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

Optimal. Leaf size=115 \[ \frac{a (c+d x)^3}{3 d}-\frac{2 b d (c+d x) \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{b (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac{2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)} \]

[Out]

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

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Rubi [A]  time = 0.268511, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{a (c+d x)^3}{3 d}-\frac{2 b d (c+d x) \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac{b (c+d x)^2 \left (F^{e g+f g x}\right )^n}{f g n \log (F)}+\frac{2 b d^2 \left (F^{e g+f g x}\right )^n}{f^3 g^3 n^3 \log ^3(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.9089, size = 105, normalized size = 0.91 \[ \frac{a \left (c + d x\right )^{3}}{3 d} + \frac{2 b d^{2} \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{3} g^{3} n^{3} \log{\left (F \right )}^{3}} - \frac{2 b d \left (c + d x\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} + \frac{b \left (c + d x\right )^{2} \left (F^{g \left (e + f x\right )}\right )^{n}}{f g n \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.145187, size = 91, normalized size = 0.79 \[ a c^2 x+a c d x^2+\frac{1}{3} a d^2 x^3+\frac{b \left (F^{g (e+f x)}\right )^n \left (f^2 g^2 n^2 \log ^2(F) (c+d x)^2-2 d f g n \log (F) (c+d x)+2 d^2\right )}{f^3 g^3 n^3 \log ^3(F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.046, size = 199, normalized size = 1.7 \[ a{c}^{2}x+acd{x}^{2}+{\frac{b{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}{c}^{2}}{ngf\ln \left ( F \right ) }}-2\,{\frac{b{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}cd}{ \left ( \ln \left ( F \right ) \right ) ^{2}{f}^{2}{g}^{2}{n}^{2}}}+2\,{\frac{b{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}{d}^{2}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{f}^{3}{g}^{3}{n}^{3}}}+{\frac{b{d}^{2}{x}^{2}{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}}{ngf\ln \left ( F \right ) }}+{\frac{a{d}^{2}{x}^{3}}{3}}+2\,{\frac{bd \left ( \ln \left ( F \right ) cfgn-d \right ) x{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}}{ \left ( \ln \left ( F \right ) \right ) ^{2}{f}^{2}{g}^{2}{n}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261793, size = 224, normalized size = 1.95 \[ \frac{{\left (a d^{2} f^{3} g^{3} n^{3} x^{3} + 3 \, a c d f^{3} g^{3} n^{3} x^{2} + 3 \, a c^{2} f^{3} g^{3} n^{3} x\right )} \log \left (F\right )^{3} + 3 \,{\left (2 \, b d^{2} +{\left (b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b c d f^{2} g^{2} n^{2} x + b c^{2} f^{2} g^{2} n^{2}\right )} \log \left (F\right )^{2} - 2 \,{\left (b d^{2} f g n x + b c d f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{3 \, f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.514271, size = 196, normalized size = 1.7 \[ a c^{2} x + a c d x^{2} + \frac{a d^{2} x^{3}}{3} + \begin{cases} \frac{\left (b c^{2} f^{2} g^{2} n^{2} \log{\left (F \right )}^{2} + 2 b c d f^{2} g^{2} n^{2} x \log{\left (F \right )}^{2} - 2 b c d f g n \log{\left (F \right )} + b d^{2} f^{2} g^{2} n^{2} x^{2} \log{\left (F \right )}^{2} - 2 b d^{2} f g n x \log{\left (F \right )} + 2 b d^{2}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{3} g^{3} n^{3} \log{\left (F \right )}^{3}} & \text{for}\: f^{3} g^{3} n^{3} \log{\left (F \right )}^{3} \neq 0 \\b c^{2} x + b c d x^{2} + \frac{b d^{2} x^{3}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.277035, 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)^2,x, algorithm="giac")

[Out]

Done