Integrand size = 25, antiderivative size = 496 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^3 \, dx=\frac {a^3 (c+d x)^4}{4 d}-\frac {18 a^2 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}-\frac {9 a b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)}-\frac {2 b^3 d^3 \left (F^{e g+f g x}\right )^{3 n}}{27 f^4 g^4 n^4 \log ^4(F)}+\frac {18 a^2 b d^2 \left (F^{e g+f g x}\right )^n (c+d x)}{f^3 g^3 n^3 \log ^3(F)}+\frac {9 a b^2 d^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{4 f^3 g^3 n^3 \log ^3(F)}+\frac {2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{9 f^3 g^3 n^3 \log ^3(F)}-\frac {9 a^2 b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}-\frac {9 a b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{4 f^2 g^2 n^2 \log ^2(F)}-\frac {b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f^2 g^2 n^2 \log ^2(F)}+\frac {3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^3}{3 f g n \log (F)} \]
1/4*a^3*(d*x+c)^4/d-18*a^2*b*d^3*(F^(f*g*x+e*g))^n/f^4/g^4/n^4/ln(F)^4-9/8 *a*b^2*d^3*(F^(f*g*x+e*g))^(2*n)/f^4/g^4/n^4/ln(F)^4-2/27*b^3*d^3*(F^(f*g* x+e*g))^(3*n)/f^4/g^4/n^4/ln(F)^4+18*a^2*b*d^2*(F^(f*g*x+e*g))^n*(d*x+c)/f ^3/g^3/n^3/ln(F)^3+9/4*a*b^2*d^2*(F^(f*g*x+e*g))^(2*n)*(d*x+c)/f^3/g^3/n^3 /ln(F)^3+2/9*b^3*d^2*(F^(f*g*x+e*g))^(3*n)*(d*x+c)/f^3/g^3/n^3/ln(F)^3-9*a ^2*b*d*(F^(f*g*x+e*g))^n*(d*x+c)^2/f^2/g^2/n^2/ln(F)^2-9/4*a*b^2*d*(F^(f*g *x+e*g))^(2*n)*(d*x+c)^2/f^2/g^2/n^2/ln(F)^2-1/3*b^3*d*(F^(f*g*x+e*g))^(3* n)*(d*x+c)^2/f^2/g^2/n^2/ln(F)^2+3*a^2*b*(F^(f*g*x+e*g))^n*(d*x+c)^3/f/g/n /ln(F)+3/2*a*b^2*(F^(f*g*x+e*g))^(2*n)*(d*x+c)^3/f/g/n/ln(F)+1/3*b^3*(F^(f *g*x+e*g))^(3*n)*(d*x+c)^3/f/g/n/ln(F)
Time = 0.74 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.69 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^3 \, dx=a^3 c^3 x+\frac {3}{2} a^3 c^2 d x^2+a^3 c d^2 x^3+\frac {1}{4} a^3 d^3 x^4+\frac {3 a^2 b \left (F^{g (e+f x)}\right )^n \left (-6 d^3+6 d^2 f g n (c+d x) \log (F)-3 d f^2 g^2 n^2 (c+d x)^2 \log ^2(F)+f^3 g^3 n^3 (c+d x)^3 \log ^3(F)\right )}{f^4 g^4 n^4 \log ^4(F)}+\frac {3 a b^2 \left (F^{g (e+f x)}\right )^{2 n} \left (-3 d^3+6 d^2 f g n (c+d x) \log (F)-6 d f^2 g^2 n^2 (c+d x)^2 \log ^2(F)+4 f^3 g^3 n^3 (c+d x)^3 \log ^3(F)\right )}{8 f^4 g^4 n^4 \log ^4(F)}+\frac {b^3 \left (F^{g (e+f x)}\right )^{3 n} \left (-2 d^3+6 d^2 f g n (c+d x) \log (F)-9 d f^2 g^2 n^2 (c+d x)^2 \log ^2(F)+9 f^3 g^3 n^3 (c+d x)^3 \log ^3(F)\right )}{27 f^4 g^4 n^4 \log ^4(F)} \]
a^3*c^3*x + (3*a^3*c^2*d*x^2)/2 + a^3*c*d^2*x^3 + (a^3*d^3*x^4)/4 + (3*a^2 *b*(F^(g*(e + f*x)))^n*(-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[F]^2 + f^3*g^3*n^3*(c + d*x)^3*Log[F]^3))/(f^4*g^4*n ^4*Log[F]^4) + (3*a*b^2*(F^(g*(e + f*x)))^(2*n)*(-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[F]^2 + 4*f^3*g^3*n^3*(c + d *x)^3*Log[F]^3))/(8*f^4*g^4*n^4*Log[F]^4) + (b^3*(F^(g*(e + f*x)))^(3*n)*( -2*d^3 + 6*d^2*f*g*n*(c + d*x)*Log[F] - 9*d*f^2*g^2*n^2*(c + d*x)^2*Log[F] ^2 + 9*f^3*g^3*n^3*(c + d*x)^3*Log[F]^3))/(27*f^4*g^4*n^4*Log[F]^4)
Time = 0.97 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2614, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (c+d x)^3 \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 \, dx\) |
| \(\Big \downarrow \) 2614 |
| \(\displaystyle \int \left (a^3 (c+d x)^3+3 a^2 b (c+d x)^3 \left (F^{e g+f g x}\right )^n+3 a b^2 (c+d x)^3 \left (F^{e g+f g x}\right )^{2 n}+b^3 (c+d x)^3 \left (F^{e g+f g x}\right )^{3 n}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 (c+d x)^4}{4 d}+\frac {18 a^2 b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {9 a^2 b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {3 a^2 b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac {18 a^2 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}+\frac {9 a b^2 d^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {9 a b^2 d (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)}+\frac {3 a b^2 (c+d x)^3 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac {9 a b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)}+\frac {2 b^3 d^2 (c+d x) \left (F^{e g+f g x}\right )^{3 n}}{9 f^3 g^3 n^3 \log ^3(F)}-\frac {b^3 d (c+d x)^2 \left (F^{e g+f g x}\right )^{3 n}}{3 f^2 g^2 n^2 \log ^2(F)}+\frac {b^3 (c+d x)^3 \left (F^{e g+f g x}\right )^{3 n}}{3 f g n \log (F)}-\frac {2 b^3 d^3 \left (F^{e g+f g x}\right )^{3 n}}{27 f^4 g^4 n^4 \log ^4(F)}\) |
(a^3*(c + d*x)^4)/(4*d) - (18*a^2*b*d^3*(F^(e*g + f*g*x))^n)/(f^4*g^4*n^4* Log[F]^4) - (9*a*b^2*d^3*(F^(e*g + f*g*x))^(2*n))/(8*f^4*g^4*n^4*Log[F]^4) - (2*b^3*d^3*(F^(e*g + f*g*x))^(3*n))/(27*f^4*g^4*n^4*Log[F]^4) + (18*a^2 *b*d^2*(F^(e*g + f*g*x))^n*(c + d*x))/(f^3*g^3*n^3*Log[F]^3) + (9*a*b^2*d^ 2*(F^(e*g + f*g*x))^(2*n)*(c + d*x))/(4*f^3*g^3*n^3*Log[F]^3) + (2*b^3*d^2 *(F^(e*g + f*g*x))^(3*n)*(c + d*x))/(9*f^3*g^3*n^3*Log[F]^3) - (9*a^2*b*d* (F^(e*g + f*g*x))^n*(c + d*x)^2)/(f^2*g^2*n^2*Log[F]^2) - (9*a*b^2*d*(F^(e *g + f*g*x))^(2*n)*(c + d*x)^2)/(4*f^2*g^2*n^2*Log[F]^2) - (b^3*d*(F^(e*g + f*g*x))^(3*n)*(c + d*x)^2)/(3*f^2*g^2*n^2*Log[F]^2) + (3*a^2*b*(F^(e*g + f*g*x))^n*(c + d*x)^3)/(f*g*n*Log[F]) + (3*a*b^2*(F^(e*g + f*g*x))^(2*n)* (c + d*x)^3)/(2*f*g*n*Log[F]) + (b^3*(F^(e*g + f*g*x))^(3*n)*(c + d*x)^3)/ (3*f*g*n*Log[F])
3.1.39.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*(F ^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1096\) vs. \(2(478)=956\).
Time = 2.32 (sec) , antiderivative size = 1097, normalized size of antiderivative = 2.21
1/216*(-16*((F^(g*(f*x+e)))^n)^3*b^3*d^3+216*ln(F)^3*x*((F^(g*(f*x+e)))^n) ^3*b^3*c^2*d*f^3*g^3*n^3-486*ln(F)^2*x^2*((F^(g*(f*x+e)))^n)^2*a*b^2*d^3*f ^2*g^2*n^2-144*ln(F)^2*x*((F^(g*(f*x+e)))^n)^3*b^3*c*d^2*f^2*g^2*n^2-1944* ln(F)^2*x^2*(F^(g*(f*x+e)))^n*a^2*b*d^3*f^2*g^2*n^2-486*ln(F)^2*((F^(g*(f* x+e)))^n)^2*a*b^2*c^2*d*f^2*g^2*n^2-1944*ln(F)^2*(F^(g*(f*x+e)))^n*a^2*b*c ^2*d*f^2*g^2*n^2+486*ln(F)*x*((F^(g*(f*x+e)))^n)^2*a*b^2*d^3*f*g*n+3888*ln (F)*x*(F^(g*(f*x+e)))^n*a^2*b*d^3*f*g*n+486*ln(F)*((F^(g*(f*x+e)))^n)^2*a* b^2*c*d^2*f*g*n+3888*ln(F)*(F^(g*(f*x+e)))^n*a^2*b*c*d^2*f*g*n+324*a*b^2*d ^3*((F^(g*(f*x+e)))^n)^2*x^3*n^3*g^3*f^3*ln(F)^3+216*ln(F)^3*x^2*((F^(g*(f *x+e)))^n)^3*b^3*c*d^2*f^3*g^3*n^3+648*a^2*b*d^3*(F^(g*(f*x+e)))^n*x^3*n^3 *g^3*f^3*ln(F)^3-972*ln(F)^2*x*((F^(g*(f*x+e)))^n)^2*a*b^2*c*d^2*f^2*g^2*n ^2-3888*ln(F)^2*x*(F^(g*(f*x+e)))^n*a^2*b*c*d^2*f^2*g^2*n^2+972*ln(F)^3*x^ 2*((F^(g*(f*x+e)))^n)^2*a*b^2*c*d^2*f^3*g^3*n^3+54*a^3*d^3*x^4*n^4*g^4*f^4 *ln(F)^4+216*a^3*c^3*x*n^4*g^4*f^4*ln(F)^4+72*ln(F)^3*((F^(g*(f*x+e)))^n)^ 3*b^3*c^3*f^3*g^3*n^3+1944*ln(F)^3*x^2*(F^(g*(f*x+e)))^n*a^2*b*c*d^2*f^3*g ^3*n^3+972*ln(F)^3*x*((F^(g*(f*x+e)))^n)^2*a*b^2*c^2*d*f^3*g^3*n^3+1944*ln (F)^3*x*(F^(g*(f*x+e)))^n*a^2*b*c^2*d*f^3*g^3*n^3+216*a^3*d^2*c*x^3*n^4*g^ 4*f^4*ln(F)^4-3888*(F^(g*(f*x+e)))^n*a^2*b*d^3-243*((F^(g*(f*x+e)))^n)^2*a *b^2*d^3+48*ln(F)*x*((F^(g*(f*x+e)))^n)^3*b^3*d^3*f*g*n+48*ln(F)*((F^(g*(f *x+e)))^n)^3*b^3*c*d^2*f*g*n+324*a^3*d*c^2*x^2*n^4*g^4*f^4*ln(F)^4+72*d...
Time = 0.26 (sec) , antiderivative size = 708, normalized size of antiderivative = 1.43 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^3 \, dx=\frac {54 \, {\left (a^{3} d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, a^{3} c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, a^{3} c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, a^{3} c^{3} f^{4} g^{4} n^{4} x\right )} \log \left (F\right )^{4} - 8 \, {\left (2 \, b^{3} d^{3} - 9 \, {\left (b^{3} d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, b^{3} c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, b^{3} c^{2} d f^{3} g^{3} n^{3} x + b^{3} c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 9 \, {\left (b^{3} d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{3} c d^{2} f^{2} g^{2} n^{2} x + b^{3} c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (b^{3} d^{3} f g n x + b^{3} c d^{2} f g n\right )} \log \left (F\right )\right )} F^{3 \, f g n x + 3 \, e g n} - 81 \, {\left (3 \, a b^{2} d^{3} - 4 \, {\left (a b^{2} d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, a b^{2} c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, a b^{2} c^{2} d f^{3} g^{3} n^{3} x + a b^{2} c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 6 \, {\left (a b^{2} d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, a b^{2} c d^{2} f^{2} g^{2} n^{2} x + a b^{2} c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (a b^{2} d^{3} f g n x + a b^{2} c d^{2} f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} - 648 \, {\left (6 \, a^{2} b d^{3} - {\left (a^{2} b d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, a^{2} b c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, a^{2} b c^{2} d f^{3} g^{3} n^{3} x + a^{2} b c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 3 \, {\left (a^{2} b d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, a^{2} b c d^{2} f^{2} g^{2} n^{2} x + a^{2} b c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (a^{2} b d^{3} f g n x + a^{2} b c d^{2} f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{216 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \]
1/216*(54*(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)*log(F)^4 - 8*(2*b^3*d^3 - 9*(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)*log(F)^3 + 9*(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)*log(F)^2 - 6*(b^3*d^3*f *g*n*x + b^3*c*d^2*f*g*n)*log(F))*F^(3*f*g*n*x + 3*e*g*n) - 81*(3*a*b^2*d^ 3 - 4*(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)*log(F)^3 + 6*(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)*log(F )^2 - 6*(a*b^2*d^3*f*g*n*x + a*b^2*c*d^2*f*g*n)*log(F))*F^(2*f*g*n*x + 2*e *g*n) - 648*(6*a^2*b*d^3 - (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)*log(F)^ 3 + 3*(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)*log(F)^2 - 6*(a^2*b*d^3*f*g*n*x + a^2*b*c*d^2*f*g*n)*log(F ))*F^(f*g*n*x + e*g*n))/(f^4*g^4*n^4*log(F)^4)
Leaf count of result is larger than twice the leaf count of optimal. 1324 vs. \(2 (498) = 996\).
Time = 11.92 (sec) , antiderivative size = 1324, normalized size of antiderivative = 2.67 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^3 \, dx=\text {Too large to display} \]
Piecewise(((a + b)**3*(c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/ 4), Eq(F, 1) & Eq(f, 0) & Eq(g, 0) & Eq(n, 0)), ((a + b*(F**(e*g))**n)**3* (c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4), Eq(f, 0)), ((a + b )**3*(c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4), Eq(F, 1) | Eq (g, 0) | Eq(n, 0)), (a**3*c**3*x + 3*a**3*c**2*d*x**2/2 + a**3*c*d**2*x**3 + a**3*d**3*x**4/4 + 3*a**2*b*c**3*(F**(e*g + f*g*x))**n/(f*g*n*log(F)) + 9*a**2*b*c**2*d*x*(F**(e*g + f*g*x))**n/(f*g*n*log(F)) - 9*a**2*b*c**2*d* (F**(e*g + f*g*x))**n/(f**2*g**2*n**2*log(F)**2) + 9*a**2*b*c*d**2*x**2*(F **(e*g + f*g*x))**n/(f*g*n*log(F)) - 18*a**2*b*c*d**2*x*(F**(e*g + f*g*x)) **n/(f**2*g**2*n**2*log(F)**2) + 18*a**2*b*c*d**2*(F**(e*g + f*g*x))**n/(f **3*g**3*n**3*log(F)**3) + 3*a**2*b*d**3*x**3*(F**(e*g + f*g*x))**n/(f*g*n *log(F)) - 9*a**2*b*d**3*x**2*(F**(e*g + f*g*x))**n/(f**2*g**2*n**2*log(F) **2) + 18*a**2*b*d**3*x*(F**(e*g + f*g*x))**n/(f**3*g**3*n**3*log(F)**3) - 18*a**2*b*d**3*(F**(e*g + f*g*x))**n/(f**4*g**4*n**4*log(F)**4) + 3*a*b** 2*c**3*(F**(e*g + f*g*x))**(2*n)/(2*f*g*n*log(F)) + 9*a*b**2*c**2*d*x*(F** (e*g + f*g*x))**(2*n)/(2*f*g*n*log(F)) - 9*a*b**2*c**2*d*(F**(e*g + f*g*x) )**(2*n)/(4*f**2*g**2*n**2*log(F)**2) + 9*a*b**2*c*d**2*x**2*(F**(e*g + f* g*x))**(2*n)/(2*f*g*n*log(F)) - 9*a*b**2*c*d**2*x*(F**(e*g + f*g*x))**(2*n )/(2*f**2*g**2*n**2*log(F)**2) + 9*a*b**2*c*d**2*(F**(e*g + f*g*x))**(2*n) /(4*f**3*g**3*n**3*log(F)**3) + 3*a*b**2*d**3*x**3*(F**(e*g + f*g*x))**...
Time = 0.23 (sec) , antiderivative size = 854, normalized size of antiderivative = 1.72 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^3 \, dx=\frac {1}{4} \, a^{3} d^{3} x^{4} + a^{3} c d^{2} x^{3} + \frac {3}{2} \, a^{3} c^{2} d x^{2} + a^{3} c^{3} x + \frac {3 \, F^{f g n x + e g n} a^{2} b c^{3}}{f g n \log \left (F\right )} + \frac {3 \, F^{2 \, f g n x + 2 \, e g n} a b^{2} c^{3}}{2 \, f g n \log \left (F\right )} + \frac {F^{3 \, f g n x + 3 \, e g n} b^{3} c^{3}}{3 \, f g n \log \left (F\right )} + \frac {9 \, {\left (F^{e g n} f g n x \log \left (F\right ) - F^{e g n}\right )} F^{f g n x} a^{2} b c^{2} d}{f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {9 \, {\left (2 \, F^{2 \, e g n} f g n x \log \left (F\right ) - F^{2 \, e g n}\right )} F^{2 \, f g n x} a b^{2} c^{2} d}{4 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {{\left (3 \, F^{3 \, e g n} f g n x \log \left (F\right ) - F^{3 \, e g n}\right )} F^{3 \, f g n x} b^{3} c^{2} d}{3 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {9 \, {\left (F^{e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{e g n} f g n x \log \left (F\right ) + 2 \, F^{e g n}\right )} F^{f g n x} a^{2} b c d^{2}}{f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {9 \, {\left (2 \, F^{2 \, e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{2 \, e g n} f g n x \log \left (F\right ) + F^{2 \, e g n}\right )} F^{2 \, f g n x} a b^{2} c d^{2}}{4 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {{\left (9 \, F^{3 \, e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 6 \, F^{3 \, e g n} f g n x \log \left (F\right ) + 2 \, F^{3 \, e g n}\right )} F^{3 \, f g n x} b^{3} c d^{2}}{9 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {3 \, {\left (F^{e g n} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{e g n} f g n x \log \left (F\right ) - 6 \, F^{e g n}\right )} F^{f g n x} a^{2} b d^{3}}{f^{4} g^{4} n^{4} \log \left (F\right )^{4}} + \frac {3 \, {\left (4 \, F^{2 \, e g n} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} - 6 \, F^{2 \, e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{2 \, e g n} f g n x \log \left (F\right ) - 3 \, F^{2 \, e g n}\right )} F^{2 \, f g n x} a b^{2} d^{3}}{8 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} + \frac {{\left (9 \, F^{3 \, e g n} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} - 9 \, F^{3 \, e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{3 \, e g n} f g n x \log \left (F\right ) - 2 \, F^{3 \, e g n}\right )} F^{3 \, f g n x} b^{3} d^{3}}{27 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \]
1/4*a^3*d^3*x^4 + a^3*c*d^2*x^3 + 3/2*a^3*c^2*d*x^2 + a^3*c^3*x + 3*F^(f*g *n*x + e*g*n)*a^2*b*c^3/(f*g*n*log(F)) + 3/2*F^(2*f*g*n*x + 2*e*g*n)*a*b^2 *c^3/(f*g*n*log(F)) + 1/3*F^(3*f*g*n*x + 3*e*g*n)*b^3*c^3/(f*g*n*log(F)) + 9*(F^(e*g*n)*f*g*n*x*log(F) - F^(e*g*n))*F^(f*g*n*x)*a^2*b*c^2*d/(f^2*g^2 *n^2*log(F)^2) + 9/4*(2*F^(2*e*g*n)*f*g*n*x*log(F) - F^(2*e*g*n))*F^(2*f*g *n*x)*a*b^2*c^2*d/(f^2*g^2*n^2*log(F)^2) + 1/3*(3*F^(3*e*g*n)*f*g*n*x*log( F) - F^(3*e*g*n))*F^(3*f*g*n*x)*b^3*c^2*d/(f^2*g^2*n^2*log(F)^2) + 9*(F^(e *g*n)*f^2*g^2*n^2*x^2*log(F)^2 - 2*F^(e*g*n)*f*g*n*x*log(F) + 2*F^(e*g*n)) *F^(f*g*n*x)*a^2*b*c*d^2/(f^3*g^3*n^3*log(F)^3) + 9/4*(2*F^(2*e*g*n)*f^2*g ^2*n^2*x^2*log(F)^2 - 2*F^(2*e*g*n)*f*g*n*x*log(F) + F^(2*e*g*n))*F^(2*f*g *n*x)*a*b^2*c*d^2/(f^3*g^3*n^3*log(F)^3) + 1/9*(9*F^(3*e*g*n)*f^2*g^2*n^2* x^2*log(F)^2 - 6*F^(3*e*g*n)*f*g*n*x*log(F) + 2*F^(3*e*g*n))*F^(3*f*g*n*x) *b^3*c*d^2/(f^3*g^3*n^3*log(F)^3) + 3*(F^(e*g*n)*f^3*g^3*n^3*x^3*log(F)^3 - 3*F^(e*g*n)*f^2*g^2*n^2*x^2*log(F)^2 + 6*F^(e*g*n)*f*g*n*x*log(F) - 6*F^ (e*g*n))*F^(f*g*n*x)*a^2*b*d^3/(f^4*g^4*n^4*log(F)^4) + 3/8*(4*F^(2*e*g*n) *f^3*g^3*n^3*x^3*log(F)^3 - 6*F^(2*e*g*n)*f^2*g^2*n^2*x^2*log(F)^2 + 6*F^( 2*e*g*n)*f*g*n*x*log(F) - 3*F^(2*e*g*n))*F^(2*f*g*n*x)*a*b^2*d^3/(f^4*g^4* n^4*log(F)^4) + 1/27*(9*F^(3*e*g*n)*f^3*g^3*n^3*x^3*log(F)^3 - 9*F^(3*e*g* n)*f^2*g^2*n^2*x^2*log(F)^2 + 6*F^(3*e*g*n)*f*g*n*x*log(F) - 2*F^(3*e*g*n) )*F^(3*f*g*n*x)*b^3*d^3/(f^4*g^4*n^4*log(F)^4)
Result contains complex when optimal does not.
Time = 0.60 (sec) , antiderivative size = 18707, normalized size of antiderivative = 37.72 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^3 \, dx=\text {Too large to display} \]
1/4*a^3*d^3*x^4 + a^3*c*d^2*x^3 + 3/2*a^3*c^2*d*x^2 + a^3*c^3*x - 1/27*((( 27*pi^2*b^3*d^3*f^3*g^3*n^3*x^3*log(abs(F))*sgn(F) - 27*pi^2*b^3*d^3*f^3*g ^3*n^3*x^3*log(abs(F)) + 18*b^3*d^3*f^3*g^3*n^3*x^3*log(abs(F))^3 + 81*pi^ 2*b^3*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))*sgn(F) - 81*pi^2*b^3*c*d^2*f^3*g^3 *n^3*x^2*log(abs(F)) + 54*b^3*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^3 + 81*pi^ 2*b^3*c^2*d*f^3*g^3*n^3*x*log(abs(F))*sgn(F) - 81*pi^2*b^3*c^2*d*f^3*g^3*n ^3*x*log(abs(F)) + 54*b^3*c^2*d*f^3*g^3*n^3*x*log(abs(F))^3 + 27*pi^2*b^3* c^3*f^3*g^3*n^3*log(abs(F))*sgn(F) - 27*pi^2*b^3*c^3*f^3*g^3*n^3*log(abs(F )) + 18*b^3*c^3*f^3*g^3*n^3*log(abs(F))^3 - 9*pi^2*b^3*d^3*f^2*g^2*n^2*x^2 *sgn(F) + 9*pi^2*b^3*d^3*f^2*g^2*n^2*x^2 - 18*b^3*d^3*f^2*g^2*n^2*x^2*log( abs(F))^2 - 18*pi^2*b^3*c*d^2*f^2*g^2*n^2*x*sgn(F) + 18*pi^2*b^3*c*d^2*f^2 *g^2*n^2*x - 36*b^3*c*d^2*f^2*g^2*n^2*x*log(abs(F))^2 - 9*pi^2*b^3*c^2*d*f ^2*g^2*n^2*sgn(F) + 9*pi^2*b^3*c^2*d*f^2*g^2*n^2 - 18*b^3*c^2*d*f^2*g^2*n^ 2*log(abs(F))^2 + 12*b^3*d^3*f*g*n*x*log(abs(F)) + 12*b^3*c*d^2*f*g*n*log( abs(F)) - 4*b^3*d^3)*(pi^4*f^4*g^4*n^4*sgn(F) - 6*pi^2*f^4*g^4*n^4*log(abs (F))^2*sgn(F) - pi^4*f^4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F))^2 - 2*f^ 4*g^4*n^4*log(abs(F))^4)/((pi^4*f^4*g^4*n^4*sgn(F) - 6*pi^2*f^4*g^4*n^4*lo g(abs(F))^2*sgn(F) - pi^4*f^4*g^4*n^4 + 6*pi^2*f^4*g^4*n^4*log(abs(F))^2 - 2*f^4*g^4*n^4*log(abs(F))^4)^2 + 16*(pi^3*f^4*g^4*n^4*log(abs(F))*sgn(F) - pi*f^4*g^4*n^4*log(abs(F))^3*sgn(F) - pi^3*f^4*g^4*n^4*log(abs(F)) + ...
Time = 0.86 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.31 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^3 \, dx=a^3\,c^3\,x-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {3\,a^2\,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 {3\,a^2\,b\,d^3\,x^3}{f\,g\,n\,\ln \left (F\right )}-\frac {9\,a^2\,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 {9\,a^2\,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 {3\,a\,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 {3\,a\,b^2\,d^3\,x^3}{2\,f\,g\,n\,\ln \left (F\right )}-\frac {9\,a\,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 {9\,a\,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 )-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{3\,n}\,\left (\frac {b^3\,\left (-9\,c^3\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3+9\,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 )+2\,d^3\right )}{27\,f^4\,g^4\,n^4\,{\ln \left (F\right )}^4}-\frac {b^3\,d^3\,x^3}{3\,f\,g\,n\,\ln \left (F\right )}-\frac {b^3\,d\,x\,\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 )}{9\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {b^3\,d^2\,x^2\,\left (d-3\,c\,f\,g\,n\,\ln \left (F\right )\right )}{3\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+\frac {a^3\,d^3\,x^4}{4}+\frac {3\,a^3\,c^2\,d\,x^2}{2}+a^3\,c\,d^2\,x^3 \]
a^3*c^3*x - (F^(f*g*x)*F^(e*g))^n*((3*a^2*b*(6*d^3 - c^3*f^3*g^3*n^3*log(F )^3 - 6*c*d^2*f*g*n*log(F) + 3*c^2*d*f^2*g^2*n^2*log(F)^2))/(f^4*g^4*n^4*l og(F)^4) - (3*a^2*b*d^3*x^3)/(f*g*n*log(F)) - (9*a^2*b*d*x*(2*d^2 + c^2*f^ 2*g^2*n^2*log(F)^2 - 2*c*d*f*g*n*log(F)))/(f^3*g^3*n^3*log(F)^3) + (9*a^2* b*d^2*x^2*(d - c*f*g*n*log(F)))/(f^2*g^2*n^2*log(F)^2)) - (F^(f*g*x)*F^(e* g))^(2*n)*((3*a*b^2*(3*d^3 - 4*c^3*f^3*g^3*n^3*log(F)^3 - 6*c*d^2*f*g*n*lo g(F) + 6*c^2*d*f^2*g^2*n^2*log(F)^2))/(8*f^4*g^4*n^4*log(F)^4) - (3*a*b^2* d^3*x^3)/(2*f*g*n*log(F)) - (9*a*b^2*d*x*(d^2 + 2*c^2*f^2*g^2*n^2*log(F)^2 - 2*c*d*f*g*n*log(F)))/(4*f^3*g^3*n^3*log(F)^3) + (9*a*b^2*d^2*x^2*(d - 2 *c*f*g*n*log(F)))/(4*f^2*g^2*n^2*log(F)^2)) - (F^(f*g*x)*F^(e*g))^(3*n)*(( b^3*(2*d^3 - 9*c^3*f^3*g^3*n^3*log(F)^3 - 6*c*d^2*f*g*n*log(F) + 9*c^2*d*f ^2*g^2*n^2*log(F)^2))/(27*f^4*g^4*n^4*log(F)^4) - (b^3*d^3*x^3)/(3*f*g*n*l og(F)) - (b^3*d*x*(2*d^2 + 9*c^2*f^2*g^2*n^2*log(F)^2 - 6*c*d*f*g*n*log(F) ))/(9*f^3*g^3*n^3*log(F)^3) + (b^3*d^2*x^2*(d - 3*c*f*g*n*log(F)))/(3*f^2* g^2*n^2*log(F)^2)) + (a^3*d^3*x^4)/4 + (3*a^3*c^2*d*x^2)/2 + a^3*c*d^2*x^3