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

Optimal. Leaf size=594 \[ \frac{9 d^2 (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{6 d^2 (c+d x) \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 d^3 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{9 d^3 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{6 d^3 \text{PolyLog}\left (4,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{3 d^2 (c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{9 d (c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^3 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f g n \log (F)}+\frac{3 d (c+d x)^2}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 (c+d x)^3}{2 a^3 f g n \log (F)}+\frac{(c+d x)^4}{4 a^3 d}-\frac{3 d (c+d x)^2}{2 a^2 f^2 g^2 n^2 \log ^2(F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{(c+d x)^3}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{(c+d x)^3}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]

[Out]

(c + d*x)^4/(4*a^3*d) + (3*d*(c + d*x)^2)/(2*a^3*f^2*g^2*n^2*Log[F]^2) - (3*d*(c
+ d*x)^2)/(2*a^2*f^2*(a + b*(F^(g*(e + f*x)))^n)*g^2*n^2*Log[F]^2) - (3*(c + d*
x)^3)/(2*a^3*f*g*n*Log[F]) + (c + d*x)^3/(2*a*f*(a + b*(F^(g*(e + f*x)))^n)^2*g*
n*Log[F]) + (c + d*x)^3/(a^2*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) - (3*d^2*
(c + d*x)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f^3*g^3*n^3*Log[F]^3) + (9*d*
(c + d*x)^2*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(2*a^3*f^2*g^2*n^2*Log[F]^2) - (
(c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f*g*n*Log[F]) - (3*d^3*Poly
Log[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F]^4) + (9*d^2*(c + d
*x)*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*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^3*f^2*g^2*n^2*Log[F]^2)
- (9*d^3*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*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^3*f^3*g^3*n^3*Log
[F]^3) - (6*d^3*PolyLog[4, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F
]^4)

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Rubi [A]  time = 3.15226, antiderivative size = 594, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{9 d^2 (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{6 d^2 (c+d x) \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac{3 d (c+d x)^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 d^3 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{9 d^3 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{6 d^3 \text{PolyLog}\left (4,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}-\frac{3 d^2 (c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac{9 d (c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^3 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^3 f g n \log (F)}+\frac{3 d (c+d x)^2}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac{3 (c+d x)^3}{2 a^3 f g n \log (F)}+\frac{(c+d x)^4}{4 a^3 d}-\frac{3 d (c+d x)^2}{2 a^2 f^2 g^2 n^2 \log ^2(F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{(c+d x)^3}{a^2 f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}+\frac{(c+d x)^3}{2 a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(c + d*x)^4/(4*a^3*d) + (3*d*(c + d*x)^2)/(2*a^3*f^2*g^2*n^2*Log[F]^2) - (3*d*(c
+ d*x)^2)/(2*a^2*f^2*(a + b*(F^(g*(e + f*x)))^n)*g^2*n^2*Log[F]^2) - (3*(c + d*
x)^3)/(2*a^3*f*g*n*Log[F]) + (c + d*x)^3/(2*a*f*(a + b*(F^(g*(e + f*x)))^n)^2*g*
n*Log[F]) + (c + d*x)^3/(a^2*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) - (3*d^2*
(c + d*x)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f^3*g^3*n^3*Log[F]^3) + (9*d*
(c + d*x)^2*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(2*a^3*f^2*g^2*n^2*Log[F]^2) - (
(c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f*g*n*Log[F]) - (3*d^3*Poly
Log[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F]^4) + (9*d^2*(c + d
*x)*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*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^3*f^2*g^2*n^2*Log[F]^2)
- (9*d^3*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*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^3*f^3*g^3*n^3*Log
[F]^3) - (6*d^3*PolyLog[4, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F
]^4)

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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)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 4.08425, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [B]  time = 0.079, size = 3116, normalized size = 5.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*d^3*polylog(3,-b*(F^(g*(f*x+e)))^n/a)/a^3/f^4/g^4/n^4/ln(F)^4-6*d^3*polylog(4
,-b*(F^(g*(f*x+e)))^n/a)/a^3/f^4/g^4/n^4/ln(F)^4-3*d^3*polylog(2,-b*(F^(g*(f*x+e
)))^n/a)/a^3/f^4/g^4/n^4/ln(F)^4-1/n/g/f/ln(F)/a^3*c^3*ln(a+b*(F^(g*(f*x+e)))^n)
+3/n/g^4/f^4/ln(F)^4/a^3*d^3*ln(F^(g*(f*x+e)))^3+1/n/g/f/ln(F)/a^3*c^3*ln((F^(g*
(f*x+e)))^n)+3/2/n^2/g^4/f^4/ln(F)^4/a^3*d^3*ln(F^(g*(f*x+e)))^2+3/2/g^2/f^2/ln(
F)^2/a^3*d^3*ln(F^(g*(f*x+e)))^2*x^2-2/g^3/f^3/ln(F)^3/a^3*d^3*ln(F^(g*(f*x+e)))
^3*x-2/g^3/f^3/ln(F)^3/a^3*c*d^2*ln(F^(g*(f*x+e)))^3+3/2/g^2/f^2/ln(F)^2/a^3*d*c
^2*ln(F^(g*(f*x+e)))^2+3/n/g^3/f^3/ln(F)^3/a^3*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^3*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^3*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^3*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^3*d^3*ln(1+b*(F^(g*(f*x+e)))^n/a)*ln(F^(g*(f*x+e)))^2
*x-9/n^2/g^3/f^3/ln(F)^3/a^3*d^3*ln(a+b*(F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))*x-9
/n^2/g^2/f^2/ln(F)^2/a^3*c*d^2*ln((F^(g*(f*x+e)))^n)*x+9/n^2/g^3/f^3/ln(F)^3/a^3
*c*d^2*ln((F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))+9/n^2/g^2/f^2/ln(F)^2/a^3*c*d^2*l
n(a+b*(F^(g*(f*x+e)))^n)*x-9/n^2/g^3/f^3/ln(F)^3/a^3*c*d^2*ln(a+b*(F^(g*(f*x+e))
)^n)*ln(F^(g*(f*x+e)))+9/n^2/g^3/f^3/ln(F)^3/a^3*d^3*ln(1+b*(F^(g*(f*x+e)))^n/a)
*ln(F^(g*(f*x+e)))*x+9/n^2/g^3/f^3/ln(F)^3/a^3*d^3*ln((F^(g*(f*x+e)))^n)*ln(F^(g
*(f*x+e)))*x+9/n^2/g^3/f^3/ln(F)^3/a^3*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^3*c^2*d*ln(a+b*(F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+
e)))-3/n/g^2/f^2/ln(F)^2/a^3*c^2*d*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^3*c^2*d*ln((F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))+3/n/g^3/f
^3/ln(F)^3/a^3*c*d^2*ln(1+b*(F^(g*(f*x+e)))^n/a)*ln(F^(g*(f*x+e)))^2-3/n/g^2/f^2
/ln(F)^2/a^3*d^3*ln((F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))*x^2-6/n^2/g^2/f^2/ln(F)
^2/a^3*c*d^2*polylog(2,-b*(F^(g*(f*x+e)))^n/a)*x-3/n/g/f/ln(F)/a^3*c^2*d*ln(a+b*
(F^(g*(f*x+e)))^n)*x+3/n/g/f/ln(F)/a^3*c^2*d*ln((F^(g*(f*x+e)))^n)*x+3/n/g/f/ln(
F)/a^3*c*d^2*ln((F^(g*(f*x+e)))^n)*x^2+3/n/g^3/f^3/ln(F)^3/a^3*c*d^2*ln((F^(g*(f
*x+e)))^n)*ln(F^(g*(f*x+e)))^2-3/n/g/f/ln(F)/a^3*c*d^2*ln(a+b*(F^(g*(f*x+e)))^n)
*x^2-3/n/g^3/f^3/ln(F)^3/a^3*c*d^2*ln(a+b*(F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))^2
+3/n^3/g^4/f^4/ln(F)^4/a^3*d^3*ln(a+b*(F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))-3/n^3
/g^4/f^4/ln(F)^4/a^3*d^3*ln(1+b*(F^(g*(f*x+e)))^n/a)*ln(F^(g*(f*x+e)))-9/2/n/g^3
/f^3/ln(F)^3/a^3*d^3*ln(F^(g*(f*x+e)))^2*x+6/n^3/g^3/f^3/ln(F)^3/a^3*d^3*polylog
(3,-b*(F^(g*(f*x+e)))^n/a)*x+9/n^3/g^3/f^3/ln(F)^3/a^3*d^3*polylog(2,-b*(F^(g*(f
*x+e)))^n/a)*x+6/n^3/g^3/f^3/ln(F)^3/a^3*c*d^2*polylog(3,-b*(F^(g*(f*x+e)))^n/a)
-3/n^3/g^3/f^3/ln(F)^3/a^3*c*d^2*ln(a+b*(F^(g*(f*x+e)))^n)+9/n^3/g^3/f^3/ln(F)^3
/a^3*c*d^2*polylog(2,-b*(F^(g*(f*x+e)))^n/a)+3/n^3/g^3/f^3/ln(F)^3/a^3*c*d^2*ln(
(F^(g*(f*x+e)))^n)-9/2/n/g^3/f^3/ln(F)^3/a^3*d^2*c*ln(F^(g*(f*x+e)))^2-9/2/n^2/g
^2/f^2/ln(F)^2/a^3*c^2*d*ln((F^(g*(f*x+e)))^n)+9/2/n^2/g^2/f^2/ln(F)^2/a^3*c^2*d
*ln(a+b*(F^(g*(f*x+e)))^n)-3/n^2/g^2/f^2/ln(F)^2/a^3*d^3*polylog(2,-b*(F^(g*(f*x
+e)))^n/a)*x^2-3/n^2/g^2/f^2/ln(F)^2/a^3*c^2*d*polylog(2,-b*(F^(g*(f*x+e)))^n/a)
-9/2/n^2/g^4/f^4/ln(F)^4/a^3*d^3*ln(1+b*(F^(g*(f*x+e)))^n/a)*ln(F^(g*(f*x+e)))^2
-9/2/n^2/g^2/f^2/ln(F)^2/a^3*d^3*ln((F^(g*(f*x+e)))^n)*x^2-9/2/n^2/g^4/f^4/ln(F)
^4/a^3*d^3*ln((F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))^2+9/2/n^2/g^2/f^2/ln(F)^2/a^3
*d^3*ln(a+b*(F^(g*(f*x+e)))^n)*x^2+9/2/n^2/g^4/f^4/ln(F)^4/a^3*d^3*ln(a+b*(F^(g*
(f*x+e)))^n)*ln(F^(g*(f*x+e)))^2-1/n/g^4/f^4/ln(F)^4/a^3*d^3*ln(1+b*(F^(g*(f*x+e
)))^n/a)*ln(F^(g*(f*x+e)))^3+1/n/g/f/ln(F)/a^3*d^3*ln((F^(g*(f*x+e)))^n)*x^3-1/n
/g^4/f^4/ln(F)^4/a^3*d^3*ln((F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))^3+3/g^2/f^2/ln(
F)^2/a^3*c*d^2*ln(F^(g*(f*x+e)))^2*x-1/n/g/f/ln(F)/a^3*d^3*ln(a+b*(F^(g*(f*x+e))
)^n)*x^3+1/n/g^4/f^4/ln(F)^4/a^3*d^3*ln(a+b*(F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))
^3+3/n^3/g^3/f^3/ln(F)^3/a^3*d^3*ln((F^(g*(f*x+e)))^n)*x-3/n^3/g^4/f^4/ln(F)^4/a
^3*d^3*ln((F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))-3/n^3/g^3/f^3/ln(F)^3/a^3*d^3*ln(
a+b*(F^(g*(f*x+e)))^n)*x+1/2*(2*ln(F)*b*d^3*f*g*n*x^3*(F^(g*(f*x+e)))^n+3*ln(F)*
a*d^3*f*g*n*x^3+6*ln(F)*b*c*d^2*f*g*n*x^2*(F^(g*(f*x+e)))^n+9*ln(F)*a*c*d^2*f*g*
n*x^2+6*ln(F)*b*c^2*d*f*g*n*x*(F^(g*(f*x+e)))^n+9*ln(F)*a*c^2*d*f*g*n*x+2*ln(F)*
b*c^3*f*g*n*(F^(g*(f*x+e)))^n+3*ln(F)*a*c^3*f*g*n-3*b*d^3*x^2*(F^(g*(f*x+e)))^n-
3*a*d^3*x^2-6*b*c*d^2*x*(F^(g*(f*x+e)))^n-6*a*c*d^2*x-3*b*c^2*d*(F^(g*(f*x+e)))^
n-3*a*c^2*d)/n^2/g^2/f^2/ln(F)^2/a^2/(a+b*(F^(g*(f*x+e)))^n)^2+3/4/g^4/f^4/ln(F)
^4/a^3*d^3*ln(F^(g*(f*x+e)))^4-6/n/g^2/f^2/ln(F)^2/a^3*c*d^2*ln(1+b*(F^(g*(f*x+e
)))^n/a)*ln(F^(g*(f*x+e)))*x-6/n/g^2/f^2/ln(F)^2/a^3*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^3*c*d^2*ln(a+b*(F^(g*(f*x+e)))^n)*ln(
F^(g*(f*x+e)))*x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{2} \, c^{3}{\left (\frac{2 \,{\left (F^{f g x + e g}\right )}^{n} b + 3 \, a}{{\left (2 \,{\left (F^{f g x + e g}\right )}^{n} a^{3} b n +{\left (F^{f g x + e g}\right )}^{2 \, n} a^{2} b^{2} n + a^{4} n\right )} f g \log \left (F\right )} + \frac{2 \, \log \left (F^{f g x + e g}\right )}{a^{3} f g \log \left (F\right )} - \frac{2 \, \log \left (\frac{{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{3} f g n \log \left (F\right )}\right )} + \frac{3 \, a d^{3} f g n x^{3} \log \left (F\right ) - 3 \, a c^{2} d + 3 \,{\left (3 \, a c d^{2} f g n \log \left (F\right ) - a d^{3}\right )} x^{2} +{\left (2 \,{\left (F^{e g}\right )}^{n} b d^{3} f g n x^{3} \log \left (F\right ) - 3 \,{\left (F^{e g}\right )}^{n} b c^{2} d + 3 \,{\left (2 \,{\left (F^{e g}\right )}^{n} b c d^{2} f g n \log \left (F\right ) -{\left (F^{e g}\right )}^{n} b d^{3}\right )} x^{2} + 6 \,{\left ({\left (F^{e g}\right )}^{n} b c^{2} d f g n \log \left (F\right ) -{\left (F^{e g}\right )}^{n} b c d^{2}\right )} x\right )}{\left (F^{f g x}\right )}^{n} + 3 \,{\left (3 \, a c^{2} d f g n \log \left (F\right ) - 2 \, a c d^{2}\right )} x}{2 \,{\left (2 \,{\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a^{3} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} +{\left (F^{f g x}\right )}^{2 \, n}{\left (F^{e g}\right )}^{2 \, n} a^{2} b^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{4} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\right )}} + \int \frac{2 \, d^{3} f^{2} g^{2} n^{2} x^{3} \log \left (F\right )^{2} - 9 \, c^{2} d f g n \log \left (F\right ) + 6 \, c d^{2} + 3 \,{\left (2 \, c d^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 3 \, d^{3} f g n \log \left (F\right )\right )} x^{2} + 6 \,{\left (c^{2} d f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 3 \, c d^{2} f g n \log \left (F\right ) + d^{3}\right )} x}{2 \,{\left ({\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a^{2} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{3} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\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)^3,x, algorithm="maxima")

[Out]

1/2*c^3*((2*(F^(f*g*x + e*g))^n*b + 3*a)/((2*(F^(f*g*x + e*g))^n*a^3*b*n + (F^(f
*g*x + e*g))^(2*n)*a^2*b^2*n + a^4*n)*f*g*log(F)) + 2*log(F^(f*g*x + e*g))/(a^3*
f*g*log(F)) - 2*log(((F^(f*g*x + e*g))^n*b + a)/b)/(a^3*f*g*n*log(F))) + 1/2*(3*
a*d^3*f*g*n*x^3*log(F) - 3*a*c^2*d + 3*(3*a*c*d^2*f*g*n*log(F) - a*d^3)*x^2 + (2
*(F^(e*g))^n*b*d^3*f*g*n*x^3*log(F) - 3*(F^(e*g))^n*b*c^2*d + 3*(2*(F^(e*g))^n*b
*c*d^2*f*g*n*log(F) - (F^(e*g))^n*b*d^3)*x^2 + 6*((F^(e*g))^n*b*c^2*d*f*g*n*log(
F) - (F^(e*g))^n*b*c*d^2)*x)*(F^(f*g*x))^n + 3*(3*a*c^2*d*f*g*n*log(F) - 2*a*c*d
^2)*x)/(2*(F^(f*g*x))^n*(F^(e*g))^n*a^3*b*f^2*g^2*n^2*log(F)^2 + (F^(f*g*x))^(2*
n)*(F^(e*g))^(2*n)*a^2*b^2*f^2*g^2*n^2*log(F)^2 + a^4*f^2*g^2*n^2*log(F)^2) + in
tegrate(1/2*(2*d^3*f^2*g^2*n^2*x^3*log(F)^2 - 9*c^2*d*f*g*n*log(F) + 6*c*d^2 + 3
*(2*c*d^2*f^2*g^2*n^2*log(F)^2 - 3*d^3*f*g*n*log(F))*x^2 + 6*(c^2*d*f^2*g^2*n^2*
log(F)^2 - 3*c*d^2*f*g*n*log(F) + d^3)*x)/((F^(f*g*x))^n*(F^(e*g))^n*a^2*b*f^2*g
^2*n^2*log(F)^2 + a^3*f^2*g^2*n^2*log(F)^2), x)

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Fricas [A]  time = 0.284597, size = 3650, normalized size = 6.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(6*(a^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*g^3*
n^3*log(F)^3 + 6*(a^2*d^3*e^2 - 2*a^2*c*d^2*e*f + a^2*c^2*d*f^2)*g^2*n^2*log(F)^
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 - (a^2*d^3*e^4 - 4*a^2*c*d^2*e^3*f + 6*a^2*c^
2*d*e^2*f^2 - 4*a^2*c^3*e*f^3)*g^4*n^4)*log(F)^4 - ((b^2*d^3*f^4*g^4*n^4*x^4 + 4
*b^2*c*d^2*f^4*g^4*n^4*x^3 + 6*b^2*c^2*d*f^4*g^4*n^4*x^2 + 4*b^2*c^3*f^4*g^4*n^4
*x - (b^2*d^3*e^4 - 4*b^2*c*d^2*e^3*f + 6*b^2*c^2*d*e^2*f^2 - 4*b^2*c^3*e*f^3)*g
^4*n^4)*log(F)^4 - 6*(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*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2)*
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*d^3*e^2 - 2*b^2*c*d^2*e*f)*g^2*n^2)*log(F)^2)*F^(2*f*g*n*x + 2*e*g*n) - 2*((a*
b*d^3*f^4*g^4*n^4*x^4 + 4*a*b*c*d^2*f^4*g^4*n^4*x^3 + 6*a*b*c^2*d*f^4*g^4*n^4*x^
2 + 4*a*b*c^3*f^4*g^4*n^4*x - (a*b*d^3*e^4 - 4*a*b*c*d^2*e^3*f + 6*a*b*c^2*d*e^2
*f^2 - 4*a*b*c^3*e*f^3)*g^4*n^4)*log(F)^4 - 2*(2*a*b*d^3*f^3*g^3*n^3*x^3 + 6*a*b
*c*d^2*f^3*g^3*n^3*x^2 + 6*a*b*c^2*d*f^3*g^3*n^3*x + (3*a*b*d^3*e^3 - 9*a*b*c*d^
2*e^2*f + 9*a*b*c^2*d*e*f^2 - 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 - (2*a*b*d^3*e^2 - 4*a*b*c*d^2*e*f + a*b*c
^2*d*f^2)*g^2*n^2)*log(F)^2)*F^(f*g*n*x + e*g*n) + 12*(a^2*d^3 + (a^2*d^3*f^2*g^
2*n^2*x^2 + 2*a^2*c*d^2*f^2*g^2*n^2*x + a^2*c^2*d*f^2*g^2*n^2)*log(F)^2 + (b^2*d
^3 + (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 - 3*(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) + 2*(a*b*d^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 - 3*(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) - 3*(a^2*d^3*f*g*n*x + a^2*c*d^2*f*g*n)*log(F))*dilog(-(F^(f*g*n*x
+ e*g*n)*b + a)/a + 1) - 2*(2*(a^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*
f^2 - a^2*c^3*f^3)*g^3*n^3*log(F)^3 + 9*(a^2*d^3*e^2 - 2*a^2*c*d^2*e*f + a^2*c^2
*d*f^2)*g^2*n^2*log(F)^2 + 6*(a^2*d^3*e - a^2*c*d^2*f)*g*n*log(F) + (2*(b^2*d^3*
e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*g^3*n^3*log(F)^3 + 9*
(b^2*d^3*e^2 - 2*b^2*c*d^2*e*f + b^2*c^2*d*f^2)*g^2*n^2*log(F)^2 + 6*(b^2*d^3*e
- b^2*c*d^2*f)*g*n*log(F))*F^(2*f*g*n*x + 2*e*g*n) + 2*(2*(a*b*d^3*e^3 - 3*a*b*c
*d^2*e^2*f + 3*a*b*c^2*d*e*f^2 - a*b*c^3*f^3)*g^3*n^3*log(F)^3 + 9*(a*b*d^3*e^2
- 2*a*b*c*d^2*e*f + a*b*c^2*d*f^2)*g^2*n^2*log(F)^2 + 6*(a*b*d^3*e - a*b*c*d^2*f
)*g*n*log(F))*F^(f*g*n*x + e*g*n))*log(F^(f*g*n*x + e*g*n)*b + a) + 2*(2*(a^2*d^
3*f^3*g^3*n^3*x^3 + 3*a^2*c*d^2*f^3*g^3*n^3*x^2 + 3*a^2*c^2*d*f^3*g^3*n^3*x + (a
^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2)*g^3*n^3)*log(F)^3 - 9*(a^2*d
^3*f^2*g^2*n^2*x^2 + 2*a^2*c*d^2*f^2*g^2*n^2*x - (a^2*d^3*e^2 - 2*a^2*c*d^2*e*f)
*g^2*n^2)*log(F)^2 + (2*(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*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^
2)*g^3*n^3)*log(F)^3 - 9*(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*d^3*e^2 - 2*b^2*c*d^2*e*f)*g^2*n^2)*log(F)^2 + 6*(b^2*d^3*f*g*n*x + b^2*d^3
*e*g*n)*log(F))*F^(2*f*g*n*x + 2*e*g*n) + 2*(2*(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*d^3*e^3 - 3*a*b*c*d^2*e
^2*f + 3*a*b*c^2*d*e*f^2)*g^3*n^3)*log(F)^3 - 9*(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*d^3*e^2 - 2*a*b*c*d^2*e*f)*g^2*n^2)*log(F)^2 + 6*(a*
b*d^3*f*g*n*x + a*b*d^3*e*g*n)*log(F))*F^(f*g*n*x + e*g*n) + 6*(a^2*d^3*f*g*n*x
+ a^2*d^3*e*g*n)*log(F))*log((F^(f*g*n*x + e*g*n)*b + a)/a) + 24*(2*F^(f*g*n*x +
e*g*n)*a*b*d^3 + F^(2*f*g*n*x + 2*e*g*n)*b^2*d^3 + a^2*d^3)*polylog(4, -F^(f*g*
n*x + e*g*n)*b/a) + 12*(3*a^2*d^3 + (3*b^2*d^3 - 2*(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) + 2*(3*a*b*d^3 - 2*(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) - 2*(a^2*d^3*f*g*n*x + a^2*c*d^2*f*g*n
)*log(F))*polylog(3, -F^(f*g*n*x + e*g*n)*b/a))/(2*F^(f*g*n*x + e*g*n)*a^4*b*f^4
*g^4*n^4*log(F)^4 + F^(2*f*g*n*x + 2*e*g*n)*a^3*b^2*f^4*g^4*n^4*log(F)^4 + a^5*f
^4*g^4*n^4*log(F)^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*a*c**3*f*g*n*log(F) + 9*a*c**2*d*f*g*n*x*log(F) - 3*a*c**2*d + 9*a*c*d**2*f*g
*n*x**2*log(F) - 6*a*c*d**2*x + 3*a*d**3*f*g*n*x**3*log(F) - 3*a*d**3*x**2 + (2*
b*c**3*f*g*n*log(F) + 6*b*c**2*d*f*g*n*x*log(F) - 3*b*c**2*d + 6*b*c*d**2*f*g*n*
x**2*log(F) - 6*b*c*d**2*x + 2*b*d**3*f*g*n*x**3*log(F) - 3*b*d**3*x**2)*(F**(g*
(e + f*x)))**n)/(2*a**4*f**2*g**2*n**2*log(F)**2 + 4*a**3*b*f**2*g**2*n**2*(F**(
g*(e + f*x)))**n*log(F)**2 + 2*a**2*b**2*f**2*g**2*n**2*(F**(g*(e + f*x)))**(2*n
)*log(F)**2) + (Integral(6*c*d**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))),
x) + Integral(6*d**3*x/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Inte
gral(2*c**3*f**2*g**2*n**2*log(F)**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F)
)), x) + Integral(-9*c**2*d*f*g*n*log(F)/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*lo
g(F))), x) + Integral(-9*d**3*f*g*n*x**2*log(F)/(a + b*exp(e*g*n*log(F))*exp(f*g
*n*x*log(F))), x) + Integral(2*d**3*f**2*g**2*n**2*x**3*log(F)**2/(a + b*exp(e*g
*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(-18*c*d**2*f*g*n*x*log(F)/(a + b*
exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(6*c*d**2*f**2*g**2*n**2*x*
*2*log(F)**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(6*c**2
*d*f**2*g**2*n**2*x*log(F)**2/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x))
/(2*a**2*f**2*g**2*n**2*log(F)**2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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