Optimal. Leaf size=239 \[ \frac {a^2 (c+d x)^3}{3 d}+\frac {4 a b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}+\frac {b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {4 a b d \left (F^{e g+f g x}\right )^n (c+d x)}{f^2 g^2 n^2 \log ^2(F)}-\frac {b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{2 f^2 g^2 n^2 \log ^2(F)}+\frac {2 a b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)} \]
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Rubi [A]
time = 0.22, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2214, 2207,
2225} \begin {gather*} \frac {a^2 (c+d x)^3}{3 d}-\frac {4 a b d (c+d x) \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)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac {4 a b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {b^2 d (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{2 f^2 g^2 n^2 \log ^2(F)}+\frac {b^2 (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}+\frac {b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2214
Rule 2225
Rubi steps
\begin {align*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2 \, dx &=\int \left (a^2 (c+d x)^2+2 a b \left (F^{e g+f g x}\right )^n (c+d x)^2+b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2\right ) \, dx\\ &=\frac {a^2 (c+d x)^3}{3 d}+(2 a b) \int \left (F^{e g+f g x}\right )^n (c+d x)^2 \, dx+b^2 \int \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2 \, dx\\ &=\frac {a^2 (c+d x)^3}{3 d}+\frac {2 a b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}-\frac {(4 a b d) \int \left (F^{e g+f g x}\right )^n (c+d x) \, dx}{f g n \log (F)}-\frac {\left (b^2 d\right ) \int \left (F^{e g+f g x}\right )^{2 n} (c+d x) \, dx}{f g n \log (F)}\\ &=\frac {a^2 (c+d x)^3}{3 d}-\frac {4 a b d \left (F^{e g+f g x}\right )^n (c+d x)}{f^2 g^2 n^2 \log ^2(F)}-\frac {b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{2 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)^2}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}+\frac {\left (4 a b d^2\right ) \int \left (F^{e g+f g x}\right )^n \, dx}{f^2 g^2 n^2 \log ^2(F)}+\frac {\left (b^2 d^2\right ) \int \left (F^{e g+f g x}\right )^{2 n} \, dx}{2 f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {a^2 (c+d x)^3}{3 d}+\frac {4 a b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}+\frac {b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {4 a b d \left (F^{e g+f g x}\right )^n (c+d x)}{f^2 g^2 n^2 \log ^2(F)}-\frac {b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{2 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)^2}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 171, normalized size = 0.72 \begin {gather*} a^2 c^2 x+a^2 c d x^2+\frac {1}{3} a^2 d^2 x^3+\frac {2 a b \left (F^{g (e+f x)}\right )^n \left (2 d^2-2 d f g n (c+d x) \log (F)+f^2 g^2 n^2 (c+d x)^2 \log ^2(F)\right )}{f^3 g^3 n^3 \log ^3(F)}+\frac {b^2 \left (F^{g (e+f x)}\right )^{2 n} \left (d^2-2 d f g n (c+d x) \log (F)+2 f^2 g^2 n^2 (c+d x)^2 \log ^2(F)\right )}{4 f^3 g^3 n^3 \log ^3(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{2} \left (d x +c \right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 359, normalized size = 1.50 \begin {gather*} \frac {1}{3} \, a^{2} d^{2} x^{3} + a^{2} c d x^{2} + a^{2} c^{2} x + \frac {2 \, F^{f g n x + g n e} a b c^{2}}{f g n \log \left (F\right )} + \frac {F^{2 \, f g n x + 2 \, g n e} b^{2} c^{2}}{2 \, f g n \log \left (F\right )} + \frac {4 \, {\left (F^{g n e} f g n x \log \left (F\right ) - F^{g n e}\right )} F^{f g n x} a b c d}{f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {{\left (2 \, F^{2 \, g n e} f g n x \log \left (F\right ) - F^{2 \, g n e}\right )} F^{2 \, f g n x} b^{2} c d}{2 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {2 \, {\left (F^{g n e} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{g n e} f g n x \log \left (F\right ) + 2 \, F^{g n e}\right )} F^{f g n x} a b d^{2}}{f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {{\left (2 \, F^{2 \, g n e} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{2 \, g n e} f g n x \log \left (F\right ) + F^{2 \, g n e}\right )} F^{2 \, f g n x} b^{2} d^{2}}{4 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 289, normalized size = 1.21 \begin {gather*} \frac {4 \, {\left (a^{2} d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, a^{2} c d f^{3} g^{3} n^{3} x^{2} + 3 \, a^{2} c^{2} f^{3} g^{3} n^{3} x\right )} \log \left (F\right )^{3} + 3 \, {\left (b^{2} d^{2} + 2 \, {\left (b^{2} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c d f^{2} g^{2} n^{2} x + b^{2} c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (b^{2} d^{2} f g n x + b^{2} c d f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, g n e} + 24 \, {\left (2 \, a b d^{2} + {\left (a b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a b c d f^{2} g^{2} n^{2} x + a b c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (a b d^{2} f g n x + a b c d f g n\right )} \log \left (F\right )\right )} F^{f g n x + g n e}}{12 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 437, normalized size = 1.83 \begin {gather*} a^{2} c^{2} x + a^{2} c d x^{2} + \frac {a^{2} d^{2} x^{3}}{3} + \begin {cases} \frac {\left (2 b^{2} c^{2} f^{5} g^{5} n^{5} \log {\left (F \right )}^{5} + 4 b^{2} c d f^{5} g^{5} n^{5} x \log {\left (F \right )}^{5} - 2 b^{2} c d f^{4} g^{4} n^{4} \log {\left (F \right )}^{4} + 2 b^{2} d^{2} f^{5} g^{5} n^{5} x^{2} \log {\left (F \right )}^{5} - 2 b^{2} d^{2} f^{4} g^{4} n^{4} x \log {\left (F \right )}^{4} + b^{2} d^{2} f^{3} g^{3} n^{3} \log {\left (F \right )}^{3}\right ) \left (F^{g \left (e + f x\right )}\right )^{2 n} + \left (8 a b c^{2} f^{5} g^{5} n^{5} \log {\left (F \right )}^{5} + 16 a b c d f^{5} g^{5} n^{5} x \log {\left (F \right )}^{5} - 16 a b c d f^{4} g^{4} n^{4} \log {\left (F \right )}^{4} + 8 a b d^{2} f^{5} g^{5} n^{5} x^{2} \log {\left (F \right )}^{5} - 16 a b d^{2} f^{4} g^{4} n^{4} x \log {\left (F \right )}^{4} + 16 a b d^{2} f^{3} g^{3} n^{3} \log {\left (F \right )}^{3}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{4 f^{6} g^{6} n^{6} \log {\left (F \right )}^{6}} & \text {for}\: f^{6} g^{6} n^{6} \log {\left (F \right )}^{6} \neq 0 \\x^{3} \cdot \left (\frac {2 a b d^{2}}{3} + \frac {b^{2} d^{2}}{3}\right ) + x^{2} \cdot \left (2 a b c d + b^{2} c d\right ) + x \left (2 a b c^{2} + b^{2} c^{2}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 2.73, size = 5695, normalized size = 23.83 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.82, size = 267, normalized size = 1.12 \begin {gather*} {\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}\,\left (\frac {b^2\,\left (2\,c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-2\,c\,d\,f\,g\,n\,\ln \left (F\right )+d^2\right )}{4\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {b^2\,d^2\,x^2}{2\,f\,g\,n\,\ln \left (F\right )}-\frac {b^2\,d\,x\,\left (d-2\,c\,f\,g\,n\,\ln \left (F\right )\right )}{2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {2\,a\,b\,\left (c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-2\,c\,d\,f\,g\,n\,\ln \left (F\right )+2\,d^2\right )}{f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {2\,a\,b\,d^2\,x^2}{f\,g\,n\,\ln \left (F\right )}-\frac {4\,a\,b\,d\,x\,\left (d-c\,f\,g\,n\,\ln \left (F\right )\right )}{f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+a^2\,c^2\,x+\frac {a^2\,d^2\,x^3}{3}+a^2\,c\,d\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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