Optimal. Leaf size=322 \[ \frac {a^2 (c+d x)^4}{4 d}-\frac {12 a b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}-\frac {3 b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)}+\frac {12 a b d^2 \left (F^{e g+f g x}\right )^n (c+d x)}{f^3 g^3 n^3 \log ^3(F)}+\frac {3 b^2 d^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {6 a b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}-\frac {3 b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{4 f^2 g^2 n^2 \log ^2(F)}+\frac {2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)} \]
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Rubi [A]
time = 0.34, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps
used = 10, 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)^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)} \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)^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*}
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Mathematica [A]
time = 0.44, size = 239, normalized size = 0.74 \begin {gather*} a^2 c^3 x+\frac {3}{2} a^2 c^2 d x^2+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^3+6 d^2 f g n (c+d x) \log (F)-3 d f^2 g^2 n^2 (c+d x)^2 \log ^2(F)+f^3 g^3 n^3 (c+d x)^3 \log ^3(F)\right )}{f^4 g^4 n^4 \log ^4(F)}+\frac {b^2 \left (F^{g (e+f x)}\right )^{2 n} \left (-3 d^3+6 d^2 f g n (c+d x) \log (F)-6 d f^2 g^2 n^2 (c+d x)^2 \log ^2(F)+4 f^3 g^3 n^3 (c+d x)^3 \log ^3(F)\right )}{8 f^4 g^4 n^4 \log ^4(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{2} \left (d x +c \right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 588, normalized size = 1.83 \begin {gather*} \frac {1}{4} \, a^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + a^{2} c^{3} x + \frac {2 \, F^{f g n x + g n e} a b c^{3}}{f g n \log \left (F\right )} + \frac {F^{2 \, f g n x + 2 \, g n e} b^{2} c^{3}}{2 \, f g n \log \left (F\right )} + \frac {6 \, {\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^{2} d}{f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {3 \, {\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^{2} d}{4 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {6 \, {\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 c d^{2}}{f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {3 \, {\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} c d^{2}}{4 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {2 \, {\left (F^{g n e} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{g n e} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{g n e} f g n x \log \left (F\right ) - 6 \, F^{g n e}\right )} F^{f g n x} a b d^{3}}{f^{4} g^{4} n^{4} \log \left (F\right )^{4}} + \frac {{\left (4 \, F^{2 \, g n e} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} - 6 \, F^{2 \, g n e} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{2 \, g n e} f g n x \log \left (F\right ) - 3 \, F^{2 \, g n e}\right )} F^{2 \, f g n x} b^{2} d^{3}}{8 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 484, normalized size = 1.50 \begin {gather*} \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 \, g n e} - 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 + g n e}}{8 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 707, normalized size = 2.20 \begin {gather*} 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}\: 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} \cdot \left (2 a b c d^{2} + b^{2} c d^{2}\right ) + x^{2} \cdot \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 {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 = 4.35, size = 12033, normalized size = 37.37 \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 = 3.97, size = 438, normalized size = 1.36 \begin {gather*} a^2\,c^3\,x-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {2\,a\,b\,\left (-c^3\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3+3\,c^2\,d\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-6\,c\,d^2\,f\,g\,n\,\ln \left (F\right )+6\,d^3\right )}{f^4\,g^4\,n^4\,{\ln \left (F\right )}^4}-\frac {2\,a\,b\,d^3\,x^3}{f\,g\,n\,\ln \left (F\right )}-\frac {6\,a\,b\,d\,x\,\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 {6\,a\,b\,d^2\,x^2\,\left (d-c\,f\,g\,n\,\ln \left (F\right )\right )}{f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}\,\left (\frac {b^2\,\left (-4\,c^3\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3+6\,c^2\,d\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-6\,c\,d^2\,f\,g\,n\,\ln \left (F\right )+3\,d^3\right )}{8\,f^4\,g^4\,n^4\,{\ln \left (F\right )}^4}-\frac {b^2\,d^3\,x^3}{2\,f\,g\,n\,\ln \left (F\right )}-\frac {3\,b^2\,d\,x\,\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 {3\,b^2\,d^2\,x^2\,\left (d-2\,c\,f\,g\,n\,\ln \left (F\right )\right )}{4\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+\frac {a^2\,d^3\,x^4}{4}+\frac {3\,a^2\,c^2\,d\,x^2}{2}+a^2\,c\,d^2\,x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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