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

Optimal. Leaf size=305 \[ -\frac{a^3}{d (c+d x)}+\frac{3 a^2 b f g n \log (F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{3 a^2 b \left (F^{e g+f g x}\right )^n}{d (c+d x)}+\frac{6 a b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)}+\frac{3 b^3 f g n \log (F) \left (F^{e g+f g x}\right )^{3 n} F^{3 g n \left (e-\frac{c f}{d}\right )-3 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{3 f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{b^3 \left (F^{e g+f g x}\right )^{3 n}}{d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.849095, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{a^3}{d (c+d x)}+\frac{3 a^2 b f g n \log (F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{3 a^2 b \left (F^{e g+f g x}\right )^n}{d (c+d x)}+\frac{6 a b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)}+\frac{3 b^3 f g n \log (F) \left (F^{e g+f g x}\right )^{3 n} F^{3 g n \left (e-\frac{c f}{d}\right )-3 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{3 f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{b^3 \left (F^{e g+f g x}\right )^{3 n}}{d (c+d x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 73.3235, size = 313, normalized size = 1.03 \[ \frac{3 F^{g n \left (- 3 e - 3 f x\right )} F^{- \frac{3 g n \left (c f - d e\right )}{d}} b^{3} f g n \left (F^{g \left (e + f x\right )}\right )^{3 n} \log{\left (F \right )} \operatorname{Ei}{\left (\frac{f g n \left (3 c + 3 d x\right ) \log{\left (F \right )}}{d} \right )}}{d^{2}} + \frac{6 F^{g n \left (- 2 e - 2 f x\right )} F^{- \frac{2 g n \left (c f - d e\right )}{d}} a b^{2} f g n \left (F^{g \left (e + f x\right )}\right )^{2 n} \log{\left (F \right )} \operatorname{Ei}{\left (\frac{f g n \left (2 c + 2 d x\right ) \log{\left (F \right )}}{d} \right )}}{d^{2}} + \frac{3 F^{g n \left (- e - f x\right )} F^{- \frac{g n \left (c f - d e\right )}{d}} a^{2} b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log{\left (F \right )} \operatorname{Ei}{\left (\frac{f g n \left (c + d x\right ) \log{\left (F \right )}}{d} \right )}}{d^{2}} - \frac{a^{3}}{d \left (c + d x\right )} - \frac{3 a^{2} b \left (F^{g \left (e + f x\right )}\right )^{n}}{d \left (c + d x\right )} - \frac{3 a b^{2} \left (F^{g \left (e + f x\right )}\right )^{2 n}}{d \left (c + d x\right )} - \frac{b^{3} \left (F^{g \left (e + f x\right )}\right )^{3 n}}{d \left (c + d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.88003, size = 250, normalized size = 0.82 \[ -\frac{a^3 d-3 a^2 b f g n \log (F) (c+d x) \left (F^{g (e+f x)}\right )^n F^{-\frac{f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )+3 a^2 b d \left (F^{g (e+f x)}\right )^n-6 a b^2 f g n \log (F) (c+d x) \left (F^{g (e+f x)}\right )^{2 n} F^{-\frac{2 f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )+3 a b^2 d \left (F^{g (e+f x)}\right )^{2 n}-3 b^3 f g n \log (F) (c+d x) \left (F^{g (e+f x)}\right )^{3 n} F^{-\frac{3 f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{3 f g n \log (F) (c+d x)}{d}\right )+b^3 d \left (F^{g (e+f x)}\right )^{3 n}}{d^2 (c+d x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{3}}{ \left ( dx+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.28413, size = 350, normalized size = 1.15 \[ -\frac{3 \, F^{f g n x + e g n} a^{2} b d + 3 \, F^{2 \, f g n x + 2 \, e g n} a b^{2} d + F^{3 \, f g n x + 3 \, e g n} b^{3} d + a^{3} d - 3 \,{\left (b^{3} d f g n x + b^{3} c f g n\right )} F^{\frac{3 \,{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{3 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right ) - 6 \,{\left (a b^{2} d f g n x + a b^{2} c f g n\right )} F^{\frac{2 \,{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{2 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right ) - 3 \,{\left (a^{2} b d f g n x + a^{2} b c f g n\right )} F^{\frac{{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right )}{d^{3} x + c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)**3/(d*x+c)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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