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

Optimal. Leaf size=200 \[ \frac{a^3 \log (c+d x)}{d}+\frac{3 a^2 b \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d}+\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )}{d}+\frac{b^3 \left (F^{e g+f g x}\right )^{3 n} F^{3 g n \left (e-\frac{c f}{d}\right )-3 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{3 f g n \log (F) (c+d x)}{d}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.591831, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{a^3 \log (c+d x)}{d}+\frac{3 a^2 b \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d}+\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )}{d}+\frac{b^3 \left (F^{e g+f g x}\right )^{3 n} F^{3 g n \left (e-\frac{c f}{d}\right )-3 g n (e+f x)} \text{ExpIntegralEi}\left (\frac{3 f g n \log (F) (c+d x)}{d}\right )}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.4856, size = 211, normalized size = 1.05 \[ \frac{F^{g n \left (- 3 e - 3 f x\right )} F^{- \frac{3 g n \left (c f - d e\right )}{d}} b^{3} \left (F^{g \left (e + f x\right )}\right )^{3 n} \operatorname{Ei}{\left (\frac{f g n \left (3 c + 3 d x\right ) \log{\left (F \right )}}{d} \right )}}{d} + \frac{3 F^{g n \left (- 2 e - 2 f x\right )} F^{- \frac{2 g n \left (c f - d e\right )}{d}} a b^{2} \left (F^{g \left (e + f x\right )}\right )^{2 n} \operatorname{Ei}{\left (\frac{f g n \left (2 c + 2 d x\right ) \log{\left (F \right )}}{d} \right )}}{d} + \frac{3 F^{g n \left (- e - f x\right )} F^{- \frac{g n \left (c f - d e\right )}{d}} a^{2} b \left (F^{g \left (e + f x\right )}\right )^{n} \operatorname{Ei}{\left (\frac{f g n \left (c + d x\right ) \log{\left (F \right )}}{d} \right )}}{d} + \frac{a^{3} \log{\left (c + d x \right )}}{d} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.48527, size = 160, normalized size = 0.8 \[ \frac{a^3 \log (c+d x)+3 a^2 b \left (F^{g (e+f x)}\right )^n F^{-\frac{f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )+3 a b^2 \left (F^{g (e+f x)}\right )^{2 n} F^{-\frac{2 f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{2 f g n \log (F) (c+d x)}{d}\right )+b^3 \left (F^{g (e+f x)}\right )^{3 n} F^{-\frac{3 f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{3 f g n \log (F) (c+d x)}{d}\right )}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{3}}{dx+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.26588, size = 189, normalized size = 0.94 \[ \frac{F^{\frac{3 \,{\left (d e - c f\right )} g n}{d}} b^{3}{\rm Ei}\left (\frac{3 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + 3 \, F^{\frac{2 \,{\left (d e - c f\right )} g n}{d}} a b^{2}{\rm Ei}\left (\frac{2 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + 3 \, F^{\frac{{\left (d e - c f\right )} g n}{d}} a^{2} b{\rm Ei}\left (\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + a^{3} \log \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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