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

Optimal. Leaf size=153 \[ \frac{a (c+d x)^4}{4 d}+\frac{6 b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac{3 b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{6 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)} \]

[Out]

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

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Rubi [A]  time = 0.412296, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{a (c+d x)^4}{4 d}+\frac{6 b d^2 (c+d x) \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)}-\frac{3 b d (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{b (c+d x)^3 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{6 b d^3 \left (F^{e g+f g x}\right )^n}{f^4 g^4 n^4 \log ^4(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 45.5903, size = 144, normalized size = 0.94 \[ \frac{a \left (c + d x\right )^{4}}{4 d} - \frac{6 b d^{3} \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{4} g^{4} n^{4} \log{\left (F \right )}^{4}} + \frac{6 b d^{2} \left (c + d x\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{3} g^{3} n^{3} \log{\left (F \right )}^{3}} - \frac{3 b d \left (c + d x\right )^{2} \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} + \frac{b \left (c + d x\right )^{3} \left (F^{g \left (e + f x\right )}\right )^{n}}{f g n \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.209358, size = 130, normalized size = 0.85 \[ a c^3 x+\frac{3}{2} a c^2 d x^2+a c d^2 x^3+\frac{1}{4} a d^3 x^4+\frac{b \left (F^{g (e+f x)}\right )^n \left (6 d^2 f g n \log (F) (c+d x)+f^3 g^3 n^3 \log ^3(F) (c+d x)^3-3 d f^2 g^2 n^2 \log ^2(F) (c+d x)^2-6 d^3\right )}{f^4 g^4 n^4 \log ^4(F)} \]

Antiderivative was successfully verified.

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

[Out]

a*c^3*x + (3*a*c^2*d*x^2)/2 + a*c*d^2*x^3 + (a*d^3*x^4)/4 + (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)

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) \left ( dx+c \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)*(d*x + c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262087, size = 360, normalized size = 2.35 \[ \frac{{\left (a d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, a c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, a c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, a c^{3} f^{4} g^{4} n^{4} x\right )} \log \left (F\right )^{4} - 4 \,{\left (6 \, b d^{3} -{\left (b d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, b c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, b c^{2} d f^{3} g^{3} n^{3} x + b c^{3} f^{3} g^{3} n^{3}\right )} \log \left (F\right )^{3} + 3 \,{\left (b d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, b c d^{2} f^{2} g^{2} n^{2} x + b c^{2} d f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 6 \,{\left (b d^{3} f g n x + b c d^{2} f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{4 \, f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.655762, size = 332, normalized size = 2.17 \[ a c^{3} x + \frac{3 a c^{2} d x^{2}}{2} + a c d^{2} x^{3} + \frac{a d^{3} x^{4}}{4} + \begin{cases} \frac{\left (b c^{3} f^{3} g^{3} n^{3} \log{\left (F \right )}^{3} + 3 b c^{2} d f^{3} g^{3} n^{3} x \log{\left (F \right )}^{3} - 3 b c^{2} d f^{2} g^{2} n^{2} \log{\left (F \right )}^{2} + 3 b c d^{2} f^{3} g^{3} n^{3} x^{2} \log{\left (F \right )}^{3} - 6 b c d^{2} f^{2} g^{2} n^{2} x \log{\left (F \right )}^{2} + 6 b c d^{2} f g n \log{\left (F \right )} + b d^{3} f^{3} g^{3} n^{3} x^{3} \log{\left (F \right )}^{3} - 3 b d^{3} f^{2} g^{2} n^{2} x^{2} \log{\left (F \right )}^{2} + 6 b d^{3} f g n x \log{\left (F \right )} - 6 b d^{3}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{4} g^{4} n^{4} \log{\left (F \right )}^{4}} & \text{for}\: f^{4} g^{4} n^{4} \log{\left (F \right )}^{4} \neq 0 \\b c^{3} x + \frac{3 b c^{2} d x^{2}}{2} + b c d^{2} x^{3} + \frac{b d^{3} x^{4}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**3*x + 3*a*c**2*d*x**2/2 + a*c*d**2*x**3 + a*d**3*x**4/4 + Piecewise(((b*c**
3*f**3*g**3*n**3*log(F)**3 + 3*b*c**2*d*f**3*g**3*n**3*x*log(F)**3 - 3*b*c**2*d*
f**2*g**2*n**2*log(F)**2 + 3*b*c*d**2*f**3*g**3*n**3*x**2*log(F)**3 - 6*b*c*d**2
*f**2*g**2*n**2*x*log(F)**2 + 6*b*c*d**2*f*g*n*log(F) + b*d**3*f**3*g**3*n**3*x*
*3*log(F)**3 - 3*b*d**3*f**2*g**2*n**2*x**2*log(F)**2 + 6*b*d**3*f*g*n*x*log(F)
- 6*b*d**3)*(F**(g*(e + f*x)))**n/(f**4*g**4*n**4*log(F)**4), Ne(f**4*g**4*n**4*
log(F)**4, 0)), (b*c**3*x + 3*b*c**2*d*x**2/2 + b*c*d**2*x**3 + b*d**3*x**4/4, T
rue))

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GIAC/XCAS [A]  time = 0.318312, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done