Integrand size = 25, antiderivative size = 366 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2 \, dx=\frac {a^3 (c+d x)^3}{3 d}+\frac {6 a^2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}+\frac {3 a b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}+\frac {2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n}}{27 f^3 g^3 n^3 \log ^3(F)}-\frac {6 a^2 b d \left (F^{e g+f g x}\right )^n (c+d x)}{f^2 g^2 n^2 \log ^2(F)}-\frac {3 a b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{2 f^2 g^2 n^2 \log ^2(F)}-\frac {2 b^3 d \left (F^{e g+f g x}\right )^{3 n} (c+d x)}{9 f^2 g^2 n^2 \log ^2(F)}+\frac {3 a^2 b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {3 a b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)}+\frac {b^3 \left (F^{e g+f g x}\right )^{3 n} (c+d x)^2}{3 f g n \log (F)} \]
1/3*a^3*(d*x+c)^3/d+6*a^2*b*d^2*(F^(f*g*x+e*g))^n/f^3/g^3/n^3/ln(F)^3+3/4* a*b^2*d^2*(F^(f*g*x+e*g))^(2*n)/f^3/g^3/n^3/ln(F)^3+2/27*b^3*d^2*(F^(f*g*x +e*g))^(3*n)/f^3/g^3/n^3/ln(F)^3-6*a^2*b*d*(F^(f*g*x+e*g))^n*(d*x+c)/f^2/g ^2/n^2/ln(F)^2-3/2*a*b^2*d*(F^(f*g*x+e*g))^(2*n)*(d*x+c)/f^2/g^2/n^2/ln(F) ^2-2/9*b^3*d*(F^(f*g*x+e*g))^(3*n)*(d*x+c)/f^2/g^2/n^2/ln(F)^2+3*a^2*b*(F^ (f*g*x+e*g))^n*(d*x+c)^2/f/g/n/ln(F)+3/2*a*b^2*(F^(f*g*x+e*g))^(2*n)*(d*x+ c)^2/f/g/n/ln(F)+1/3*b^3*(F^(f*g*x+e*g))^(3*n)*(d*x+c)^2/f/g/n/ln(F)
Time = 0.56 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.68 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2 \, dx=a^3 c^2 x+a^3 c d x^2+\frac {1}{3} a^3 d^2 x^3+\frac {3 a^2 b \left (F^{g (e+f x)}\right )^n \left (2 d^2-2 d f g n (c+d x) \log (F)+f^2 g^2 n^2 (c+d x)^2 \log ^2(F)\right )}{f^3 g^3 n^3 \log ^3(F)}+\frac {3 a b^2 \left (F^{g (e+f x)}\right )^{2 n} \left (d^2-2 d f g n (c+d x) \log (F)+2 f^2 g^2 n^2 (c+d x)^2 \log ^2(F)\right )}{4 f^3 g^3 n^3 \log ^3(F)}+\frac {b^3 \left (F^{g (e+f x)}\right )^{3 n} \left (2 d^2-6 d f g n (c+d x) \log (F)+9 f^2 g^2 n^2 (c+d x)^2 \log ^2(F)\right )}{27 f^3 g^3 n^3 \log ^3(F)} \]
a^3*c^2*x + a^3*c*d*x^2 + (a^3*d^2*x^3)/3 + (3*a^2*b*(F^(g*(e + f*x)))^n*( 2*d^2 - 2*d*f*g*n*(c + d*x)*Log[F] + f^2*g^2*n^2*(c + d*x)^2*Log[F]^2))/(f ^3*g^3*n^3*Log[F]^3) + (3*a*b^2*(F^(g*(e + f*x)))^(2*n)*(d^2 - 2*d*f*g*n*( c + d*x)*Log[F] + 2*f^2*g^2*n^2*(c + d*x)^2*Log[F]^2))/(4*f^3*g^3*n^3*Log[ F]^3) + (b^3*(F^(g*(e + f*x)))^(3*n)*(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[F]^2))/(27*f^3*g^3*n^3*Log[F]^3)
Time = 0.70 (sec) , antiderivative size = 366, 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.
| \(\displaystyle \int (c+d x)^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 \, dx\) |
| \(\Big \downarrow \) 2614 |
| \(\displaystyle \int \left (a^3 (c+d x)^2+3 a^2 b (c+d x)^2 \left (F^{e g+f g x}\right )^n+3 a b^2 (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}+b^3 (c+d x)^2 \left (F^{e g+f g x}\right )^{3 n}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 (c+d x)^3}{3 d}-\frac {6 a^2 b d (c+d x) \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {3 a^2 b (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac {6 a^2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {3 a b^2 d (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{2 f^2 g^2 n^2 \log ^2(F)}+\frac {3 a b^2 (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}+\frac {3 a b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {2 b^3 d (c+d x) \left (F^{e g+f g x}\right )^{3 n}}{9 f^2 g^2 n^2 \log ^2(F)}+\frac {b^3 (c+d x)^2 \left (F^{e g+f g x}\right )^{3 n}}{3 f g n \log (F)}+\frac {2 b^3 d^2 \left (F^{e g+f g x}\right )^{3 n}}{27 f^3 g^3 n^3 \log ^3(F)}\) |
(a^3*(c + d*x)^3)/(3*d) + (6*a^2*b*d^2*(F^(e*g + f*g*x))^n)/(f^3*g^3*n^3*L og[F]^3) + (3*a*b^2*d^2*(F^(e*g + f*g*x))^(2*n))/(4*f^3*g^3*n^3*Log[F]^3) + (2*b^3*d^2*(F^(e*g + f*g*x))^(3*n))/(27*f^3*g^3*n^3*Log[F]^3) - (6*a^2*b *d*(F^(e*g + f*g*x))^n*(c + d*x))/(f^2*g^2*n^2*Log[F]^2) - (3*a*b^2*d*(F^( e*g + f*g*x))^(2*n)*(c + d*x))/(2*f^2*g^2*n^2*Log[F]^2) - (2*b^3*d*(F^(e*g + f*g*x))^(3*n)*(c + d*x))/(9*f^2*g^2*n^2*Log[F]^2) + (3*a^2*b*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f*g*n*Log[F]) + (3*a*b^2*(F^(e*g + f*g*x))^(2*n)*( c + d*x)^2)/(2*f*g*n*Log[F]) + (b^3*(F^(e*g + f*g*x))^(3*n)*(c + d*x)^2)/( 3*f*g*n*Log[F])
3.1.40.3.1 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]
Time = 1.26 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.69
| method | result | size |
| parallelrisch | \(\frac {36 a^{3} d^{2} x^{3} n^{3} g^{3} f^{3} \ln \left (F \right )^{3}+108 a^{3} d c \,x^{2} n^{3} g^{3} f^{3} \ln \left (F \right )^{3}+108 a^{3} c^{2} x \,n^{3} g^{3} f^{3} \ln \left (F \right )^{3}+36 d^{2} b^{3} \left (F^{g \left (f x +e \right )}\right )^{3 n} x^{2} n^{2} g^{2} f^{2} \ln \left (F \right )^{2}+162 a \,b^{2} d^{2} \left (F^{g \left (f x +e \right )}\right )^{2 n} x^{2} n^{2} g^{2} f^{2} \ln \left (F \right )^{2}+72 \ln \left (F \right )^{2} x \left (F^{g \left (f x +e \right )}\right )^{3 n} b^{3} c d \,f^{2} g^{2} n^{2}+324 a^{2} b \,d^{2} \left (F^{g \left (f x +e \right )}\right )^{n} x^{2} n^{2} g^{2} f^{2} \ln \left (F \right )^{2}+324 \ln \left (F \right )^{2} x \left (F^{g \left (f x +e \right )}\right )^{2 n} a \,b^{2} c d \,f^{2} g^{2} n^{2}+36 \ln \left (F \right )^{2} \left (F^{g \left (f x +e \right )}\right )^{3 n} b^{3} c^{2} f^{2} g^{2} n^{2}+648 \ln \left (F \right )^{2} x \left (F^{g \left (f x +e \right )}\right )^{n} a^{2} b c d \,f^{2} g^{2} n^{2}+162 \ln \left (F \right )^{2} \left (F^{g \left (f x +e \right )}\right )^{2 n} a \,b^{2} c^{2} f^{2} g^{2} n^{2}+324 \ln \left (F \right )^{2} \left (F^{g \left (f x +e \right )}\right )^{n} a^{2} b \,c^{2} f^{2} g^{2} n^{2}-24 \ln \left (F \right ) x \left (F^{g \left (f x +e \right )}\right )^{3 n} b^{3} d^{2} f g n -162 \ln \left (F \right ) x \left (F^{g \left (f x +e \right )}\right )^{2 n} a \,b^{2} d^{2} f g n -24 \ln \left (F \right ) \left (F^{g \left (f x +e \right )}\right )^{3 n} b^{3} c d f g n -648 \ln \left (F \right ) x \left (F^{g \left (f x +e \right )}\right )^{n} a^{2} b \,d^{2} f g n -162 \ln \left (F \right ) \left (F^{g \left (f x +e \right )}\right )^{2 n} a \,b^{2} c d f g n -648 \ln \left (F \right ) \left (F^{g \left (f x +e \right )}\right )^{n} a^{2} b c d f g n +8 \left (F^{g \left (f x +e \right )}\right )^{3 n} b^{3} d^{2}+81 \left (F^{g \left (f x +e \right )}\right )^{2 n} a \,b^{2} d^{2}+648 \left (F^{g \left (f x +e \right )}\right )^{n} a^{2} b \,d^{2}}{108 n^{3} g^{3} f^{3} \ln \left (F \right )^{3}}\) | \(620\) |
1/108*(36*a^3*d^2*x^3*n^3*g^3*f^3*ln(F)^3+108*a^3*d*c*x^2*n^3*g^3*f^3*ln(F )^3+108*a^3*c^2*x*n^3*g^3*f^3*ln(F)^3+36*d^2*b^3*((F^(g*(f*x+e)))^n)^3*x^2 *n^2*g^2*f^2*ln(F)^2+162*a*b^2*d^2*((F^(g*(f*x+e)))^n)^2*x^2*n^2*g^2*f^2*l n(F)^2+72*ln(F)^2*x*((F^(g*(f*x+e)))^n)^3*b^3*c*d*f^2*g^2*n^2+324*a^2*b*d^ 2*(F^(g*(f*x+e)))^n*x^2*n^2*g^2*f^2*ln(F)^2+324*ln(F)^2*x*((F^(g*(f*x+e))) ^n)^2*a*b^2*c*d*f^2*g^2*n^2+36*ln(F)^2*((F^(g*(f*x+e)))^n)^3*b^3*c^2*f^2*g ^2*n^2+648*ln(F)^2*x*(F^(g*(f*x+e)))^n*a^2*b*c*d*f^2*g^2*n^2+162*ln(F)^2*( (F^(g*(f*x+e)))^n)^2*a*b^2*c^2*f^2*g^2*n^2+324*ln(F)^2*(F^(g*(f*x+e)))^n*a ^2*b*c^2*f^2*g^2*n^2-24*ln(F)*x*((F^(g*(f*x+e)))^n)^3*b^3*d^2*f*g*n-162*ln (F)*x*((F^(g*(f*x+e)))^n)^2*a*b^2*d^2*f*g*n-24*ln(F)*((F^(g*(f*x+e)))^n)^3 *b^3*c*d*f*g*n-648*ln(F)*x*(F^(g*(f*x+e)))^n*a^2*b*d^2*f*g*n-162*ln(F)*((F ^(g*(f*x+e)))^n)^2*a*b^2*c*d*f*g*n-648*ln(F)*(F^(g*(f*x+e)))^n*a^2*b*c*d*f *g*n+8*((F^(g*(f*x+e)))^n)^3*b^3*d^2+81*((F^(g*(f*x+e)))^n)^2*a*b^2*d^2+64 8*(F^(g*(f*x+e)))^n*a^2*b*d^2)/n^3/g^3/f^3/ln(F)^3
Time = 0.26 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.13 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2 \, dx=\frac {36 \, {\left (a^{3} d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, a^{3} c d f^{3} g^{3} n^{3} x^{2} + 3 \, a^{3} c^{2} f^{3} g^{3} n^{3} x\right )} \log \left (F\right )^{3} + 4 \, {\left (2 \, b^{3} d^{2} + 9 \, {\left (b^{3} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{3} c d f^{2} g^{2} n^{2} x + b^{3} c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (b^{3} d^{2} f g n x + b^{3} c d f g n\right )} \log \left (F\right )\right )} F^{3 \, f g n x + 3 \, e g n} + 81 \, {\left (a b^{2} d^{2} + 2 \, {\left (a b^{2} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a b^{2} c d f^{2} g^{2} n^{2} x + a b^{2} c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (a b^{2} d^{2} f g n x + a b^{2} c d f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} + 324 \, {\left (2 \, a^{2} b d^{2} + {\left (a^{2} b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a^{2} b c d f^{2} g^{2} n^{2} x + a^{2} b c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (a^{2} b d^{2} f g n x + a^{2} b c d f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{108 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \]
1/108*(36*(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)*log(F)^3 + 4*(2*b^3*d^2 + 9*(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)*log(F)^2 - 6*(b^3*d^2*f*g*n*x + b^3*c*d*f*g*n)*log(F))*F^(3*f*g*n*x + 3*e*g*n) + 81*(a*b^2*d^2 + 2*(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) *log(F)^2 - 2*(a*b^2*d^2*f*g*n*x + a*b^2*c*d*f*g*n)*log(F))*F^(2*f*g*n*x + 2*e*g*n) + 324*(2*a^2*b*d^2 + (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)*log(F)^2 - 2*(a^2*b*d^2*f*g*n*x + a^2 *b*c*d*f*g*n)*log(F))*F^(f*g*n*x + e*g*n))/(f^3*g^3*n^3*log(F)^3)
Leaf count of result is larger than twice the leaf count of optimal. 775 vs. \(2 (362) = 724\).
Time = 5.04 (sec) , antiderivative size = 775, normalized size of antiderivative = 2.12 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2 \, dx=\begin {cases} \left (a + b\right )^{3} \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {for}\: F = 1 \wedge f = 0 \wedge g = 0 \wedge n = 0 \\\left (a + b \left (F^{e g}\right )^{n}\right )^{3} \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {for}\: f = 0 \\\left (a + b\right )^{3} \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {for}\: F = 1 \vee g = 0 \vee n = 0 \\a^{3} c^{2} x + a^{3} c d x^{2} + \frac {a^{3} d^{2} x^{3}}{3} + \frac {3 a^{2} b c^{2} \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} + \frac {6 a^{2} b c d x \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} - \frac {6 a^{2} b c d \left (F^{e g + f g x}\right )^{n}}{f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {3 a^{2} b d^{2} x^{2} \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} - \frac {6 a^{2} b d^{2} x \left (F^{e g + f g x}\right )^{n}}{f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {6 a^{2} b d^{2} \left (F^{e g + f g x}\right )^{n}}{f^{3} g^{3} n^{3} \log {\left (F \right )}^{3}} + \frac {3 a b^{2} c^{2} \left (F^{e g + f g x}\right )^{2 n}}{2 f g n \log {\left (F \right )}} + \frac {3 a b^{2} c d x \left (F^{e g + f g x}\right )^{2 n}}{f g n \log {\left (F \right )}} - \frac {3 a b^{2} c d \left (F^{e g + f g x}\right )^{2 n}}{2 f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {3 a b^{2} d^{2} x^{2} \left (F^{e g + f g x}\right )^{2 n}}{2 f g n \log {\left (F \right )}} - \frac {3 a b^{2} d^{2} x \left (F^{e g + f g x}\right )^{2 n}}{2 f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \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 {\left (F \right )}^{3}} + \frac {b^{3} c^{2} \left (F^{e g + f g x}\right )^{3 n}}{3 f g n \log {\left (F \right )}} + \frac {2 b^{3} c d x \left (F^{e g + f g x}\right )^{3 n}}{3 f g n \log {\left (F \right )}} - \frac {2 b^{3} c d \left (F^{e g + f g x}\right )^{3 n}}{9 f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {b^{3} d^{2} x^{2} \left (F^{e g + f g x}\right )^{3 n}}{3 f g n \log {\left (F \right )}} - \frac {2 b^{3} d^{2} x \left (F^{e g + f g x}\right )^{3 n}}{9 f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {2 b^{3} d^{2} \left (F^{e g + f g x}\right )^{3 n}}{27 f^{3} g^{3} n^{3} \log {\left (F \right )}^{3}} & \text {otherwise} \end {cases} \]
Piecewise(((a + b)**3*(c**2*x + c*d*x**2 + d**2*x**3/3), Eq(F, 1) & Eq(f, 0) & Eq(g, 0) & Eq(n, 0)), ((a + b*(F**(e*g))**n)**3*(c**2*x + c*d*x**2 + d**2*x**3/3), Eq(f, 0)), ((a + b)**3*(c**2*x + c*d*x**2 + d**2*x**3/3), Eq (F, 1) | Eq(g, 0) | Eq(n, 0)), (a**3*c**2*x + a**3*c*d*x**2 + a**3*d**2*x* *3/3 + 3*a**2*b*c**2*(F**(e*g + f*g*x))**n/(f*g*n*log(F)) + 6*a**2*b*c*d*x *(F**(e*g + f*g*x))**n/(f*g*n*log(F)) - 6*a**2*b*c*d*(F**(e*g + f*g*x))**n /(f**2*g**2*n**2*log(F)**2) + 3*a**2*b*d**2*x**2*(F**(e*g + f*g*x))**n/(f* g*n*log(F)) - 6*a**2*b*d**2*x*(F**(e*g + f*g*x))**n/(f**2*g**2*n**2*log(F) **2) + 6*a**2*b*d**2*(F**(e*g + f*g*x))**n/(f**3*g**3*n**3*log(F)**3) + 3* a*b**2*c**2*(F**(e*g + f*g*x))**(2*n)/(2*f*g*n*log(F)) + 3*a*b**2*c*d*x*(F **(e*g + f*g*x))**(2*n)/(f*g*n*log(F)) - 3*a*b**2*c*d*(F**(e*g + f*g*x))** (2*n)/(2*f**2*g**2*n**2*log(F)**2) + 3*a*b**2*d**2*x**2*(F**(e*g + f*g*x)) **(2*n)/(2*f*g*n*log(F)) - 3*a*b**2*d**2*x*(F**(e*g + f*g*x))**(2*n)/(2*f* *2*g**2*n**2*log(F)**2) + 3*a*b**2*d**2*(F**(e*g + f*g*x))**(2*n)/(4*f**3* g**3*n**3*log(F)**3) + b**3*c**2*(F**(e*g + f*g*x))**(3*n)/(3*f*g*n*log(F) ) + 2*b**3*c*d*x*(F**(e*g + f*g*x))**(3*n)/(3*f*g*n*log(F)) - 2*b**3*c*d*( F**(e*g + f*g*x))**(3*n)/(9*f**2*g**2*n**2*log(F)**2) + b**3*d**2*x**2*(F* *(e*g + f*g*x))**(3*n)/(3*f*g*n*log(F)) - 2*b**3*d**2*x*(F**(e*g + f*g*x)) **(3*n)/(9*f**2*g**2*n**2*log(F)**2) + 2*b**3*d**2*(F**(e*g + f*g*x))**(3* n)/(27*f**3*g**3*n**3*log(F)**3), True))
Time = 0.21 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.43 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2 \, dx=\frac {1}{3} \, a^{3} d^{2} x^{3} + a^{3} c d x^{2} + a^{3} c^{2} x + \frac {3 \, F^{f g n x + e g n} a^{2} b c^{2}}{f g n \log \left (F\right )} + \frac {3 \, F^{2 \, f g n x + 2 \, e g n} a b^{2} c^{2}}{2 \, f g n \log \left (F\right )} + \frac {F^{3 \, f g n x + 3 \, e g n} b^{3} c^{2}}{3 \, f g n \log \left (F\right )} + \frac {6 \, {\left (F^{e g n} f g n x \log \left (F\right ) - F^{e g n}\right )} F^{f g n x} a^{2} b c d}{f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {3 \, {\left (2 \, F^{2 \, e g n} f g n x \log \left (F\right ) - F^{2 \, e g n}\right )} F^{2 \, f g n x} a b^{2} c d}{2 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {2 \, {\left (3 \, F^{3 \, e g n} f g n x \log \left (F\right ) - F^{3 \, e g n}\right )} F^{3 \, f g n x} b^{3} c d}{9 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {3 \, {\left (F^{e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{e g n} f g n x \log \left (F\right ) + 2 \, F^{e g n}\right )} F^{f g n x} a^{2} b d^{2}}{f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {3 \, {\left (2 \, F^{2 \, e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{2 \, e g n} f g n x \log \left (F\right ) + F^{2 \, e g n}\right )} F^{2 \, f g n x} a b^{2} d^{2}}{4 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {{\left (9 \, F^{3 \, e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 6 \, F^{3 \, e g n} f g n x \log \left (F\right ) + 2 \, F^{3 \, e g n}\right )} F^{3 \, f g n x} b^{3} d^{2}}{27 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \]
1/3*a^3*d^2*x^3 + a^3*c*d*x^2 + a^3*c^2*x + 3*F^(f*g*n*x + e*g*n)*a^2*b*c^ 2/(f*g*n*log(F)) + 3/2*F^(2*f*g*n*x + 2*e*g*n)*a*b^2*c^2/(f*g*n*log(F)) + 1/3*F^(3*f*g*n*x + 3*e*g*n)*b^3*c^2/(f*g*n*log(F)) + 6*(F^(e*g*n)*f*g*n*x* log(F) - F^(e*g*n))*F^(f*g*n*x)*a^2*b*c*d/(f^2*g^2*n^2*log(F)^2) + 3/2*(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/(f^2*g^2 *n^2*log(F)^2) + 2/9*(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*d/(f^2*g^2*n^2*log(F)^2) + 3*(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*d^2/(f^3* g^3*n^3*log(F)^3) + 3/4*(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*d^2/(f^3*g^3*n^3*l og(F)^3) + 1/27*(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^2/(f^3*g^3*n^3*log(F)^3)
Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 8820, normalized size of antiderivative = 24.10 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2 \, dx=\text {Too large to display} \]
1/3*a^3*d^2*x^3 + a^3*c*d*x^2 + a^3*c^2*x - 1/27*((6*(3*pi*b^3*d^2*f^2*g^2 *n^2*x^2*log(abs(F))*sgn(F) - 3*pi*b^3*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 6 *pi*b^3*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 6*pi*b^3*c*d*f^2*g^2*n^2*x* log(abs(F)) + 3*pi*b^3*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 3*pi*b^3*c^2*f ^2*g^2*n^2*log(abs(F)) - pi*b^3*d^2*f*g*n*x*sgn(F) + pi*b^3*d^2*f*g*n*x - pi*b^3*c*d*f*g*n*sgn(F) + pi*b^3*c*d*f*g*n)*(pi^3*f^3*g^3*n^3*sgn(F) - 3*p i*f^3*g^3*n^3*log(abs(F))^2*sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*l og(abs(F))^2)/((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2*s gn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*f^3 *g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n ^3*log(abs(F))^3)^2) - (9*pi^2*b^3*d^2*f^2*g^2*n^2*x^2*sgn(F) - 9*pi^2*b^3 *d^2*f^2*g^2*n^2*x^2 + 18*b^3*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 18*pi^2* b^3*c*d*f^2*g^2*n^2*x*sgn(F) - 18*pi^2*b^3*c*d*f^2*g^2*n^2*x + 36*b^3*c*d* f^2*g^2*n^2*x*log(abs(F))^2 + 9*pi^2*b^3*c^2*f^2*g^2*n^2*sgn(F) - 9*pi^2*b ^3*c^2*f^2*g^2*n^2 + 18*b^3*c^2*f^2*g^2*n^2*log(abs(F))^2 - 12*b^3*d^2*f*g *n*x*log(abs(F)) - 12*b^3*c*d*f*g*n*log(abs(F)) + 4*b^3*d^2)*(3*pi^2*f^3*g ^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*g^3*n^3 *log(abs(F))^3)/((pi^3*f^3*g^3*n^3*sgn(F) - 3*pi*f^3*g^3*n^3*log(abs(F))^2 *sgn(F) - pi^3*f^3*g^3*n^3 + 3*pi*f^3*g^3*n^3*log(abs(F))^2)^2 + (3*pi^2*f ^3*g^3*n^3*log(abs(F))*sgn(F) - 3*pi^2*f^3*g^3*n^3*log(abs(F)) + 2*f^3*...
Time = 0.56 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.09 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2 \, dx={\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{3\,n}\,\left (\frac {b^3\,\left (9\,c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-6\,c\,d\,f\,g\,n\,\ln \left (F\right )+2\,d^2\right )}{27\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {b^3\,d^2\,x^2}{3\,f\,g\,n\,\ln \left (F\right )}-\frac {2\,b^3\,d\,x\,\left (d-3\,c\,f\,g\,n\,\ln \left (F\right )\right )}{9\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {3\,a^2\,b\,\left (c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-2\,c\,d\,f\,g\,n\,\ln \left (F\right )+2\,d^2\right )}{f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {3\,a^2\,b\,d^2\,x^2}{f\,g\,n\,\ln \left (F\right )}-\frac {6\,a^2\,b\,d\,x\,\left (d-c\,f\,g\,n\,\ln \left (F\right )\right )}{f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}\,\left (\frac {3\,a\,b^2\,\left (2\,c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-2\,c\,d\,f\,g\,n\,\ln \left (F\right )+d^2\right )}{4\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {3\,a\,b^2\,d^2\,x^2}{2\,f\,g\,n\,\ln \left (F\right )}-\frac {3\,a\,b^2\,d\,x\,\left (d-2\,c\,f\,g\,n\,\ln \left (F\right )\right )}{2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+a^3\,c^2\,x+\frac {a^3\,d^2\,x^3}{3}+a^3\,c\,d\,x^2 \]
(F^(f*g*x)*F^(e*g))^(3*n)*((b^3*(2*d^2 + 9*c^2*f^2*g^2*n^2*log(F)^2 - 6*c* d*f*g*n*log(F)))/(27*f^3*g^3*n^3*log(F)^3) + (b^3*d^2*x^2)/(3*f*g*n*log(F) ) - (2*b^3*d*x*(d - 3*c*f*g*n*log(F)))/(9*f^2*g^2*n^2*log(F)^2)) + (F^(f*g *x)*F^(e*g))^n*((3*a^2*b*(2*d^2 + c^2*f^2*g^2*n^2*log(F)^2 - 2*c*d*f*g*n*l og(F)))/(f^3*g^3*n^3*log(F)^3) + (3*a^2*b*d^2*x^2)/(f*g*n*log(F)) - (6*a^2 *b*d*x*(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*(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) + (3*a*b^2*d^2*x^2)/(2*f*g*n*log(F)) - (3*a*b^2*d *x*(d - 2*c*f*g*n*log(F)))/(2*f^2*g^2*n^2*log(F)^2)) + a^3*c^2*x + (a^3*d^ 2*x^3)/3 + a^3*c*d*x^2