Integrand size = 25, antiderivative size = 322 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^3 \, dx=\frac {a^2 (c+d x)^4}{4 d}-\frac {12 a b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}-\frac {3 b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)}+\frac {12 a b d^2 \left (F^{e g+f g x}\right )^n (c+d x)}{f^3 g^3 n^3 \log ^3(F)}+\frac {3 b^2 d^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {6 a b d \left (F^{e g+f g x}\right )^n (c+d x)^2}{f^2 g^2 n^2 \log ^2(F)}-\frac {3 b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{4 f^2 g^2 n^2 \log ^2(F)}+\frac {2 a b \left (F^{e g+f g x}\right )^n (c+d x)^3}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^3}{2 f g n \log (F)} \]
1/4*a^2*(d*x+c)^4/d-12*a*b*d^3*(F^(f*g*x+e*g))^n/f^4/g^4/n^4/ln(F)^4-3/8*b ^2*d^3*(F^(f*g*x+e*g))^(2*n)/f^4/g^4/n^4/ln(F)^4+12*a*b*d^2*(F^(f*g*x+e*g) )^n*(d*x+c)/f^3/g^3/n^3/ln(F)^3+3/4*b^2*d^2*(F^(f*g*x+e*g))^(2*n)*(d*x+c)/ f^3/g^3/n^3/ln(F)^3-6*a*b*d*(F^(f*g*x+e*g))^n*(d*x+c)^2/f^2/g^2/n^2/ln(F)^ 2-3/4*b^2*d*(F^(f*g*x+e*g))^(2*n)*(d*x+c)^2/f^2/g^2/n^2/ln(F)^2+2*a*b*(F^( f*g*x+e*g))^n*(d*x+c)^3/f/g/n/ln(F)+1/2*b^2*(F^(f*g*x+e*g))^(2*n)*(d*x+c)^ 3/f/g/n/ln(F)
Time = 0.55 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.74 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^3 \, dx=a^2 c^3 x+\frac {3}{2} a^2 c^2 d x^2+a^2 c d^2 x^3+\frac {1}{4} a^2 d^3 x^4+\frac {2 a b \left (F^{g (e+f x)}\right )^n \left (-6 d^3+6 d^2 f g n (c+d x) \log (F)-3 d f^2 g^2 n^2 (c+d x)^2 \log ^2(F)+f^3 g^3 n^3 (c+d x)^3 \log ^3(F)\right )}{f^4 g^4 n^4 \log ^4(F)}+\frac {b^2 \left (F^{g (e+f x)}\right )^{2 n} \left (-3 d^3+6 d^2 f g n (c+d x) \log (F)-6 d f^2 g^2 n^2 (c+d x)^2 \log ^2(F)+4 f^3 g^3 n^3 (c+d x)^3 \log ^3(F)\right )}{8 f^4 g^4 n^4 \log ^4(F)} \]
a^2*c^3*x + (3*a^2*c^2*d*x^2)/2 + a^2*c*d^2*x^3 + (a^2*d^3*x^4)/4 + (2*a*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) + (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)
Time = 0.69 (sec) , antiderivative size = 322, 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.
| \(\displaystyle \int (c+d x)^3 \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 \, dx\) |
| \(\Big \downarrow \) 2614 |
| \(\displaystyle \int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \left (F^{e g+f g x}\right )^n+b^2 (c+d x)^3 \left (F^{e g+f g x}\right )^{2 n}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 (c+d x)^4}{4 d}+\frac {12 a b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {6 a b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {2 a b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac {12 a b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}+\frac {3 b^2 d^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {3 b^2 d (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)}+\frac {b^2 (c+d x)^3 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac {3 b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)}\) |
(a^2*(c + d*x)^4)/(4*d) - (12*a*b*d^3*(F^(e*g + f*g*x))^n)/(f^4*g^4*n^4*Lo g[F]^4) - (3*b^2*d^3*(F^(e*g + f*g*x))^(2*n))/(8*f^4*g^4*n^4*Log[F]^4) + ( 12*a*b*d^2*(F^(e*g + f*g*x))^n*(c + d*x))/(f^3*g^3*n^3*Log[F]^3) + (3*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) - (6*a*b*d *(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f^2*g^2*n^2*Log[F]^2) - (3*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) + (2*a*b*(F^(e*g + f*g*x))^n*(c + d*x)^3)/(f*g*n*Log[F]) + (b^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^3)/(2*f*g*n*Log[F])
3.1.32.3.1 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. \(738\) vs. \(2(312)=624\).
Time = 1.09 (sec) , antiderivative size = 739, normalized size of antiderivative = 2.30
| method | result | size |
| parallelrisch | \(\frac {12 \ln \left (F \right )^{3} x^{2} \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} c \,d^{2} f^{3} g^{3} n^{3}+12 \ln \left (F \right )^{3} x \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} c^{2} d \,f^{3} g^{3} n^{3}-48 \ln \left (F \right )^{2} x^{2} \left (F^{g \left (f x +e \right )}\right )^{n} a b \,d^{3} f^{2} g^{2} n^{2}-12 \ln \left (F \right )^{2} x \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} c \,d^{2} f^{2} g^{2} n^{2}-48 \ln \left (F \right )^{2} \left (F^{g \left (f x +e \right )}\right )^{n} a b \,c^{2} d \,f^{2} g^{2} n^{2}+96 \ln \left (F \right ) x \left (F^{g \left (f x +e \right )}\right )^{n} a b \,d^{3} f g n +96 \ln \left (F \right ) \left (F^{g \left (f x +e \right )}\right )^{n} a b c \,d^{2} f g n +16 a b \,d^{3} \left (F^{g \left (f x +e \right )}\right )^{n} x^{3} n^{3} g^{3} f^{3} \ln \left (F \right )^{3}-3 \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} d^{3}-96 \left (F^{g \left (f x +e \right )}\right )^{n} a b \,d^{3}+6 \ln \left (F \right ) \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} c \,d^{2} f g n -96 \ln \left (F \right )^{2} x \left (F^{g \left (f x +e \right )}\right )^{n} a b c \,d^{2} f^{2} g^{2} n^{2}+4 \ln \left (F \right )^{3} \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} c^{3} f^{3} g^{3} n^{3}+48 \ln \left (F \right )^{3} x^{2} \left (F^{g \left (f x +e \right )}\right )^{n} a b c \,d^{2} f^{3} g^{3} n^{3}+48 \ln \left (F \right )^{3} x \left (F^{g \left (f x +e \right )}\right )^{n} a b \,c^{2} d \,f^{3} g^{3} n^{3}+8 a^{2} d^{2} c \,x^{3} n^{4} g^{4} f^{4} \ln \left (F \right )^{4}+12 a^{2} d \,c^{2} x^{2} n^{4} g^{4} f^{4} \ln \left (F \right )^{4}+4 b^{2} d^{3} \left (F^{g \left (f x +e \right )}\right )^{2 n} x^{3} n^{3} g^{3} f^{3} \ln \left (F \right )^{3}+16 \ln \left (F \right )^{3} \left (F^{g \left (f x +e \right )}\right )^{n} a b \,c^{3} f^{3} g^{3} n^{3}-6 \ln \left (F \right )^{2} x^{2} \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} d^{3} f^{2} g^{2} n^{2}-6 \ln \left (F \right )^{2} \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} c^{2} d \,f^{2} g^{2} n^{2}+6 \ln \left (F \right ) x \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} d^{3} f g n +2 a^{2} d^{3} x^{4} n^{4} g^{4} f^{4} \ln \left (F \right )^{4}+8 a^{2} c^{3} x \,n^{4} g^{4} f^{4} \ln \left (F \right )^{4}}{8 n^{4} g^{4} f^{4} \ln \left (F \right )^{4}}\) | \(739\) |
1/8*(12*ln(F)^3*x^2*((F^(g*(f*x+e)))^n)^2*b^2*c*d^2*f^3*g^3*n^3+12*ln(F)^3 *x*((F^(g*(f*x+e)))^n)^2*b^2*c^2*d*f^3*g^3*n^3-48*ln(F)^2*x^2*(F^(g*(f*x+e )))^n*a*b*d^3*f^2*g^2*n^2-12*ln(F)^2*x*((F^(g*(f*x+e)))^n)^2*b^2*c*d^2*f^2 *g^2*n^2-48*ln(F)^2*(F^(g*(f*x+e)))^n*a*b*c^2*d*f^2*g^2*n^2+96*ln(F)*x*(F^ (g*(f*x+e)))^n*a*b*d^3*f*g*n+96*ln(F)*(F^(g*(f*x+e)))^n*a*b*c*d^2*f*g*n+16 *a*b*d^3*(F^(g*(f*x+e)))^n*x^3*n^3*g^3*f^3*ln(F)^3-3*((F^(g*(f*x+e)))^n)^2 *b^2*d^3-96*(F^(g*(f*x+e)))^n*a*b*d^3+6*ln(F)*((F^(g*(f*x+e)))^n)^2*b^2*c* d^2*f*g*n-96*ln(F)^2*x*(F^(g*(f*x+e)))^n*a*b*c*d^2*f^2*g^2*n^2+4*ln(F)^3*( (F^(g*(f*x+e)))^n)^2*b^2*c^3*f^3*g^3*n^3+48*ln(F)^3*x^2*(F^(g*(f*x+e)))^n* a*b*c*d^2*f^3*g^3*n^3+48*ln(F)^3*x*(F^(g*(f*x+e)))^n*a*b*c^2*d*f^3*g^3*n^3 +8*a^2*d^2*c*x^3*n^4*g^4*f^4*ln(F)^4+12*a^2*d*c^2*x^2*n^4*g^4*f^4*ln(F)^4+ 4*b^2*d^3*((F^(g*(f*x+e)))^n)^2*x^3*n^3*g^3*f^3*ln(F)^3+16*ln(F)^3*(F^(g*( f*x+e)))^n*a*b*c^3*f^3*g^3*n^3-6*ln(F)^2*x^2*((F^(g*(f*x+e)))^n)^2*b^2*d^3 *f^2*g^2*n^2-6*ln(F)^2*((F^(g*(f*x+e)))^n)^2*b^2*c^2*d*f^2*g^2*n^2+6*ln(F) *x*((F^(g*(f*x+e)))^n)^2*b^2*d^3*f*g*n+2*a^2*d^3*x^4*n^4*g^4*f^4*ln(F)^4+8 *a^2*c^3*x*n^4*g^4*f^4*ln(F)^4)/n^4/g^4/f^4/ln(F)^4
Time = 0.27 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.50 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^3 \, dx=\frac {2 \, {\left (a^{2} d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, a^{2} c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, a^{2} c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, a^{2} c^{3} f^{4} g^{4} n^{4} x\right )} \log \left (F\right )^{4} - {\left (3 \, b^{2} d^{3} - 4 \, {\left (b^{2} d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, b^{2} c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, b^{2} c^{2} d f^{3} g^{3} n^{3} x + b^{2} c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 6 \, {\left (b^{2} d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c d^{2} f^{2} g^{2} n^{2} x + b^{2} c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (b^{2} d^{3} f g n x + b^{2} c d^{2} f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} - 16 \, {\left (6 \, a b d^{3} - {\left (a b d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, a b c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, a b c^{2} d f^{3} g^{3} n^{3} x + a b c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 3 \, {\left (a b d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, a b c d^{2} f^{2} g^{2} n^{2} x + a b c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (a b d^{3} f g n x + a b c d^{2} f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{8 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \]
1/8*(2*(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)*log(F)^4 - (3*b^2*d^3 - 4*(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)*log(F)^3 + 6*(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)*log(F)^2 - 6*(b^2*d^3*f*g*n* x + b^2*c*d^2*f*g*n)*log(F))*F^(2*f*g*n*x + 2*e*g*n) - 16*(6*a*b*d^3 - (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)*log(F)^3 + 3*(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)*log(F)^2 - 6*(a*b*d^3*f*g*n* x + a*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. 913 vs. \(2 (323) = 646\).
Time = 4.99 (sec) , antiderivative size = 913, normalized size of antiderivative = 2.84 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^3 \, dx=\begin {cases} \left (a + b\right )^{2} \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) & \text {for}\: F = 1 \wedge f = 0 \wedge g = 0 \wedge n = 0 \\\left (a + b \left (F^{e g}\right )^{n}\right )^{2} \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) & \text {for}\: f = 0 \\\left (a + b\right )^{2} \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) & \text {for}\: F = 1 \vee g = 0 \vee n = 0 \\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} + \frac {2 a b c^{3} \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} + \frac {6 a b c^{2} d x \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} - \frac {6 a b c^{2} d \left (F^{e g + f g x}\right )^{n}}{f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {6 a b c d^{2} x^{2} \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} - \frac {12 a b c d^{2} x \left (F^{e g + f g x}\right )^{n}}{f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {12 a b c d^{2} \left (F^{e g + f g x}\right )^{n}}{f^{3} g^{3} n^{3} \log {\left (F \right )}^{3}} + \frac {2 a b d^{3} x^{3} \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} - \frac {6 a b d^{3} x^{2} \left (F^{e g + f g x}\right )^{n}}{f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {12 a b d^{3} x \left (F^{e g + f g x}\right )^{n}}{f^{3} g^{3} n^{3} \log {\left (F \right )}^{3}} - \frac {12 a b d^{3} \left (F^{e g + f g x}\right )^{n}}{f^{4} g^{4} n^{4} \log {\left (F \right )}^{4}} + \frac {b^{2} c^{3} \left (F^{e g + f g x}\right )^{2 n}}{2 f g n \log {\left (F \right )}} + \frac {3 b^{2} c^{2} d x \left (F^{e g + f g x}\right )^{2 n}}{2 f g n \log {\left (F \right )}} - \frac {3 b^{2} c^{2} d \left (F^{e g + f g x}\right )^{2 n}}{4 f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {3 b^{2} c d^{2} x^{2} \left (F^{e g + f g x}\right )^{2 n}}{2 f g n \log {\left (F \right )}} - \frac {3 b^{2} c 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 b^{2} c 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^{2} d^{3} x^{3} \left (F^{e g + f g x}\right )^{2 n}}{2 f g n \log {\left (F \right )}} - \frac {3 b^{2} d^{3} x^{2} \left (F^{e g + f g x}\right )^{2 n}}{4 f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {3 b^{2} d^{3} x \left (F^{e g + f g x}\right )^{2 n}}{4 f^{3} g^{3} n^{3} \log {\left (F \right )}^{3}} - \frac {3 b^{2} d^{3} \left (F^{e g + f g x}\right )^{2 n}}{8 f^{4} g^{4} n^{4} \log {\left (F \right )}^{4}} & \text {otherwise} \end {cases} \]
Piecewise(((a + b)**2*(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)**2* (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 )**2*(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**2*c**3*x + 3*a**2*c**2*d*x**2/2 + a**2*c*d**2*x**3 + a**2*d**3*x**4/4 + 2*a*b*c**3*(F**(e*g + f*g*x))**n/(f*g*n*log(F)) + 6* a*b*c**2*d*x*(F**(e*g + f*g*x))**n/(f*g*n*log(F)) - 6*a*b*c**2*d*(F**(e*g + f*g*x))**n/(f**2*g**2*n**2*log(F)**2) + 6*a*b*c*d**2*x**2*(F**(e*g + f*g *x))**n/(f*g*n*log(F)) - 12*a*b*c*d**2*x*(F**(e*g + f*g*x))**n/(f**2*g**2* n**2*log(F)**2) + 12*a*b*c*d**2*(F**(e*g + f*g*x))**n/(f**3*g**3*n**3*log( F)**3) + 2*a*b*d**3*x**3*(F**(e*g + f*g*x))**n/(f*g*n*log(F)) - 6*a*b*d**3 *x**2*(F**(e*g + f*g*x))**n/(f**2*g**2*n**2*log(F)**2) + 12*a*b*d**3*x*(F* *(e*g + f*g*x))**n/(f**3*g**3*n**3*log(F)**3) - 12*a*b*d**3*(F**(e*g + f*g *x))**n/(f**4*g**4*n**4*log(F)**4) + b**2*c**3*(F**(e*g + f*g*x))**(2*n)/( 2*f*g*n*log(F)) + 3*b**2*c**2*d*x*(F**(e*g + f*g*x))**(2*n)/(2*f*g*n*log(F )) - 3*b**2*c**2*d*(F**(e*g + f*g*x))**(2*n)/(4*f**2*g**2*n**2*log(F)**2) + 3*b**2*c*d**2*x**2*(F**(e*g + f*g*x))**(2*n)/(2*f*g*n*log(F)) - 3*b**2*c *d**2*x*(F**(e*g + f*g*x))**(2*n)/(2*f**2*g**2*n**2*log(F)**2) + 3*b**2*c* d**2*(F**(e*g + f*g*x))**(2*n)/(4*f**3*g**3*n**3*log(F)**3) + b**2*d**3*x* *3*(F**(e*g + f*g*x))**(2*n)/(2*f*g*n*log(F)) - 3*b**2*d**3*x**2*(F**(e...
Time = 0.22 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.76 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^3 \, dx=\frac {1}{4} \, a^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + a^{2} c^{3} x + \frac {2 \, F^{f g n x + e g n} a b c^{3}}{f g n \log \left (F\right )} + \frac {F^{2 \, f g n x + 2 \, e g n} b^{2} c^{3}}{2 \, f g n \log \left (F\right )} + \frac {6 \, {\left (F^{e g n} f g n x \log \left (F\right ) - F^{e g n}\right )} F^{f g n x} a b c^{2} d}{f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {3 \, {\left (2 \, F^{2 \, e g n} f g n x \log \left (F\right ) - F^{2 \, e g n}\right )} F^{2 \, f g n x} b^{2} c^{2} d}{4 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {6 \, {\left (F^{e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{e g n} f g n x \log \left (F\right ) + 2 \, F^{e g n}\right )} F^{f g n x} a b c d^{2}}{f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {3 \, {\left (2 \, F^{2 \, e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{2 \, e g n} f g n x \log \left (F\right ) + F^{2 \, e g n}\right )} F^{2 \, f g n x} b^{2} c d^{2}}{4 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {2 \, {\left (F^{e g n} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{e g n} f g n x \log \left (F\right ) - 6 \, F^{e g n}\right )} F^{f g n x} a b d^{3}}{f^{4} g^{4} n^{4} \log \left (F\right )^{4}} + \frac {{\left (4 \, F^{2 \, e g n} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} - 6 \, F^{2 \, e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{2 \, e g n} f g n x \log \left (F\right ) - 3 \, F^{2 \, e g n}\right )} F^{2 \, f g n x} b^{2} d^{3}}{8 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \]
1/4*a^2*d^3*x^4 + a^2*c*d^2*x^3 + 3/2*a^2*c^2*d*x^2 + a^2*c^3*x + 2*F^(f*g *n*x + e*g*n)*a*b*c^3/(f*g*n*log(F)) + 1/2*F^(2*f*g*n*x + 2*e*g*n)*b^2*c^3 /(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*b *c^2*d/(f^2*g^2*n^2*log(F)^2) + 3/4*(2*F^(2*e*g*n)*f*g*n*x*log(F) - F^(2*e *g*n))*F^(2*f*g*n*x)*b^2*c^2*d/(f^2*g^2*n^2*log(F)^2) + 6*(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*b*c*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*lo g(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)*b^2*c*d ^2/(f^3*g^3*n^3*log(F)^3) + 2*(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*b*d^3/(f^4*g^4*n^4*log(F)^4) + 1/8*(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)*b^2*d^3/(f^4*g^4*n^4*log(F)^4 )
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 12013, normalized size of antiderivative = 37.31 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^3 \, dx=\text {Too large to display} \]
1/4*a^2*d^3*x^4 + a^2*c*d^2*x^3 + 3/2*a^2*c^2*d*x^2 + a^2*c^3*x - 1/4*(((6 *pi^2*b^2*d^3*f^3*g^3*n^3*x^3*log(abs(F))*sgn(F) - 6*pi^2*b^2*d^3*f^3*g^3* n^3*x^3*log(abs(F)) + 4*b^2*d^3*f^3*g^3*n^3*x^3*log(abs(F))^3 + 18*pi^2*b^ 2*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))*sgn(F) - 18*pi^2*b^2*c*d^2*f^3*g^3*n^3 *x^2*log(abs(F)) + 12*b^2*c*d^2*f^3*g^3*n^3*x^2*log(abs(F))^3 + 18*pi^2*b^ 2*c^2*d*f^3*g^3*n^3*x*log(abs(F))*sgn(F) - 18*pi^2*b^2*c^2*d*f^3*g^3*n^3*x *log(abs(F)) + 12*b^2*c^2*d*f^3*g^3*n^3*x*log(abs(F))^3 + 6*pi^2*b^2*c^3*f ^3*g^3*n^3*log(abs(F))*sgn(F) - 6*pi^2*b^2*c^3*f^3*g^3*n^3*log(abs(F)) + 4 *b^2*c^3*f^3*g^3*n^3*log(abs(F))^3 - 3*pi^2*b^2*d^3*f^2*g^2*n^2*x^2*sgn(F) + 3*pi^2*b^2*d^3*f^2*g^2*n^2*x^2 - 6*b^2*d^3*f^2*g^2*n^2*x^2*log(abs(F))^ 2 - 6*pi^2*b^2*c*d^2*f^2*g^2*n^2*x*sgn(F) + 6*pi^2*b^2*c*d^2*f^2*g^2*n^2*x - 12*b^2*c*d^2*f^2*g^2*n^2*x*log(abs(F))^2 - 3*pi^2*b^2*c^2*d*f^2*g^2*n^2 *sgn(F) + 3*pi^2*b^2*c^2*d*f^2*g^2*n^2 - 6*b^2*c^2*d*f^2*g^2*n^2*log(abs(F ))^2 + 6*b^2*d^3*f*g*n*x*log(abs(F)) + 6*b^2*c*d^2*f*g*n*log(abs(F)) - 3*b ^2*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*log(abs(F))^2*s gn(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)) + pi*f^4*g^4*n^...
Time = 0.59 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.36 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^3 \, dx=a^2\,c^3\,x-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {2\,a\,b\,\left (-c^3\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3+3\,c^2\,d\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-6\,c\,d^2\,f\,g\,n\,\ln \left (F\right )+6\,d^3\right )}{f^4\,g^4\,n^4\,{\ln \left (F\right )}^4}-\frac {2\,a\,b\,d^3\,x^3}{f\,g\,n\,\ln \left (F\right )}-\frac {6\,a\,b\,d\,x\,\left (c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-2\,c\,d\,f\,g\,n\,\ln \left (F\right )+2\,d^2\right )}{f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {6\,a\,b\,d^2\,x^2\,\left (d-c\,f\,g\,n\,\ln \left (F\right )\right )}{f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )-{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}\,\left (\frac {b^2\,\left (-4\,c^3\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3+6\,c^2\,d\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-6\,c\,d^2\,f\,g\,n\,\ln \left (F\right )+3\,d^3\right )}{8\,f^4\,g^4\,n^4\,{\ln \left (F\right )}^4}-\frac {b^2\,d^3\,x^3}{2\,f\,g\,n\,\ln \left (F\right )}-\frac {3\,b^2\,d\,x\,\left (2\,c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-2\,c\,d\,f\,g\,n\,\ln \left (F\right )+d^2\right )}{4\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {3\,b^2\,d^2\,x^2\,\left (d-2\,c\,f\,g\,n\,\ln \left (F\right )\right )}{4\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+\frac {a^2\,d^3\,x^4}{4}+\frac {3\,a^2\,c^2\,d\,x^2}{2}+a^2\,c\,d^2\,x^3 \]
a^2*c^3*x - (F^(f*g*x)*F^(e*g))^n*((2*a*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*log (F)^4) - (2*a*b*d^3*x^3)/(f*g*n*log(F)) - (6*a*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) + (6*a*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 )*((b^2*(3*d^3 - 4*c^3*f^3*g^3*n^3*log(F)^3 - 6*c*d^2*f*g*n*log(F) + 6*c^2 *d*f^2*g^2*n^2*log(F)^2))/(8*f^4*g^4*n^4*log(F)^4) - (b^2*d^3*x^3)/(2*f*g* n*log(F)) - (3*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) + (3*b^2*d^2*x^2*(d - 2*c*f*g*n*log(F)))/(4 *f^2*g^2*n^2*log(F)^2)) + (a^2*d^3*x^4)/4 + (3*a^2*c^2*d*x^2)/2 + a^2*c*d^ 2*x^3