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

Optimal. Leaf size=100 \[ -\frac{a}{d (c+d x)}+\frac{b f g n \log (F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{b \left (F^{e g+f g x}\right )^n}{d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.286518, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{a}{d (c+d x)}+\frac{b f g n \log (F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d^2}-\frac{b \left (F^{e g+f g x}\right )^n}{d (c+d x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.44, size = 92, normalized size = 0.92 \[ \frac{F^{g n \left (- e - f x\right )} F^{- \frac{g n \left (c f - d e\right )}{d}} b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log{\left (F \right )} \operatorname{Ei}{\left (\frac{f g n \left (c + d x\right ) \log{\left (F \right )}}{d} \right )}}{d^{2}} - \frac{a}{d \left (c + d x\right )} - \frac{b \left (F^{g \left (e + f x\right )}\right )^{n}}{d \left (c + d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

F**(g*n*(-e - f*x))*F**(-g*n*(c*f - d*e)/d)*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/(d*(c + d*x)) - b*(F**(g*(e + f*x)))**n/(
d*(c + d*x))

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Mathematica [A]  time = 0.303208, size = 78, normalized size = 0.78 \[ \frac{b f g n \log (F) \left (F^{g (e+f x)}\right )^n F^{-\frac{f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )-\frac{d \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{c+d x}}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n}}{ \left ( dx+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.27773, size = 117, normalized size = 1.17 \[ \frac{{\left (b d f g n x + b c f g n\right )} F^{\frac{{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right ) - F^{f g n x + e g n} b d - a d}{d^{3} x + c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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