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

Optimal. Leaf size=322 \[ \frac{a^2 (c+d x)^4}{4 d}+\frac{12 a b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac{6 a b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{2 a b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{12 a b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}+\frac{3 b^2 d^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac{3 b^2 d (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)}+\frac{b^2 (c+d x)^3 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac{3 b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)} \]

[Out]

(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*Log[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])

_______________________________________________________________________________________

Rubi [A]  time = 0.831167, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{a^2 (c+d x)^4}{4 d}+\frac{12 a b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac{6 a b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{2 a b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{12 a b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)}+\frac{3 b^2 d^2 (c+d x) \left (F^{e g+f g x}\right )^{2 n}}{4 f^3 g^3 n^3 \log ^3(F)}-\frac{3 b^2 d (c+d x)^2 \left (F^{e g+f g x}\right )^{2 n}}{4 f^2 g^2 n^2 \log ^2(F)}+\frac{b^2 (c+d x)^3 \left (F^{e g+f g x}\right )^{2 n}}{2 f g n \log (F)}-\frac{3 b^2 d^3 \left (F^{e g+f g x}\right )^{2 n}}{8 f^4 g^4 n^4 \log ^4(F)} \]

Antiderivative was successfully verified.

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

[Out]

(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*Log[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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 109.006, size = 309, normalized size = 0.96 \[ \frac{a^{2} \left (c + d x\right )^{4}}{4 d} - \frac{12 a b d^{3} \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{4} g^{4} n^{4} \log{\left (F \right )}^{4}} + \frac{12 a b d^{2} \left (c + d x\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{3} g^{3} n^{3} \log{\left (F \right )}^{3}} - \frac{6 a b d \left (c + d x\right )^{2} \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} + \frac{2 a b \left (c + d x\right )^{3} \left (F^{g \left (e + f x\right )}\right )^{n}}{f g n \log{\left (F \right )}} - \frac{3 b^{2} d^{3} \left (F^{g \left (e + f x\right )}\right )^{2 n}}{8 f^{4} g^{4} n^{4} \log{\left (F \right )}^{4}} + \frac{3 b^{2} d^{2} \left (c + d x\right ) \left (F^{g \left (e + f x\right )}\right )^{2 n}}{4 f^{3} g^{3} n^{3} \log{\left (F \right )}^{3}} - \frac{3 b^{2} d \left (c + d x\right )^{2} \left (F^{g \left (e + f x\right )}\right )^{2 n}}{4 f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} + \frac{b^{2} \left (c + d x\right )^{3} \left (F^{g \left (e + f x\right )}\right )^{2 n}}{2 f g n \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(c + d*x)**4/(4*d) - 12*a*b*d**3*(F**(g*(e + f*x)))**n/(f**4*g**4*n**4*log(
F)**4) + 12*a*b*d**2*(c + d*x)*(F**(g*(e + f*x)))**n/(f**3*g**3*n**3*log(F)**3)
- 6*a*b*d*(c + d*x)**2*(F**(g*(e + f*x)))**n/(f**2*g**2*n**2*log(F)**2) + 2*a*b*
(c + d*x)**3*(F**(g*(e + f*x)))**n/(f*g*n*log(F)) - 3*b**2*d**3*(F**(g*(e + f*x)
))**(2*n)/(8*f**4*g**4*n**4*log(F)**4) + 3*b**2*d**2*(c + d*x)*(F**(g*(e + f*x))
)**(2*n)/(4*f**3*g**3*n**3*log(F)**3) - 3*b**2*d*(c + d*x)**2*(F**(g*(e + f*x)))
**(2*n)/(4*f**2*g**2*n**2*log(F)**2) + b**2*(c + d*x)**3*(F**(g*(e + f*x)))**(2*
n)/(2*f*g*n*log(F))

_______________________________________________________________________________________

Mathematica [A]  time = 0.3766, size = 239, normalized size = 0.74 \[ a^2 c^3 x+\frac{3}{2} a^2 c^2 d x^2+a^2 c d^2 x^3+\frac{1}{4} a^2 d^3 x^4+\frac{2 a b \left (F^{g (e+f x)}\right )^n \left (6 d^2 f g n \log (F) (c+d x)+f^3 g^3 n^3 \log ^3(F) (c+d x)^3-3 d f^2 g^2 n^2 \log ^2(F) (c+d x)^2-6 d^3\right )}{f^4 g^4 n^4 \log ^4(F)}+\frac{b^2 \left (F^{g (e+f x)}\right )^{2 n} \left (6 d^2 f g n \log (F) (c+d x)+4 f^3 g^3 n^3 \log ^3(F) (c+d x)^3-6 d f^2 g^2 n^2 \log ^2(F) (c+d x)^2-3 d^3\right )}{8 f^4 g^4 n^4 \log ^4(F)} \]

Antiderivative was successfully verified.

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

[Out]

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)

_______________________________________________________________________________________

Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{2} \left ( dx+c \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.267489, size = 651, normalized size = 2.02 \[ \frac{2 \,{\left (a^{2} d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, a^{2} c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, a^{2} c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, a^{2} c^{3} f^{4} g^{4} n^{4} x\right )} \log \left (F\right )^{4} -{\left (3 \, b^{2} d^{3} - 4 \,{\left (b^{2} d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, b^{2} c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, b^{2} c^{2} d f^{3} g^{3} n^{3} x + b^{2} c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 6 \,{\left (b^{2} d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c d^{2} f^{2} g^{2} n^{2} x + b^{2} c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \,{\left (b^{2} d^{3} f g n x + b^{2} c d^{2} f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} - 16 \,{\left (6 \, a b d^{3} -{\left (a b d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, a b c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, a b c^{2} d f^{3} g^{3} n^{3} x + a b c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 3 \,{\left (a b d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, a b c d^{2} f^{2} g^{2} n^{2} x + a b c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \,{\left (a b d^{3} f g n x + a b c d^{2} f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{8 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)*lo
g(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)

_______________________________________________________________________________________

Sympy [A]  time = 1.15785, size = 709, normalized size = 2.2 \[ a^{2} c^{3} x + \frac{3 a^{2} c^{2} d x^{2}}{2} + a^{2} c d^{2} x^{3} + \frac{a^{2} d^{3} x^{4}}{4} + \begin{cases} \frac{\left (4 b^{2} c^{3} f^{7} g^{7} n^{7} \log{\left (F \right )}^{7} + 12 b^{2} c^{2} d f^{7} g^{7} n^{7} x \log{\left (F \right )}^{7} - 6 b^{2} c^{2} d f^{6} g^{6} n^{6} \log{\left (F \right )}^{6} + 12 b^{2} c d^{2} f^{7} g^{7} n^{7} x^{2} \log{\left (F \right )}^{7} - 12 b^{2} c d^{2} f^{6} g^{6} n^{6} x \log{\left (F \right )}^{6} + 6 b^{2} c d^{2} f^{5} g^{5} n^{5} \log{\left (F \right )}^{5} + 4 b^{2} d^{3} f^{7} g^{7} n^{7} x^{3} \log{\left (F \right )}^{7} - 6 b^{2} d^{3} f^{6} g^{6} n^{6} x^{2} \log{\left (F \right )}^{6} + 6 b^{2} d^{3} f^{5} g^{5} n^{5} x \log{\left (F \right )}^{5} - 3 b^{2} d^{3} f^{4} g^{4} n^{4} \log{\left (F \right )}^{4}\right ) \left (F^{g \left (e + f x\right )}\right )^{2 n} + \left (16 a b c^{3} f^{7} g^{7} n^{7} \log{\left (F \right )}^{7} + 48 a b c^{2} d f^{7} g^{7} n^{7} x \log{\left (F \right )}^{7} - 48 a b c^{2} d f^{6} g^{6} n^{6} \log{\left (F \right )}^{6} + 48 a b c d^{2} f^{7} g^{7} n^{7} x^{2} \log{\left (F \right )}^{7} - 96 a b c d^{2} f^{6} g^{6} n^{6} x \log{\left (F \right )}^{6} + 96 a b c d^{2} f^{5} g^{5} n^{5} \log{\left (F \right )}^{5} + 16 a b d^{3} f^{7} g^{7} n^{7} x^{3} \log{\left (F \right )}^{7} - 48 a b d^{3} f^{6} g^{6} n^{6} x^{2} \log{\left (F \right )}^{6} + 96 a b d^{3} f^{5} g^{5} n^{5} x \log{\left (F \right )}^{5} - 96 a b d^{3} f^{4} g^{4} n^{4} \log{\left (F \right )}^{4}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{8 f^{8} g^{8} n^{8} \log{\left (F \right )}^{8}} & \text{for}\: 8 f^{8} g^{8} n^{8} \log{\left (F \right )}^{8} \neq 0 \\x^{4} \left (\frac{a b d^{3}}{2} + \frac{b^{2} d^{3}}{4}\right ) + x^{3} \left (2 a b c d^{2} + b^{2} c d^{2}\right ) + x^{2} \left (3 a b c^{2} d + \frac{3 b^{2} c^{2} d}{2}\right ) + x \left (2 a b c^{3} + b^{2} c^{3}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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 + Piece
wise((((4*b**2*c**3*f**7*g**7*n**7*log(F)**7 + 12*b**2*c**2*d*f**7*g**7*n**7*x*l
og(F)**7 - 6*b**2*c**2*d*f**6*g**6*n**6*log(F)**6 + 12*b**2*c*d**2*f**7*g**7*n**
7*x**2*log(F)**7 - 12*b**2*c*d**2*f**6*g**6*n**6*x*log(F)**6 + 6*b**2*c*d**2*f**
5*g**5*n**5*log(F)**5 + 4*b**2*d**3*f**7*g**7*n**7*x**3*log(F)**7 - 6*b**2*d**3*
f**6*g**6*n**6*x**2*log(F)**6 + 6*b**2*d**3*f**5*g**5*n**5*x*log(F)**5 - 3*b**2*
d**3*f**4*g**4*n**4*log(F)**4)*(F**(g*(e + f*x)))**(2*n) + (16*a*b*c**3*f**7*g**
7*n**7*log(F)**7 + 48*a*b*c**2*d*f**7*g**7*n**7*x*log(F)**7 - 48*a*b*c**2*d*f**6
*g**6*n**6*log(F)**6 + 48*a*b*c*d**2*f**7*g**7*n**7*x**2*log(F)**7 - 96*a*b*c*d*
*2*f**6*g**6*n**6*x*log(F)**6 + 96*a*b*c*d**2*f**5*g**5*n**5*log(F)**5 + 16*a*b*
d**3*f**7*g**7*n**7*x**3*log(F)**7 - 48*a*b*d**3*f**6*g**6*n**6*x**2*log(F)**6 +
96*a*b*d**3*f**5*g**5*n**5*x*log(F)**5 - 96*a*b*d**3*f**4*g**4*n**4*log(F)**4)*
(F**(g*(e + f*x)))**n)/(8*f**8*g**8*n**8*log(F)**8), Ne(8*f**8*g**8*n**8*log(F)*
*8, 0)), (x**4*(a*b*d**3/2 + b**2*d**3/4) + x**3*(2*a*b*c*d**2 + b**2*c*d**2) +
x**2*(3*a*b*c**2*d + 3*b**2*c**2*d/2) + x*(2*a*b*c**3 + b**2*c**3), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.446292, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done