3.52 \(\int \frac{(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx\)

Optimal. Leaf size=388 \[ \frac{6 d^2 (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac{6 d^2 (c+d x) \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{6 d^3 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}-\frac{6 d^3 \text{PolyLog}\left (4,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}+\frac{3 d (c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^3 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}-\frac{(c+d x)^3}{a^2 f g n \log (F)}+\frac{(c+d x)^4}{4 a^2 d}+\frac{(c+d x)^3}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]

[Out]

(c + d*x)^4/(4*a^2*d) - (c + d*x)^3/(a^2*f*g*n*Log[F]) + (c + d*x)^3/(a*f*(a + b
*(F^(g*(e + f*x)))^n)*g*n*Log[F]) + (3*d*(c + d*x)^2*Log[1 + (b*(F^(g*(e + f*x))
)^n)/a])/(a^2*f^2*g^2*n^2*Log[F]^2) - ((c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^
n)/a])/(a^2*f*g*n*Log[F]) + (6*d^2*(c + d*x)*PolyLog[2, -((b*(F^(g*(e + f*x)))^n
)/a)])/(a^2*f^3*g^3*n^3*Log[F]^3) - (3*d*(c + d*x)^2*PolyLog[2, -((b*(F^(g*(e +
f*x)))^n)/a)])/(a^2*f^2*g^2*n^2*Log[F]^2) - (6*d^3*PolyLog[3, -((b*(F^(g*(e + f*
x)))^n)/a)])/(a^2*f^4*g^4*n^4*Log[F]^4) + (6*d^2*(c + d*x)*PolyLog[3, -((b*(F^(g
*(e + f*x)))^n)/a)])/(a^2*f^3*g^3*n^3*Log[F]^3) - (6*d^3*PolyLog[4, -((b*(F^(g*(
e + f*x)))^n)/a)])/(a^2*f^4*g^4*n^4*Log[F]^4)

_______________________________________________________________________________________

Rubi [A]  time = 1.42057, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ \frac{6 d^2 (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac{6 d^2 (c+d x) \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{6 d^3 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}-\frac{6 d^3 \text{PolyLog}\left (4,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}+\frac{3 d (c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^3 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}-\frac{(c+d x)^3}{a^2 f g n \log (F)}+\frac{(c+d x)^4}{4 a^2 d}+\frac{(c+d x)^3}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^3/(a + b*(F^(g*(e + f*x)))^n)^2,x]

[Out]

(c + d*x)^4/(4*a^2*d) - (c + d*x)^3/(a^2*f*g*n*Log[F]) + (c + d*x)^3/(a*f*(a + b
*(F^(g*(e + f*x)))^n)*g*n*Log[F]) + (3*d*(c + d*x)^2*Log[1 + (b*(F^(g*(e + f*x))
)^n)/a])/(a^2*f^2*g^2*n^2*Log[F]^2) - ((c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^
n)/a])/(a^2*f*g*n*Log[F]) + (6*d^2*(c + d*x)*PolyLog[2, -((b*(F^(g*(e + f*x)))^n
)/a)])/(a^2*f^3*g^3*n^3*Log[F]^3) - (3*d*(c + d*x)^2*PolyLog[2, -((b*(F^(g*(e +
f*x)))^n)/a)])/(a^2*f^2*g^2*n^2*Log[F]^2) - (6*d^3*PolyLog[3, -((b*(F^(g*(e + f*
x)))^n)/a)])/(a^2*f^4*g^4*n^4*Log[F]^4) + (6*d^2*(c + d*x)*PolyLog[3, -((b*(F^(g
*(e + f*x)))^n)/a)])/(a^2*f^3*g^3*n^3*Log[F]^3) - (6*d^3*PolyLog[4, -((b*(F^(g*(
e + f*x)))^n)/a)])/(a^2*f^4*g^4*n^4*Log[F]^4)

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**3/(a+b*(F**(g*(f*x+e)))**n)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 3.57524, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(c + d*x)^3/(a + b*(F^(g*(e + f*x)))^n)^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.072, size = 2553, normalized size = 6.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3/(a+b*(F^(g*(f*x+e)))^n)^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ c^{3}{\left (\frac{1}{{\left ({\left (F^{f g x + e g}\right )}^{n} a b n + a^{2} n\right )} f g \log \left (F\right )} + \frac{\log \left (F^{f g x + e g}\right )}{a^{2} f g \log \left (F\right )} - \frac{\log \left (\frac{{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{2} f g n \log \left (F\right )}\right )} + \frac{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x}{{\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a b f g n \log \left (F\right ) + a^{2} f g n \log \left (F\right )} + \int \frac{d^{3} f g n x^{3} \log \left (F\right ) - 3 \, c^{2} d + 3 \,{\left (c d^{2} f g n \log \left (F\right ) - d^{3}\right )} x^{2} + 3 \,{\left (c^{2} d f g n \log \left (F\right ) - 2 \, c d^{2}\right )} x}{{\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a b f g n \log \left (F\right ) + a^{2} f g n \log \left (F\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/((F^((f*x + e)*g))^n*b + a)^2,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.291846, size = 1877, normalized size = 4.84 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/((F^((f*x + e)*g))^n*b + a)^2,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}{a^{2} f g n \log{\left (F \right )} + a b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log{\left (F \right )}} + \frac{\int \left (- \frac{3 c^{2} d}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\right )\, dx + \int \left (- \frac{3 d^{3} x^{2}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\right )\, dx + \int \left (- \frac{6 c d^{2} x}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\right )\, dx + \int \frac{c^{3} f g n \log{\left (F \right )}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\, dx + \int \frac{d^{3} f g n x^{3} \log{\left (F \right )}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\, dx + \int \frac{3 c d^{2} f g n x^{2} \log{\left (F \right )}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\, dx + \int \frac{3 c^{2} d f g n x \log{\left (F \right )}}{a + b e^{e g n \log{\left (F \right )}} e^{f g n x \log{\left (F \right )}}}\, dx}{a f g n \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**3/(a+b*(F**(g*(f*x+e)))**n)**2,x)

[Out]

(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3)/(a**2*f*g*n*log(F) + a*b*f*g*n*(
F**(g*(e + f*x)))**n*log(F)) + (Integral(-3*c**2*d/(a + b*exp(e*g*n*log(F))*exp(
f*g*n*x*log(F))), x) + Integral(-3*d**3*x**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*
x*log(F))), x) + Integral(-6*c*d**2*x/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F
))), x) + Integral(c**3*f*g*n*log(F)/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)
)), x) + Integral(d**3*f*g*n*x**3*log(F)/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*lo
g(F))), x) + Integral(3*c*d**2*f*g*n*x**2*log(F)/(a + b*exp(e*g*n*log(F))*exp(f*
g*n*x*log(F))), x) + Integral(3*c**2*d*f*g*n*x*log(F)/(a + b*exp(e*g*n*log(F))*e
xp(f*g*n*x*log(F))), x))/(a*f*g*n*log(F))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{3}}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/((F^((f*x + e)*g))^n*b + a)^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^3/((F^((f*x + e)*g))^n*b + a)^2, x)