\(\int (d+e (F^{c (a+b x)})^n) (f+g x)^3 \, dx\) [19]

Optimal result

Integrand size = 23, antiderivative size = 153 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right ) (f+g x)^3 \, dx=\frac {d (f+g x)^4}{4 g}-\frac {6 e \left (F^{a c+b c x}\right )^n g^3}{b^4 c^4 n^4 \log ^4(F)}+\frac {6 e \left (F^{a c+b c x}\right )^n g^2 (f+g x)}{b^3 c^3 n^3 \log ^3(F)}-\frac {3 e \left (F^{a c+b c x}\right )^n g (f+g x)^2}{b^2 c^2 n^2 \log ^2(F)}+\frac {e \left (F^{a c+b c x}\right )^n (f+g x)^3}{b c n \log (F)} \] Output:

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

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.85 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right ) (f+g x)^3 \, dx=d f^3 x+\frac {3}{2} d f^2 g x^2+d f g^2 x^3+\frac {1}{4} d g^3 x^4+\frac {e \left (F^{c (a+b x)}\right )^n \left (-6 g^3+6 b c g^2 n (f+g x) \log (F)-3 b^2 c^2 g n^2 (f+g x)^2 \log ^2(F)+b^3 c^3 n^3 (f+g x)^3 \log ^3(F)\right )}{b^4 c^4 n^4 \log ^4(F)} \] Input:

Integrate[(d + e*(F^(c*(a + b*x)))^n)*(f + g*x)^3,x]
 

Output:

d*f^3*x + (3*d*f^2*g*x^2)/2 + d*f*g^2*x^3 + (d*g^3*x^4)/4 + (e*(F^(c*(a + 
b*x)))^n*(-6*g^3 + 6*b*c*g^2*n*(f + g*x)*Log[F] - 3*b^2*c^2*g*n^2*(f + g*x 
)^2*Log[F]^2 + b^3*c^3*n^3*(f + g*x)^3*Log[F]^3))/(b^4*c^4*n^4*Log[F]^4)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.

\(\displaystyle \int (f+g x)^3 \left (e \left (F^{c (a+b x)}\right )^n+d\right ) \, dx\)

\(\Big \downarrow \) 2614

\(\displaystyle \int \left (e (f+g x)^3 \left (F^{a c+b c x}\right )^n+d (f+g x)^3\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 e g^3 \left (F^{a c+b c x}\right )^n}{b^4 c^4 n^4 \log ^4(F)}+\frac {6 e g^2 (f+g x) \left (F^{a c+b c x}\right )^n}{b^3 c^3 n^3 \log ^3(F)}-\frac {3 e g (f+g x)^2 \left (F^{a c+b c x}\right )^n}{b^2 c^2 n^2 \log ^2(F)}+\frac {e (f+g x)^3 \left (F^{a c+b c x}\right )^n}{b c n \log (F)}+\frac {d (f+g x)^4}{4 g}\)

Input:

Int[(d + e*(F^(c*(a + b*x)))^n)*(f + g*x)^3,x]
 

Output:

(d*(f + g*x)^4)/(4*g) - (6*e*(F^(a*c + b*c*x))^n*g^3)/(b^4*c^4*n^4*Log[F]^ 
4) + (6*e*(F^(a*c + b*c*x))^n*g^2*(f + g*x))/(b^3*c^3*n^3*Log[F]^3) - (3*e 
*(F^(a*c + b*c*x))^n*g*(f + g*x)^2)/(b^2*c^2*n^2*Log[F]^2) + (e*(F^(a*c + 
b*c*x))^n*(f + g*x)^3)/(b*c*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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(391\) vs. \(2(151)=302\).

Time = 0.46 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.56

method	result	size
parallelrisch	\(\frac {d \,g^{3} x^{4} n^{4} c^{4} b^{4} \ln \left (F \right )^{4}+4 d \,g^{2} f \,x^{3} n^{4} c^{4} b^{4} \ln \left (F \right )^{4}+6 d g \,f^{2} x^{2} n^{4} c^{4} b^{4} \ln \left (F \right )^{4}+4 d \,f^{3} x \,n^{4} c^{4} b^{4} \ln \left (F \right )^{4}+4 x^{3} \left (F^{c \left (b x +a \right )}\right )^{n} e \,g^{3} n^{3} c^{3} b^{3} \ln \left (F \right )^{3}+12 \ln \left (F \right )^{3} x^{2} \left (F^{c \left (b x +a \right )}\right )^{n} b^{3} c^{3} e f \,g^{2} n^{3}+12 \ln \left (F \right )^{3} x \left (F^{c \left (b x +a \right )}\right )^{n} b^{3} c^{3} e \,f^{2} g \,n^{3}+4 \ln \left (F \right )^{3} \left (F^{c \left (b x +a \right )}\right )^{n} b^{3} c^{3} e \,f^{3} n^{3}-12 \ln \left (F \right )^{2} x^{2} \left (F^{c \left (b x +a \right )}\right )^{n} b^{2} c^{2} e \,g^{3} n^{2}-24 \ln \left (F \right )^{2} x \left (F^{c \left (b x +a \right )}\right )^{n} b^{2} c^{2} e f \,g^{2} n^{2}-12 \ln \left (F \right )^{2} \left (F^{c \left (b x +a \right )}\right )^{n} b^{2} c^{2} e \,f^{2} g \,n^{2}+24 \ln \left (F \right ) x \left (F^{c \left (b x +a \right )}\right )^{n} b c e \,g^{3} n +24 \ln \left (F \right ) \left (F^{c \left (b x +a \right )}\right )^{n} b c e f \,g^{2} n -24 \left (F^{c \left (b x +a \right )}\right )^{n} e \,g^{3}}{4 n^{4} c^{4} b^{4} \ln \left (F \right )^{4}}\)	\(392\)
orering	\(\frac {\left (\ln \left (F \right )^{4} b^{4} c^{4} g^{4} n^{4} x^{5}+5 \ln \left (F \right )^{4} b^{4} c^{4} f \,g^{3} n^{4} x^{4}+10 \ln \left (F \right )^{4} b^{4} c^{4} f^{2} g^{2} n^{4} x^{3}+10 \ln \left (F \right )^{4} b^{4} c^{4} f^{3} g \,n^{4} x^{2}+4 \ln \left (F \right )^{4} b^{4} c^{4} f^{4} n^{4} x +3 \ln \left (F \right )^{3} b^{3} c^{3} g^{4} n^{3} x^{4}+12 \ln \left (F \right )^{3} b^{3} c^{3} f \,g^{3} n^{3} x^{3}+18 \ln \left (F \right )^{3} b^{3} c^{3} f^{2} g^{2} n^{3} x^{2}+16 \ln \left (F \right )^{3} b^{3} c^{3} f^{3} g \,n^{3} x +4 \ln \left (F \right )^{3} b^{3} c^{3} f^{4} n^{3}-12 \ln \left (F \right )^{2} b^{2} c^{2} g^{4} n^{2} x^{3}-36 \ln \left (F \right )^{2} b^{2} c^{2} f \,g^{3} n^{2} x^{2}-48 \ln \left (F \right )^{2} b^{2} c^{2} f^{2} g^{2} n^{2} x -12 \ln \left (F \right )^{2} b^{2} c^{2} f^{3} g \,n^{2}+36 \ln \left (F \right ) b c \,g^{4} n \,x^{2}+96 \ln \left (F \right ) b c f \,g^{3} n x +24 \ln \left (F \right ) b c \,f^{2} g^{2} n -96 g^{4} x -24 f \,g^{3}\right ) \left (d +e \left (F^{c \left (b x +a \right )}\right )^{n}\right )}{4 n^{4} c^{4} b^{4} \ln \left (F \right )^{4} \left (g x +f \right )}-\frac {x \left (g^{3} x^{3} n^{3} c^{3} b^{3} \ln \left (F \right )^{3}+4 \ln \left (F \right )^{3} b^{3} c^{3} f \,g^{2} n^{3} x^{2}+6 \ln \left (F \right )^{3} b^{3} c^{3} f^{2} g \,n^{3} x +4 n^{3} c^{3} b^{3} \ln \left (F \right )^{3} f^{3}-4 \ln \left (F \right )^{2} b^{2} c^{2} g^{3} n^{2} x^{2}-12 \ln \left (F \right )^{2} b^{2} c^{2} f \,g^{2} n^{2} x -12 n^{2} c^{2} b^{2} \ln \left (F \right )^{2} f^{2} g +12 \ln \left (F \right ) b c \,g^{3} n x +24 f \,g^{2} n c b \ln \left (F \right )-24 g^{3}\right ) \left (e \left (F^{c \left (b x +a \right )}\right )^{n} \ln \left (F \right ) b c n \left (g x +f \right )^{3}+3 \left (d +e \left (F^{c \left (b x +a \right )}\right )^{n}\right ) \left (g x +f \right )^{2} g \right )}{4 n^{4} c^{4} b^{4} \ln \left (F \right )^{4} \left (g x +f \right )^{3}}\)	\(628\)

Input:

int((d+e*(F^(c*(b*x+a)))^n)*(g*x+f)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.75 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right ) (f+g x)^3 \, dx=\frac {{\left (b^{4} c^{4} d g^{3} n^{4} x^{4} + 4 \, b^{4} c^{4} d f g^{2} n^{4} x^{3} + 6 \, b^{4} c^{4} d f^{2} g n^{4} x^{2} + 4 \, b^{4} c^{4} d f^{3} n^{4} x\right )} \log \left (F\right )^{4} - 4 \, {\left (6 \, e g^{3} - {\left (b^{3} c^{3} e g^{3} n^{3} x^{3} + 3 \, b^{3} c^{3} e f g^{2} n^{3} x^{2} + 3 \, b^{3} c^{3} e f^{2} g n^{3} x + b^{3} c^{3} e f^{3} n^{3}\right )} \log \left (F\right )^{3} + 3 \, {\left (b^{2} c^{2} e g^{3} n^{2} x^{2} + 2 \, b^{2} c^{2} e f g^{2} n^{2} x + b^{2} c^{2} e f^{2} g n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (b c e g^{3} n x + b c e f g^{2} n\right )} \log \left (F\right )\right )} F^{b c n x + a c n}}{4 \, b^{4} c^{4} n^{4} \log \left (F\right )^{4}} \] Input:

integrate((d+e*(F^((b*x+a)*c))^n)*(g*x+f)^3,x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (151) = 302\).

Time = 1.69 (sec) , antiderivative size = 496, normalized size of antiderivative = 3.24 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right ) (f+g x)^3 \, dx=\begin {cases} \left (d + e\right ) \left (f^{3} x + \frac {3 f^{2} g x^{2}}{2} + f g^{2} x^{3} + \frac {g^{3} x^{4}}{4}\right ) & \text {for}\: F = 1 \wedge b = 0 \wedge c = 0 \wedge n = 0 \\\left (d + e \left (F^{a c}\right )^{n}\right ) \left (f^{3} x + \frac {3 f^{2} g x^{2}}{2} + f g^{2} x^{3} + \frac {g^{3} x^{4}}{4}\right ) & \text {for}\: b = 0 \\\left (d + e\right ) \left (f^{3} x + \frac {3 f^{2} g x^{2}}{2} + f g^{2} x^{3} + \frac {g^{3} x^{4}}{4}\right ) & \text {for}\: F = 1 \vee c = 0 \vee n = 0 \\d f^{3} x + \frac {3 d f^{2} g x^{2}}{2} + d f g^{2} x^{3} + \frac {d g^{3} x^{4}}{4} + \frac {e f^{3} \left (F^{a c + b c x}\right )^{n}}{b c n \log {\left (F \right )}} + \frac {3 e f^{2} g x \left (F^{a c + b c x}\right )^{n}}{b c n \log {\left (F \right )}} + \frac {3 e f g^{2} x^{2} \left (F^{a c + b c x}\right )^{n}}{b c n \log {\left (F \right )}} + \frac {e g^{3} x^{3} \left (F^{a c + b c x}\right )^{n}}{b c n \log {\left (F \right )}} - \frac {3 e f^{2} g \left (F^{a c + b c x}\right )^{n}}{b^{2} c^{2} n^{2} \log {\left (F \right )}^{2}} - \frac {6 e f g^{2} x \left (F^{a c + b c x}\right )^{n}}{b^{2} c^{2} n^{2} \log {\left (F \right )}^{2}} - \frac {3 e g^{3} x^{2} \left (F^{a c + b c x}\right )^{n}}{b^{2} c^{2} n^{2} \log {\left (F \right )}^{2}} + \frac {6 e f g^{2} \left (F^{a c + b c x}\right )^{n}}{b^{3} c^{3} n^{3} \log {\left (F \right )}^{3}} + \frac {6 e g^{3} x \left (F^{a c + b c x}\right )^{n}}{b^{3} c^{3} n^{3} \log {\left (F \right )}^{3}} - \frac {6 e g^{3} \left (F^{a c + b c x}\right )^{n}}{b^{4} c^{4} n^{4} \log {\left (F \right )}^{4}} & \text {otherwise} \end {cases} \] Input:

integrate((d+e*(F**((b*x+a)*c))**n)*(g*x+f)**3,x)
 

Output:

Piecewise(((d + e)*(f**3*x + 3*f**2*g*x**2/2 + f*g**2*x**3 + g**3*x**4/4), 
 Eq(F, 1) & Eq(b, 0) & Eq(c, 0) & Eq(n, 0)), ((d + e*(F**(a*c))**n)*(f**3* 
x + 3*f**2*g*x**2/2 + f*g**2*x**3 + g**3*x**4/4), Eq(b, 0)), ((d + e)*(f** 
3*x + 3*f**2*g*x**2/2 + f*g**2*x**3 + g**3*x**4/4), Eq(F, 1) | Eq(c, 0) | 
Eq(n, 0)), (d*f**3*x + 3*d*f**2*g*x**2/2 + d*f*g**2*x**3 + d*g**3*x**4/4 + 
 e*f**3*(F**(a*c + b*c*x))**n/(b*c*n*log(F)) + 3*e*f**2*g*x*(F**(a*c + b*c 
*x))**n/(b*c*n*log(F)) + 3*e*f*g**2*x**2*(F**(a*c + b*c*x))**n/(b*c*n*log( 
F)) + e*g**3*x**3*(F**(a*c + b*c*x))**n/(b*c*n*log(F)) - 3*e*f**2*g*(F**(a 
*c + b*c*x))**n/(b**2*c**2*n**2*log(F)**2) - 6*e*f*g**2*x*(F**(a*c + b*c*x 
))**n/(b**2*c**2*n**2*log(F)**2) - 3*e*g**3*x**2*(F**(a*c + b*c*x))**n/(b* 
*2*c**2*n**2*log(F)**2) + 6*e*f*g**2*(F**(a*c + b*c*x))**n/(b**3*c**3*n**3 
*log(F)**3) + 6*e*g**3*x*(F**(a*c + b*c*x))**n/(b**3*c**3*n**3*log(F)**3) 
- 6*e*g**3*(F**(a*c + b*c*x))**n/(b**4*c**4*n**4*log(F)**4), True))
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.84 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right ) (f+g x)^3 \, dx=\frac {1}{4} \, d g^{3} x^{4} + d f g^{2} x^{3} + \frac {3}{2} \, d f^{2} g x^{2} + d f^{3} x + \frac {F^{b c n x + a c n} e f^{3}}{b c n \log \left (F\right )} + \frac {3 \, {\left (F^{a c n} b c n x \log \left (F\right ) - F^{a c n}\right )} F^{b c n x} e f^{2} g}{b^{2} c^{2} n^{2} \log \left (F\right )^{2}} + \frac {3 \, {\left (F^{a c n} b^{2} c^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c n} b c n x \log \left (F\right ) + 2 \, F^{a c n}\right )} F^{b c n x} e f g^{2}}{b^{3} c^{3} n^{3} \log \left (F\right )^{3}} + \frac {{\left (F^{a c n} b^{3} c^{3} n^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c n} b^{2} c^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c n} b c n x \log \left (F\right ) - 6 \, F^{a c n}\right )} F^{b c n x} e g^{3}}{b^{4} c^{4} n^{4} \log \left (F\right )^{4}} \] Input:

integrate((d+e*(F^((b*x+a)*c))^n)*(g*x+f)^3,x, algorithm="maxima")
 

Output:

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 5716, normalized size of antiderivative = 37.36 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right ) (f+g x)^3 \, dx=\text {Too large to display} \] Input:

integrate((d+e*(F^((b*x+a)*c))^n)*(g*x+f)^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/4*d*g^3*x^4 + d*f*g^2*x^3 + 3/2*d*f^2*g*x^2 + d*f^3*x - (((3*pi^2*b^3*c^ 
3*e*g^3*n^3*x^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*e*g^3*n^3*x^3*log(abs( 
F)) + 2*b^3*c^3*e*g^3*n^3*x^3*log(abs(F))^3 + 9*pi^2*b^3*c^3*e*f*g^2*n^3*x 
^2*log(abs(F))*sgn(F) - 9*pi^2*b^3*c^3*e*f*g^2*n^3*x^2*log(abs(F)) + 6*b^3 
*c^3*e*f*g^2*n^3*x^2*log(abs(F))^3 + 9*pi^2*b^3*c^3*e*f^2*g*n^3*x*log(abs( 
F))*sgn(F) - 9*pi^2*b^3*c^3*e*f^2*g*n^3*x*log(abs(F)) + 6*b^3*c^3*e*f^2*g* 
n^3*x*log(abs(F))^3 + 3*pi^2*b^3*c^3*e*f^3*n^3*log(abs(F))*sgn(F) - 3*pi^2 
*b^3*c^3*e*f^3*n^3*log(abs(F)) + 2*b^3*c^3*e*f^3*n^3*log(abs(F))^3 - 3*pi^ 
2*b^2*c^2*e*g^3*n^2*x^2*sgn(F) + 3*pi^2*b^2*c^2*e*g^3*n^2*x^2 - 6*b^2*c^2* 
e*g^3*n^2*x^2*log(abs(F))^2 - 6*pi^2*b^2*c^2*e*f*g^2*n^2*x*sgn(F) + 6*pi^2 
*b^2*c^2*e*f*g^2*n^2*x - 12*b^2*c^2*e*f*g^2*n^2*x*log(abs(F))^2 - 3*pi^2*b 
^2*c^2*e*f^2*g*n^2*sgn(F) + 3*pi^2*b^2*c^2*e*f^2*g*n^2 - 6*b^2*c^2*e*f^2*g 
*n^2*log(abs(F))^2 + 12*b*c*e*g^3*n*x*log(abs(F)) + 12*b*c*e*f*g^2*n*log(a 
bs(F)) - 12*e*g^3)*(pi^4*b^4*c^4*n^4*sgn(F) - 6*pi^2*b^4*c^4*n^4*log(abs(F 
))^2*sgn(F) - pi^4*b^4*c^4*n^4 + 6*pi^2*b^4*c^4*n^4*log(abs(F))^2 - 2*b^4* 
c^4*n^4*log(abs(F))^4)/((pi^4*b^4*c^4*n^4*sgn(F) - 6*pi^2*b^4*c^4*n^4*log( 
abs(F))^2*sgn(F) - pi^4*b^4*c^4*n^4 + 6*pi^2*b^4*c^4*n^4*log(abs(F))^2 - 2 
*b^4*c^4*n^4*log(abs(F))^4)^2 + 16*(pi^3*b^4*c^4*n^4*log(abs(F))*sgn(F) - 
pi*b^4*c^4*n^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*n^4*log(abs(F)) + pi*b^ 
4*c^4*n^4*log(abs(F))^3)^2) - 4*(pi^3*b^3*c^3*e*g^3*n^3*x^3*sgn(F) - 3*...
 

Mupad [B] (verification not implemented)

Time = 23.05 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.47 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right ) (f+g x)^3 \, dx=d\,f^3\,x-{\left (F^{b\,c\,x}\,F^{a\,c}\right )}^n\,\left (\frac {e\,\left (-b^3\,c^3\,f^3\,n^3\,{\ln \left (F\right )}^3+3\,b^2\,c^2\,f^2\,g\,n^2\,{\ln \left (F\right )}^2-6\,b\,c\,f\,g^2\,n\,\ln \left (F\right )+6\,g^3\right )}{b^4\,c^4\,n^4\,{\ln \left (F\right )}^4}-\frac {e\,g^3\,x^3}{b\,c\,n\,\ln \left (F\right )}-\frac {3\,e\,g\,x\,\left (b^2\,c^2\,f^2\,n^2\,{\ln \left (F\right )}^2-2\,b\,c\,f\,g\,n\,\ln \left (F\right )+2\,g^2\right )}{b^3\,c^3\,n^3\,{\ln \left (F\right )}^3}+\frac {3\,e\,g^2\,x^2\,\left (g-b\,c\,f\,n\,\ln \left (F\right )\right )}{b^2\,c^2\,n^2\,{\ln \left (F\right )}^2}\right )+\frac {d\,g^3\,x^4}{4}+\frac {3\,d\,f^2\,g\,x^2}{2}+d\,f\,g^2\,x^3 \] Input:

int((f + g*x)^3*(d + e*(F^(c*(a + b*x)))^n),x)
 

Output:

d*f^3*x - (F^(b*c*x)*F^(a*c))^n*((e*(6*g^3 - b^3*c^3*f^3*n^3*log(F)^3 - 6* 
b*c*f*g^2*n*log(F) + 3*b^2*c^2*f^2*g*n^2*log(F)^2))/(b^4*c^4*n^4*log(F)^4) 
 - (e*g^3*x^3)/(b*c*n*log(F)) - (3*e*g*x*(2*g^2 + b^2*c^2*f^2*n^2*log(F)^2 
 - 2*b*c*f*g*n*log(F)))/(b^3*c^3*n^3*log(F)^3) + (3*e*g^2*x^2*(g - b*c*f*n 
*log(F)))/(b^2*c^2*n^2*log(F)^2)) + (d*g^3*x^4)/4 + (3*d*f^2*g*x^2)/2 + d* 
f*g^2*x^3
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.62 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right ) (f+g x)^3 \, dx=\frac {4 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e \,f^{3} n^{3}+12 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e \,f^{2} g \,n^{3} x +12 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e f \,g^{2} n^{3} x^{2}+4 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e \,g^{3} n^{3} x^{3}-12 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e \,f^{2} g \,n^{2}-24 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e f \,g^{2} n^{2} x -12 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e \,g^{3} n^{2} x^{2}+24 f^{b c n x +a c n} \mathrm {log}\left (f \right ) b c e f \,g^{2} n +24 f^{b c n x +a c n} \mathrm {log}\left (f \right ) b c e \,g^{3} n x -24 f^{b c n x +a c n} e \,g^{3}+4 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d \,f^{3} n^{4} x +6 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d \,f^{2} g \,n^{4} x^{2}+4 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d f \,g^{2} n^{4} x^{3}+\mathrm {log}\left (f \right )^{4} b^{4} c^{4} d \,g^{3} n^{4} x^{4}}{4 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} n^{4}} \] Input:

int((d+e*(F^((b*x+a)*c))^n)*(g*x+f)^3,x)
 

Output:

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