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

Optimal. Leaf size=286 \[ -\frac{a^2}{2 d (c+d x)^2}+\frac{a b f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d^3}-\frac{a b f g n \log (F) \left (F^{e g+f g x}\right )^n}{d^2 (c+d x)}-\frac{a b \left (F^{e g+f g x}\right )^n}{d (c+d x)^2}+\frac{2 b^2 f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )}{d^3}-\frac{b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n}}{d^2 (c+d x)}-\frac{b^2 \left (F^{e g+f g x}\right )^{2 n}}{2 d (c+d x)^2} \]

[Out]

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

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Rubi [A]  time = 0.81929, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{a^2}{2 d (c+d x)^2}+\frac{a b f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d^3}-\frac{a b f g n \log (F) \left (F^{e g+f g x}\right )^n}{d^2 (c+d x)}-\frac{a b \left (F^{e g+f g x}\right )^n}{d (c+d x)^2}+\frac{2 b^2 f^2 g^2 n^2 \log ^2(F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )}{d^3}-\frac{b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n}}{d^2 (c+d x)}-\frac{b^2 \left (F^{e g+f g x}\right )^{2 n}}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 78.4537, size = 284, normalized size = 0.99 \[ \frac{2 F^{g n \left (- 2 e - 2 f x\right )} F^{- \frac{2 g n \left (c f - d e\right )}{d}} b^{2} f^{2} g^{2} n^{2} \left (F^{g \left (e + f x\right )}\right )^{2 n} \log{\left (F \right )}^{2} \operatorname{Ei}{\left (\frac{f g n \left (2 c + 2 d x\right ) \log{\left (F \right )}}{d} \right )}}{d^{3}} + \frac{F^{g n \left (- e - f x\right )} F^{- \frac{g n \left (c f - d e\right )}{d}} a b f^{2} g^{2} n^{2} \left (F^{g \left (e + f x\right )}\right )^{n} \log{\left (F \right )}^{2} \operatorname{Ei}{\left (\frac{f g n \left (c + d x\right ) \log{\left (F \right )}}{d} \right )}}{d^{3}} - \frac{a^{2}}{2 d \left (c + d x\right )^{2}} - \frac{a b \left (F^{g \left (e + f x\right )}\right )^{n}}{d \left (c + d x\right )^{2}} - \frac{a b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log{\left (F \right )}}{d^{2} \left (c + d x\right )} - \frac{b^{2} \left (F^{g \left (e + f x\right )}\right )^{2 n}}{2 d \left (c + d x\right )^{2}} - \frac{b^{2} f g n \left (F^{g \left (e + f x\right )}\right )^{2 n} \log{\left (F \right )}}{d^{2} \left (c + d x\right )} \]

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]

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

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Mathematica [A]  time = 0.601048, size = 217, normalized size = 0.76 \[ -\frac{a^2 d^2-2 a b f^2 g^2 n^2 \log ^2(F) (c+d x)^2 \left (F^{g (e+f x)}\right )^n F^{-\frac{f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )+2 a b d \left (F^{g (e+f x)}\right )^n (f g n \log (F) (c+d x)+d)-4 b^2 f^2 g^2 n^2 \log ^2(F) (c+d x)^2 \left (F^{g (e+f x)}\right )^{2 n} F^{-\frac{2 f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )+b^2 d \left (F^{g (e+f x)}\right )^{2 n} (2 f g n \log (F) (c+d x)+d)}{2 d^3 (c+d x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{2}}{ \left ( dx+c \right ) ^{3}}}\, dx \]

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)

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

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]

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

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Fricas [A]  time = 0.282474, size = 424, normalized size = 1.48 \[ \frac{4 \,{\left (b^{2} d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, b^{2} c d f^{2} g^{2} n^{2} x + b^{2} c^{2} f^{2} g^{2} n^{2}\right )} F^{\frac{2 \,{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{2 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right )^{2} + 2 \,{\left (a b d^{2} f^{2} g^{2} n^{2} x^{2} + 2 \, a b c d f^{2} g^{2} n^{2} x + a b c^{2} f^{2} g^{2} n^{2}\right )} F^{\frac{{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right )^{2} - a^{2} d^{2} -{\left (b^{2} d^{2} + 2 \,{\left (b^{2} d^{2} f g n x + b^{2} c d f g n\right )} \log \left (F\right )\right )} F^{2 \, f g n x + 2 \, e g n} - 2 \,{\left (a b d^{2} +{\left (a b d^{2} f g n x + a b c d f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{2 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]

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]

1/2*(4*(b^2*d^2*f^2*g^2*n^2*x^2 + 2*b^2*c*d*f^2*g^2*n^2*x + b^2*c^2*f^2*g^2*n^2)
*F^(2*(d*e - c*f)*g*n/d)*Ei(2*(d*f*g*n*x + c*f*g*n)*log(F)/d)*log(F)^2 + 2*(a*b*
d^2*f^2*g^2*n^2*x^2 + 2*a*b*c*d*f^2*g^2*n^2*x + a*b*c^2*f^2*g^2*n^2)*F^((d*e - c
*f)*g*n/d)*Ei((d*f*g*n*x + c*f*g*n)*log(F)/d)*log(F)^2 - a^2*d^2 - (b^2*d^2 + 2*
(b^2*d^2*f*g*n*x + b^2*c*d*f*g*n)*log(F))*F^(2*f*g*n*x + 2*e*g*n) - 2*(a*b*d^2 +
(a*b*d^2*f*g*n*x + a*b*c*d*f*g*n)*log(F))*F^(f*g*n*x + e*g*n))/(d^5*x^2 + 2*c*d
^4*x + c^2*d^3)

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

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]

Timed out

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

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]

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