\(\int (a+b (F^{g (e+f x)})^n)^2 (c+d x)^2 \, dx\) [33]

Optimal result

Integrand size = 25, antiderivative size = 239 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2 \, dx=\frac {a^2 (c+d x)^3}{3 d}+\frac {4 a b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}+\frac {b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac {4 a b d \left (F^{e g+f g x}\right )^n (c+d x)}{f^2 g^2 n^2 \log ^2(F)}-\frac {b^2 d \left (F^{e g+f g x}\right )^{2 n} (c+d x)}{2 f^2 g^2 n^2 \log ^2(F)}+\frac {2 a b \left (F^{e g+f g x}\right )^n (c+d x)^2}{f g n \log (F)}+\frac {b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^2}{2 f g n \log (F)} \] Output:

1/3*a^2*(d*x+c)^3/d+4*a*b*d^2*(F^(f*g*x+e*g))^n/f^3/g^3/n^3/ln(F)^3+1/4*b^ 
2*d^2*(F^(f*g*x+e*g))^(2*n)/f^3/g^3/n^3/ln(F)^3-4*a*b*d*(F^(f*g*x+e*g))^n* 
(d*x+c)/f^2/g^2/n^2/ln(F)^2-1/2*b^2*d*(F^(f*g*x+e*g))^(2*n)*(d*x+c)/f^2/g^ 
2/n^2/ln(F)^2+2*a*b*(F^(f*g*x+e*g))^n*(d*x+c)^2/f/g/n/ln(F)+1/2*b^2*(F^(f* 
g*x+e*g))^(2*n)*(d*x+c)^2/f/g/n/ln(F)
 

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.72 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2 \, dx=a^2 c^2 x+a^2 c d x^2+\frac {1}{3} a^2 d^2 x^3+\frac {2 a b \left (F^{g (e+f x)}\right )^n \left (2 d^2-2 d f g n (c+d x) \log (F)+f^2 g^2 n^2 (c+d x)^2 \log ^2(F)\right )}{f^3 g^3 n^3 \log ^3(F)}+\frac {b^2 \left (F^{g (e+f x)}\right )^{2 n} \left (d^2-2 d f g n (c+d x) \log (F)+2 f^2 g^2 n^2 (c+d x)^2 \log ^2(F)\right )}{4 f^3 g^3 n^3 \log ^3(F)} \] Input:

Integrate[(a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^2,x]
 

Output:

a^2*c^2*x + a^2*c*d*x^2 + (a^2*d^2*x^3)/3 + (2*a*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) + (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)
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 239, 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.

\(\displaystyle \int (c+d x)^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 \, dx\)

\(\Big \downarrow \) 2614

\(\displaystyle \int \left (a^2 (c+d x)^2+2 a b (c+d x)^2 \left (F^{e g+f g x}\right )^n+b^2 (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^2 (c+d x)^3}{3 d}-\frac {4 a b d (c+d x) \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {2 a b (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac {4 a b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac {b^2 d (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{2 f^2 g^2 n^2 \log ^2(F)}+\frac {b^2 (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}+\frac {b^2 d^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}\)

Input:

Int[(a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^2,x]
 

Output:

(a^2*(c + d*x)^3)/(3*d) + (4*a*b*d^2*(F^(e*g + f*g*x))^n)/(f^3*g^3*n^3*Log 
[F]^3) + (b^2*d^2*(F^(e*g + f*g*x))^(2*n))/(4*f^3*g^3*n^3*Log[F]^3) - (4*a 
*b*d*(F^(e*g + f*g*x))^n*(c + d*x))/(f^2*g^2*n^2*Log[F]^2) - (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*a*b*(F^(e*g + f 
*g*x))^n*(c + d*x)^2)/(f*g*n*Log[F]) + (b^2*(F^(e*g + f*g*x))^(2*n)*(c + d 
*x)^2)/(2*f*g*n*Log[F])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2614

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]
 

Maple [A] (warning: unable to verify)

Time = 0.47 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.77

method	result	size
parallelrisch	\(\frac {4 a^{2} d^{2} x^{3} n^{3} g^{3} f^{3} \ln \left (F \right )^{3}+12 a^{2} d c \,x^{2} n^{3} g^{3} f^{3} \ln \left (F \right )^{3}+12 a^{2} c^{2} x \,n^{3} g^{3} f^{3} \ln \left (F \right )^{3}+6 x^{2} \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} d^{2} n^{2} g^{2} f^{2} \ln \left (F \right )^{2}+24 x^{2} \left (F^{g \left (f x +e \right )}\right )^{n} a b \,d^{2} n^{2} g^{2} f^{2} \ln \left (F \right )^{2}+12 \ln \left (F \right )^{2} x \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} c d \,f^{2} g^{2} n^{2}+48 \ln \left (F \right )^{2} x \left (F^{g \left (f x +e \right )}\right )^{n} a b c d \,f^{2} g^{2} n^{2}+6 \ln \left (F \right )^{2} \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} c^{2} f^{2} g^{2} n^{2}+24 \ln \left (F \right )^{2} \left (F^{g \left (f x +e \right )}\right )^{n} a b \,c^{2} f^{2} g^{2} n^{2}-6 \ln \left (F \right ) x \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} d^{2} f g n -48 \ln \left (F \right ) x \left (F^{g \left (f x +e \right )}\right )^{n} a b \,d^{2} f g n -6 \ln \left (F \right ) \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} c d f g n -48 \ln \left (F \right ) \left (F^{g \left (f x +e \right )}\right )^{n} a b c d f g n +3 \left (F^{g \left (f x +e \right )}\right )^{2 n} b^{2} d^{2}+48 \left (F^{g \left (f x +e \right )}\right )^{n} a b \,d^{2}}{12 n^{3} g^{3} f^{3} \ln \left (F \right )^{3}}\)	\(424\)
orering	\(\frac {\left (4 \ln \left (F \right )^{4} d^{4} f^{4} g^{4} n^{4} x^{5}+20 \ln \left (F \right )^{4} c \,d^{3} f^{4} g^{4} n^{4} x^{4}+40 \ln \left (F \right )^{4} c^{2} d^{2} f^{4} g^{4} n^{4} x^{3}+36 \ln \left (F \right )^{4} c^{3} d \,f^{4} g^{4} n^{4} x^{2}+12 \ln \left (F \right )^{4} c^{4} f^{4} g^{4} n^{4} x +12 \ln \left (F \right )^{3} d^{4} f^{3} g^{3} n^{3} x^{4}+48 \ln \left (F \right )^{3} c \,d^{3} f^{3} g^{3} n^{3} x^{3}+90 \ln \left (F \right )^{3} c^{2} d^{2} f^{3} g^{3} n^{3} x^{2}+72 \ln \left (F \right )^{3} c^{3} d \,f^{3} g^{3} n^{3} x +18 \ln \left (F \right )^{3} c^{4} f^{3} g^{3} n^{3}-30 \ln \left (F \right )^{2} d^{4} f^{2} g^{2} n^{2} x^{3}-132 \ln \left (F \right )^{2} c \,d^{3} f^{2} g^{2} n^{2} x^{2}-120 \ln \left (F \right )^{2} c^{2} d^{2} f^{2} g^{2} n^{2} x -30 \ln \left (F \right )^{2} c^{3} d \,f^{2} g^{2} n^{2}+81 \ln \left (F \right ) d^{4} f g n \,x^{2}+36 \ln \left (F \right ) c \,d^{3} f g n x +9 \ln \left (F \right ) c^{2} d^{2} f g n +168 d^{4} x +42 c \,d^{3}\right ) {\left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}^{2}}{12 g^{4} n^{4} f^{4} \ln \left (F \right )^{4} \left (d x +c \right )^{2}}-\frac {\left (6 \ln \left (F \right )^{3} d^{3} f^{3} g^{3} n^{3} x^{4}+24 \ln \left (F \right )^{3} c \,d^{2} f^{3} g^{3} n^{3} x^{3}+36 \ln \left (F \right )^{3} c^{2} d \,f^{3} g^{3} n^{3} x^{2}+18 \ln \left (F \right )^{3} c^{3} f^{3} g^{3} n^{3} x -13 \ln \left (F \right )^{2} d^{3} f^{2} g^{2} n^{2} x^{3}-39 \ln \left (F \right )^{2} c \,d^{2} f^{2} g^{2} n^{2} x^{2}-12 \ln \left (F \right )^{2} c^{2} d \,f^{2} g^{2} n^{2} x +6 \ln \left (F \right )^{2} c^{3} f^{2} g^{2} n^{2}+9 d^{3} \ln \left (F \right ) f g n \,x^{2}-45 d^{2} \ln \left (F \right ) f g n x c -18 d \ln \left (F \right ) c^{2} f g n +105 d^{3} x +21 d^{2} c \right ) \left (2 \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right ) \left (d x +c \right )^{2} b \left (F^{g \left (f x +e \right )}\right )^{n} n g f \ln \left (F \right )+2 {\left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}^{2} \left (d x +c \right ) d \right )}{12 g^{4} n^{4} f^{4} \ln \left (F \right )^{4} \left (d x +c \right )^{3}}+\frac {x \left (2 d^{2} x^{2} \ln \left (F \right )^{2} f^{2} g^{2} n^{2}+6 \ln \left (F \right )^{2} c d \,f^{2} g^{2} n^{2} x +6 \ln \left (F \right )^{2} c^{2} f^{2} g^{2} n^{2}-9 \ln \left (F \right ) d^{2} f g n x -18 \ln \left (F \right ) c d f g n +21 d^{2}\right ) \left (2 b^{2} \left (F^{g \left (f x +e \right )}\right )^{2 n} n^{2} g^{2} f^{2} \ln \left (F \right )^{2} \left (d x +c \right )^{2}+8 \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right ) \left (d x +c \right ) b \left (F^{g \left (f x +e \right )}\right )^{n} n g f \ln \left (F \right ) d +2 \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right ) \left (d x +c \right )^{2} b \left (F^{g \left (f x +e \right )}\right )^{n} n^{2} g^{2} f^{2} \ln \left (F \right )^{2}+2 {\left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}^{2} d^{2}\right )}{12 g^{4} n^{4} f^{4} \ln \left (F \right )^{4} \left (d x +c \right )^{2}}\)	\(948\)

Input:

int((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c)^2,x,method=_RETURNVERBOSE)
 

Output:

1/12*(4*a^2*d^2*x^3*n^3*g^3*f^3*ln(F)^3+12*a^2*d*c*x^2*n^3*g^3*f^3*ln(F)^3 
+12*a^2*c^2*x*n^3*g^3*f^3*ln(F)^3+6*x^2*((F^(g*(f*x+e)))^n)^2*b^2*d^2*n^2* 
g^2*f^2*ln(F)^2+24*x^2*(F^(g*(f*x+e)))^n*a*b*d^2*n^2*g^2*f^2*ln(F)^2+12*ln 
(F)^2*x*((F^(g*(f*x+e)))^n)^2*b^2*c*d*f^2*g^2*n^2+48*ln(F)^2*x*(F^(g*(f*x+ 
e)))^n*a*b*c*d*f^2*g^2*n^2+6*ln(F)^2*((F^(g*(f*x+e)))^n)^2*b^2*c^2*f^2*g^2 
*n^2+24*ln(F)^2*(F^(g*(f*x+e)))^n*a*b*c^2*f^2*g^2*n^2-6*ln(F)*x*((F^(g*(f* 
x+e)))^n)^2*b^2*d^2*f*g*n-48*ln(F)*x*(F^(g*(f*x+e)))^n*a*b*d^2*f*g*n-6*ln( 
F)*((F^(g*(f*x+e)))^n)^2*b^2*c*d*f*g*n-48*ln(F)*(F^(g*(f*x+e)))^n*a*b*c*d* 
f*g*n+3*((F^(g*(f*x+e)))^n)^2*b^2*d^2+48*(F^(g*(f*x+e)))^n*a*b*d^2)/n^3/g^ 
3/f^3/ln(F)^3
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.20 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2 \, dx=\frac {4 \, {\left (a^{2} d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, a^{2} c d f^{3} g^{3} n^{3} x^{2} + 3 \, a^{2} c^{2} f^{3} g^{3} n^{3} x\right )} \log \left (F\right )^{3} + 3 \, {\left (b^{2} d^{2} + 2 \, {\left (b^{2} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c d f^{2} g^{2} n^{2} x + b^{2} c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (b^{2} d^{2} f g n x + b^{2} c d f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} + 24 \, {\left (2 \, a b d^{2} + {\left (a b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a b c d f^{2} g^{2} n^{2} x + a b c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (a b d^{2} f g n x + a b c d f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{12 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \] Input:

integrate((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c)^2,x, algorithm="fricas")
 

Output:

1/12*(4*(a^2*d^2*f^3*g^3*n^3*x^3 + 3*a^2*c*d*f^3*g^3*n^3*x^2 + 3*a^2*c^2*f 
^3*g^3*n^3*x)*log(F)^3 + 3*(b^2*d^2 + 2*(b^2*d^2*f^2*g^2*n^2*x^2 + 2*b^2*c 
*d*f^2*g^2*n^2*x + b^2*c^2*f^2*g^2*n^2)*log(F)^2 - 2*(b^2*d^2*f*g*n*x + b^ 
2*c*d*f*g*n)*log(F))*F^(2*f*g*n*x + 2*e*g*n) + 24*(2*a*b*d^2 + (a*b*d^2*f^ 
2*g^2*n^2*x^2 + 2*a*b*c*d*f^2*g^2*n^2*x + a*b*c^2*f^2*g^2*n^2)*log(F)^2 - 
2*(a*b*d^2*f*g*n*x + a*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)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (231) = 462\).

Time = 2.09 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.21 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2 \, dx=\begin {cases} \left (a + b\right )^{2} \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {for}\: F = 1 \wedge f = 0 \wedge g = 0 \wedge n = 0 \\\left (a + b \left (F^{e g}\right )^{n}\right )^{2} \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {for}\: f = 0 \\\left (a + b\right )^{2} \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {for}\: F = 1 \vee g = 0 \vee n = 0 \\a^{2} c^{2} x + a^{2} c d x^{2} + \frac {a^{2} d^{2} x^{3}}{3} + \frac {2 a b c^{2} \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} + \frac {4 a b c d x \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} - \frac {4 a b c d \left (F^{e g + f g x}\right )^{n}}{f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {2 a b d^{2} x^{2} \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} - \frac {4 a b d^{2} x \left (F^{e g + f g x}\right )^{n}}{f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {4 a b d^{2} \left (F^{e g + f g x}\right )^{n}}{f^{3} g^{3} n^{3} \log {\left (F \right )}^{3}} + \frac {b^{2} c^{2} \left (F^{e g + f g x}\right )^{2 n}}{2 f g n \log {\left (F \right )}} + \frac {b^{2} c d x \left (F^{e g + f g x}\right )^{2 n}}{f g n \log {\left (F \right )}} - \frac {b^{2} c d \left (F^{e g + f g x}\right )^{2 n}}{2 f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {b^{2} d^{2} x^{2} \left (F^{e g + f g x}\right )^{2 n}}{2 f g n \log {\left (F \right )}} - \frac {b^{2} d^{2} x \left (F^{e g + f g x}\right )^{2 n}}{2 f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {b^{2} d^{2} \left (F^{e g + f g x}\right )^{2 n}}{4 f^{3} g^{3} n^{3} \log {\left (F \right )}^{3}} & \text {otherwise} \end {cases} \] Input:

integrate((a+b*(F**(g*(f*x+e)))**n)**2*(d*x+c)**2,x)
                                                                                    
                                                                                    
 

Output:

Piecewise(((a + b)**2*(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)**2*(c**2*x + c*d*x**2 + 
d**2*x**3/3), Eq(f, 0)), ((a + b)**2*(c**2*x + c*d*x**2 + d**2*x**3/3), Eq 
(F, 1) | Eq(g, 0) | Eq(n, 0)), (a**2*c**2*x + a**2*c*d*x**2 + a**2*d**2*x* 
*3/3 + 2*a*b*c**2*(F**(e*g + f*g*x))**n/(f*g*n*log(F)) + 4*a*b*c*d*x*(F**( 
e*g + f*g*x))**n/(f*g*n*log(F)) - 4*a*b*c*d*(F**(e*g + f*g*x))**n/(f**2*g* 
*2*n**2*log(F)**2) + 2*a*b*d**2*x**2*(F**(e*g + f*g*x))**n/(f*g*n*log(F)) 
- 4*a*b*d**2*x*(F**(e*g + f*g*x))**n/(f**2*g**2*n**2*log(F)**2) + 4*a*b*d* 
*2*(F**(e*g + f*g*x))**n/(f**3*g**3*n**3*log(F)**3) + b**2*c**2*(F**(e*g + 
 f*g*x))**(2*n)/(2*f*g*n*log(F)) + b**2*c*d*x*(F**(e*g + f*g*x))**(2*n)/(f 
*g*n*log(F)) - b**2*c*d*(F**(e*g + f*g*x))**(2*n)/(2*f**2*g**2*n**2*log(F) 
**2) + b**2*d**2*x**2*(F**(e*g + f*g*x))**(2*n)/(2*f*g*n*log(F)) - b**2*d* 
*2*x*(F**(e*g + f*g*x))**(2*n)/(2*f**2*g**2*n**2*log(F)**2) + b**2*d**2*(F 
**(e*g + f*g*x))**(2*n)/(4*f**3*g**3*n**3*log(F)**3), True))
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.45 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2 \, dx=\frac {1}{3} \, a^{2} d^{2} x^{3} + a^{2} c d x^{2} + a^{2} c^{2} x + \frac {2 \, F^{f g n x + e g n} a b c^{2}}{f g n \log \left (F\right )} + \frac {F^{2 \, f g n x + 2 \, e g n} b^{2} c^{2}}{2 \, f g n \log \left (F\right )} + \frac {4 \, {\left (F^{e g n} f g n x \log \left (F\right ) - F^{e g n}\right )} F^{f g n x} a b c d}{f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {{\left (2 \, F^{2 \, e g n} f g n x \log \left (F\right ) - F^{2 \, e g n}\right )} F^{2 \, f g n x} b^{2} c d}{2 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {2 \, {\left (F^{e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{e g n} f g n x \log \left (F\right ) + 2 \, F^{e g n}\right )} F^{f g n x} a b d^{2}}{f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {{\left (2 \, F^{2 \, e g n} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{2 \, e g n} f g n x \log \left (F\right ) + F^{2 \, e g n}\right )} F^{2 \, f g n x} b^{2} d^{2}}{4 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \] Input:

integrate((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c)^2,x, algorithm="maxima")
 

Output:

1/3*a^2*d^2*x^3 + a^2*c*d*x^2 + a^2*c^2*x + 2*F^(f*g*n*x + e*g*n)*a*b*c^2/ 
(f*g*n*log(F)) + 1/2*F^(2*f*g*n*x + 2*e*g*n)*b^2*c^2/(f*g*n*log(F)) + 4*(F 
^(e*g*n)*f*g*n*x*log(F) - F^(e*g*n))*F^(f*g*n*x)*a*b*c*d/(f^2*g^2*n^2*log( 
F)^2) + 1/2*(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/(f^2*g^2*n^2*log(F)^2) + 2*(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*d^2/(f^3*g^3*n^3*log 
(F)^3) + 1/4*(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)*b^2*d^2/(f^3*g^3*n^3*log(F)^3)
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 5675, normalized size of antiderivative = 23.74 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2 \, dx=\text {Too large to display} \] Input:

integrate((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c)^2,x, algorithm="giac")
 

Output:

1/3*a^2*d^2*x^3 + a^2*c*d*x^2 + a^2*c^2*x - 1/2*(((2*pi*b^2*d^2*f^2*g^2*n^ 
2*x^2*log(abs(F))*sgn(F) - 2*pi*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F)) + 4*pi 
*b^2*c*d*f^2*g^2*n^2*x*log(abs(F))*sgn(F) - 4*pi*b^2*c*d*f^2*g^2*n^2*x*log 
(abs(F)) + 2*pi*b^2*c^2*f^2*g^2*n^2*log(abs(F))*sgn(F) - 2*pi*b^2*c^2*f^2* 
g^2*n^2*log(abs(F)) - pi*b^2*d^2*f*g*n*x*sgn(F) + pi*b^2*d^2*f*g*n*x - pi* 
b^2*c*d*f*g*n*sgn(F) + pi*b^2*c*d*f*g*n)*(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)/((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*g^3*n^3* 
log(abs(F))^3)^2) - (pi^2*b^2*d^2*f^2*g^2*n^2*x^2*sgn(F) - pi^2*b^2*d^2*f^ 
2*g^2*n^2*x^2 + 2*b^2*d^2*f^2*g^2*n^2*x^2*log(abs(F))^2 + 2*pi^2*b^2*c*d*f 
^2*g^2*n^2*x*sgn(F) - 2*pi^2*b^2*c*d*f^2*g^2*n^2*x + 4*b^2*c*d*f^2*g^2*n^2 
*x*log(abs(F))^2 + pi^2*b^2*c^2*f^2*g^2*n^2*sgn(F) - pi^2*b^2*c^2*f^2*g^2* 
n^2 + 2*b^2*c^2*f^2*g^2*n^2*log(abs(F))^2 - 2*b^2*d^2*f*g*n*x*log(abs(F)) 
- 2*b^2*c*d*f*g*n*log(abs(F)) + b^2*d^2)*(3*pi^2*f^3*g^3*n^3*log(abs(F))*s 
gn(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*g^3*n^3*log(abs(F))^...
 

Mupad [B] (verification not implemented)

Time = 23.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.12 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2 \, dx={\left (F^{f\,g\,x}\,F^{e\,g}\right )}^{2\,n}\,\left (\frac {b^2\,\left (2\,c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-2\,c\,d\,f\,g\,n\,\ln \left (F\right )+d^2\right )}{4\,f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {b^2\,d^2\,x^2}{2\,f\,g\,n\,\ln \left (F\right )}-\frac {b^2\,d\,x\,\left (d-2\,c\,f\,g\,n\,\ln \left (F\right )\right )}{2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n\,\left (\frac {2\,a\,b\,\left (c^2\,f^2\,g^2\,n^2\,{\ln \left (F\right )}^2-2\,c\,d\,f\,g\,n\,\ln \left (F\right )+2\,d^2\right )}{f^3\,g^3\,n^3\,{\ln \left (F\right )}^3}+\frac {2\,a\,b\,d^2\,x^2}{f\,g\,n\,\ln \left (F\right )}-\frac {4\,a\,b\,d\,x\,\left (d-c\,f\,g\,n\,\ln \left (F\right )\right )}{f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}\right )+a^2\,c^2\,x+\frac {a^2\,d^2\,x^3}{3}+a^2\,c\,d\,x^2 \] Input:

int((a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^2,x)
 

Output:

(F^(f*g*x)*F^(e*g))^(2*n)*((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) + (b^2*d^2*x^2)/(2*f*g*n*log(F)) - 
 (b^2*d*x*(d - 2*c*f*g*n*log(F)))/(2*f^2*g^2*n^2*log(F)^2)) + (F^(f*g*x)*F 
^(e*g))^n*((2*a*b*(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) + (2*a*b*d^2*x^2)/(f*g*n*log(F)) - (4*a*b*d*x*(d - 
 c*f*g*n*log(F)))/(f^2*g^2*n^2*log(F)^2)) + a^2*c^2*x + (a^2*d^2*x^3)/3 + 
a^2*c*d*x^2
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.82 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^2 \, dx=\frac {6 f^{2 f g n x +2 e g n} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} f^{2} g^{2} n^{2}+12 f^{2 f g n x +2 e g n} \mathrm {log}\left (f \right )^{2} b^{2} c d \,f^{2} g^{2} n^{2} x +6 f^{2 f g n x +2 e g n} \mathrm {log}\left (f \right )^{2} b^{2} d^{2} f^{2} g^{2} n^{2} x^{2}-6 f^{2 f g n x +2 e g n} \mathrm {log}\left (f \right ) b^{2} c d f g n -6 f^{2 f g n x +2 e g n} \mathrm {log}\left (f \right ) b^{2} d^{2} f g n x +3 f^{2 f g n x +2 e g n} b^{2} d^{2}+24 f^{f g n x +e g n} \mathrm {log}\left (f \right )^{2} a b \,c^{2} f^{2} g^{2} n^{2}+48 f^{f g n x +e g n} \mathrm {log}\left (f \right )^{2} a b c d \,f^{2} g^{2} n^{2} x +24 f^{f g n x +e g n} \mathrm {log}\left (f \right )^{2} a b \,d^{2} f^{2} g^{2} n^{2} x^{2}-48 f^{f g n x +e g n} \mathrm {log}\left (f \right ) a b c d f g n -48 f^{f g n x +e g n} \mathrm {log}\left (f \right ) a b \,d^{2} f g n x +48 f^{f g n x +e g n} a b \,d^{2}+12 \mathrm {log}\left (f \right )^{3} a^{2} c^{2} f^{3} g^{3} n^{3} x +12 \mathrm {log}\left (f \right )^{3} a^{2} c d \,f^{3} g^{3} n^{3} x^{2}+4 \mathrm {log}\left (f \right )^{3} a^{2} d^{2} f^{3} g^{3} n^{3} x^{3}}{12 \mathrm {log}\left (f \right )^{3} f^{3} g^{3} n^{3}} \] Input:

int((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c)^2,x)
 

Output:

(6*f**(2*e*g*n + 2*f*g*n*x)*log(f)**2*b**2*c**2*f**2*g**2*n**2 + 12*f**(2* 
e*g*n + 2*f*g*n*x)*log(f)**2*b**2*c*d*f**2*g**2*n**2*x + 6*f**(2*e*g*n + 2 
*f*g*n*x)*log(f)**2*b**2*d**2*f**2*g**2*n**2*x**2 - 6*f**(2*e*g*n + 2*f*g* 
n*x)*log(f)*b**2*c*d*f*g*n - 6*f**(2*e*g*n + 2*f*g*n*x)*log(f)*b**2*d**2*f 
*g*n*x + 3*f**(2*e*g*n + 2*f*g*n*x)*b**2*d**2 + 24*f**(e*g*n + f*g*n*x)*lo 
g(f)**2*a*b*c**2*f**2*g**2*n**2 + 48*f**(e*g*n + f*g*n*x)*log(f)**2*a*b*c* 
d*f**2*g**2*n**2*x + 24*f**(e*g*n + f*g*n*x)*log(f)**2*a*b*d**2*f**2*g**2* 
n**2*x**2 - 48*f**(e*g*n + f*g*n*x)*log(f)*a*b*c*d*f*g*n - 48*f**(e*g*n + 
f*g*n*x)*log(f)*a*b*d**2*f*g*n*x + 48*f**(e*g*n + f*g*n*x)*a*b*d**2 + 12*l 
og(f)**3*a**2*c**2*f**3*g**3*n**3*x + 12*log(f)**3*a**2*c*d*f**3*g**3*n**3 
*x**2 + 4*log(f)**3*a**2*d**2*f**3*g**3*n**3*x**3)/(12*log(f)**3*f**3*g**3 
*n**3)