Optimal. Leaf size=366 \[ \frac {a^3 (c+d x)^3}{3 d}+\frac {6 a^2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}+\frac {3 a b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}+\frac {2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n}}{27 f^3 g^3 n^3 \log ^3(F)}-\frac {6 a^2 b d \left (F^{e g+f g x}\right )^n (c+d x)}{f^2 g^2 n^2 \log ^2(F)}-\frac {3 a b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{2 f^2 g^2 n^2 \log ^2(F)}-\frac {2 b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{9 f^2 g^2 n^2 \log ^2(F)}+\frac {3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f g n \log (F)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.32, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps
used = 11, 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^3 (c+d x)^3}{3 d}-\frac {6 a^2 b d (c+d x) \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)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac {6 a^2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {3 a 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 {3 a b^2 (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}+\frac {3 a 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 {2 b^3 d (c+d x) \left (F^{e g+f g x}\right )^{3 n}}{9 f^2 g^2 n^2 \log ^2(F)}+\frac {b^3 (c+d x)^2 \left (F^{e g+f g x}\right )^{3 n}}{3 f g n \log (F)}+\frac {2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n}}{27 f^3 g^3 n^3 \log ^3(F)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2207
Rule 2214
Rule 2225
Rubi steps
\begin {align*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2 \, dx &=\int \left (a^3 (c+d x)^2+3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^2+3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2+b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2\right ) \, dx\\ &=\frac {a^3 (c+d x)^3}{3 d}+\left (3 a^2 b\right ) \int \left (F^{e g+f g x}\right )^n (c+d x)^2 \, dx+\left (3 a b^2\right ) \int \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2 \, dx+b^3 \int \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2 \, dx\\ &=\frac {a^3 (c+d x)^3}{3 d}+\frac {3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f g n \log (F)}-\frac {\left (6 a^2 b d\right ) \int \left (F^{e g+f g x}\right )^n (c+d x) \, dx}{f g n \log (F)}-\frac {\left (3 a 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 {\left (2 b^3 d\right ) \int \left (F^{e g+f g x}\right )^{3 n} (c+d x) \, dx}{3 f g n \log (F)}\\ &=\frac {a^3 (c+d x)^3}{3 d}-\frac {6 a^2 b d \left (F^{e g+f g x}\right )^n (c+d x)}{f^2 g^2 n^2 \log ^2(F)}-\frac {3 a 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 b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{9 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)^2}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f g n \log (F)}+\frac {\left (6 a^2 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 (3 a 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 {\left (2 b^3 d^2\right ) \int \left (F^{e g+f g x}\right )^{3 n} \, dx}{9 f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {a^3 (c+d x)^3}{3 d}+\frac {6 a^2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}+\frac {3 a 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 {2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n}}{27 f^3 g^3 n^3 \log ^3(F)}-\frac {6 a^2 b d \left (F^{e g+f g x}\right )^n (c+d x)}{f^2 g^2 n^2 \log ^2(F)}-\frac {3 a 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 b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{9 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)^2}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f g n \log (F)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.51, size = 248, normalized size = 0.68 \begin {gather*} a^3 c^2 x+a^3 c d x^2+\frac {1}{3} a^3 d^2 x^3+\frac {3 a^2 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 {3 a 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)}+\frac {b^3 \left (F^{g (e+f x)}\right )^{3 n} \left (2 d^2-6 d f g n (c+d x) \log (F)+9 f^2 g^2 n^2 (c+d x)^2 \log ^2(F)\right )}{27 f^3 g^3 n^3 \log ^3(F)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{3} \left (d x +c \right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 541, normalized size = 1.48 \begin {gather*} \frac {1}{3} \, a^{3} d^{2} x^{3} + a^{3} c d x^{2} + a^{3} c^{2} x + \frac {3 \, F^{f g n x + g n e} a^{2} b c^{2}}{f g n \log \left (F\right )} + \frac {3 \, F^{2 \, f g n x + 2 \, g n e} a b^{2} c^{2}}{2 \, f g n \log \left (F\right )} + \frac {F^{3 \, f g n x + 3 \, g n e} b^{3} c^{2}}{3 \, 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^{2} b c 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} a b^{2} c d}{2 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {2 \, {\left (3 \, F^{3 \, g n e} f g n x \log \left (F\right ) - F^{3 \, g n e}\right )} F^{3 \, f g n x} b^{3} c d}{9 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {3 \, {\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^{2} b 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} a b^{2} d^{2}}{4 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {{\left (9 \, F^{3 \, g n e} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 6 \, F^{3 \, g n e} f g n x \log \left (F\right ) + 2 \, F^{3 \, g n e}\right )} F^{3 \, f g n x} b^{3} d^{2}}{27 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 417, normalized size = 1.14 \begin {gather*} \frac {36 \, {\left (a^{3} d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, a^{3} c d f^{3} g^{3} n^{3} x^{2} + 3 \, a^{3} c^{2} f^{3} g^{3} n^{3} x\right )} \log \left (F\right )^{3} + 4 \, {\left (2 \, b^{3} d^{2} + 9 \, {\left (b^{3} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{3} c d f^{2} g^{2} n^{2} x + b^{3} c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (b^{3} d^{2} f g n x + b^{3} c d f g n\right )} \log \left (F\right )\right )} F^{3 \, f g n x + 3 \, g n e} + 81 \, {\left (a b^{2} d^{2} + 2 \, {\left (a b^{2} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a b^{2} c d f^{2} g^{2} n^{2} x + a b^{2} c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (a b^{2} d^{2} f g n x + a b^{2} c d f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, g n e} + 324 \, {\left (2 \, a^{2} b d^{2} + {\left (a^{2} b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a^{2} b c d f^{2} g^{2} n^{2} x + a^{2} b c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (a^{2} b d^{2} f g n x + a^{2} b c d f g n\right )} \log \left (F\right )\right )} F^{f g n x + g n e}}{108 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.25, size = 651, normalized size = 1.78 \begin {gather*} a^{3} c^{2} x + a^{3} c d x^{2} + \frac {a^{3} d^{2} x^{3}}{3} + \begin {cases} \frac {\left (36 b^{3} c^{2} f^{8} g^{8} n^{8} \log {\left (F \right )}^{8} + 72 b^{3} c d f^{8} g^{8} n^{8} x \log {\left (F \right )}^{8} - 24 b^{3} c d f^{7} g^{7} n^{7} \log {\left (F \right )}^{7} + 36 b^{3} d^{2} f^{8} g^{8} n^{8} x^{2} \log {\left (F \right )}^{8} - 24 b^{3} d^{2} f^{7} g^{7} n^{7} x \log {\left (F \right )}^{7} + 8 b^{3} d^{2} f^{6} g^{6} n^{6} \log {\left (F \right )}^{6}\right ) \left (F^{g \left (e + f x\right )}\right )^{3 n} + \left (162 a b^{2} c^{2} f^{8} g^{8} n^{8} \log {\left (F \right )}^{8} + 324 a b^{2} c d f^{8} g^{8} n^{8} x \log {\left (F \right )}^{8} - 162 a b^{2} c d f^{7} g^{7} n^{7} \log {\left (F \right )}^{7} + 162 a b^{2} d^{2} f^{8} g^{8} n^{8} x^{2} \log {\left (F \right )}^{8} - 162 a b^{2} d^{2} f^{7} g^{7} n^{7} x \log {\left (F \right )}^{7} + 81 a b^{2} d^{2} f^{6} g^{6} n^{6} \log {\left (F \right )}^{6}\right ) \left (F^{g \left (e + f x\right )}\right )^{2 n} + \left (324 a^{2} b c^{2} f^{8} g^{8} n^{8} \log {\left (F \right )}^{8} + 648 a^{2} b c d f^{8} g^{8} n^{8} x \log {\left (F \right )}^{8} - 648 a^{2} b c d f^{7} g^{7} n^{7} \log {\left (F \right )}^{7} + 324 a^{2} b d^{2} f^{8} g^{8} n^{8} x^{2} \log {\left (F \right )}^{8} - 648 a^{2} b d^{2} f^{7} g^{7} n^{7} x \log {\left (F \right )}^{7} + 648 a^{2} b d^{2} f^{6} g^{6} n^{6} \log {\left (F \right )}^{6}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{108 f^{9} g^{9} n^{9} \log {\left (F \right )}^{9}} & \text {for}\: f^{9} g^{9} n^{9} \log {\left (F \right )}^{9} \neq 0 \\x^{3} \left (a^{2} b d^{2} + a b^{2} d^{2} + \frac {b^{3} d^{2}}{3}\right ) + x^{2} \cdot \left (3 a^{2} b c d + 3 a b^{2} c d + b^{3} c d\right ) + x \left (3 a^{2} b c^{2} + 3 a b^{2} c^{2} + b^{3} c^{2}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] Result contains complex when optimal does not.
time = 2.72, size = 8850, normalized size = 24.18 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.99, size = 399, normalized size = 1.09 \begin {gather*} {\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{3\,n}\,\left (\frac {b^3\,\left (9\,c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-6\,c\,d\,f\,g\,n\,\ln \left (F\right )+2\,d^2\right )}{27\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {b^3\,d^2\,x^2}{3\,f\,g\,n\,\ln \left (F\right )}-\frac {2\,b^3\,d\,x\,\left (d-3\,c\,f\,g\,n\,\ln \left (F\right )\right )}{9\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {3\,a^2\,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 {3\,a^2\,b\,d^2\,x^2}{f\,g\,n\,\ln \left (F\right )}-\frac {6\,a^2\,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 )+{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}\,\left (\frac {3\,a\,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 {3\,a\,b^2\,d^2\,x^2}{2\,f\,g\,n\,\ln \left (F\right )}-\frac {3\,a\,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 )+a^3\,c^2\,x+\frac {a^3\,d^2\,x^3}{3}+a^3\,c\,d\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________