (a b d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, a b c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, a b c^{2} d f^{3} g^{3} n^{3} x + a b c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 3 \,{\left (a b d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, a b c d^{2} f^{2} g^{2} n^{2} x + a b c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \,{\left (a b d^{3} f g n x + a b c d^{2} f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{8 \, 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 = 0.56303, size = 709, normalized size = 2.2 \begin{align*} a^{2} c^{3} x + \frac{3 a^{2} c^{2} d x^{2}}{2} + a^{2} c d^{2} x^{3} + \frac{a^{2} d^{3} x^{4}}{4} + \begin{cases} \frac{\left (4 b^{2} c^{3} f^{7} g^{7} n^{7} \log{\left (F \right )}^{7} + 12 b^{2} c^{2} d f^{7} g^{7} n^{7} x \log{\left (F \right )}^{7} - 6 b^{2} c^{2} d f^{6} g^{6} n^{6} \log{\left (F \right )}^{6} + 12 b^{2} c d^{2} f^{7} g^{7} n^{7} x^{2} \log{\left (F \right )}^{7} - 12 b^{2} c d^{2} f^{6} g^{6} n^{6} x \log{\left (F \right )}^{6} + 6 b^{2} c d^{2} f^{5} g^{5} n^{5} \log{\left (F \right )}^{5} + 4 b^{2} d^{3} f^{7} g^{7} n^{7} x^{3} \log{\left (F \right )}^{7} - 6 b^{2} d^{3} f^{6} g^{6} n^{6} x^{2} \log{\left (F \right )}^{6} + 6 b^{2} d^{3} f^{5} g^{5} n^{5} x \log{\left (F \right )}^{5} - 3 b^{2} d^{3} f^{4} g^{4} n^{4} \log{\left (F \right )}^{4}\right ) \left (F^{g \left (e + f x\right )}\right )^{2 n} + \left (16 a b c^{3} f^{7} g^{7} n^{7} \log{\left (F \right )}^{7} + 48 a b c^{2} d f^{7} g^{7} n^{7} x \log{\left (F \right )}^{7} - 48 a b c^{2} d f^{6} g^{6} n^{6} \log{\left (F \right )}^{6} + 48 a b c d^{2} f^{7} g^{7} n^{7} x^{2} \log{\left (F \right )}^{7} - 96 a b c d^{2} f^{6} g^{6} n^{6} x \log{\left (F \right )}^{6} + 96 a b c d^{2} f^{5} g^{5} n^{5} \log{\left (F \right )}^{5} + 16 a b d^{3} f^{7} g^{7} n^{7} x^{3} \log{\left (F \right )}^{7} - 48 a b d^{3} f^{6} g^{6} n^{6} x^{2} \log{\left (F \right )}^{6} + 96 a b d^{3} f^{5} g^{5} n^{5} x \log{\left (F \right )}^{5} - 96 a b d^{3} f^{4} g^{4} n^{4} \log{\left (F \right )}^{4}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{8 f^{8} g^{8} n^{8} \log{\left (F \right )}^{8}} & \text{for}\: 8 f^{8} g^{8} n^{8} \log{\left (F \right )}^{8} \neq 0 \\x^{4} \left (\frac{a b d^{3}}{2} + \frac{b^{2} d^{3}}{4}\right ) + x^{3} \left (2 a b c d^{2} + b^{2} c d^{2}\right ) + x^{2} \left (3 a b c^{2} d + \frac{3 b^{2} c^{2} d}{2}\right ) + x \left (2 a b c^{3} + b^{2} c^{3}\right ) & \text{otherwise} \end{cases} \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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