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

Optimal. Leaf size=68 \[ \frac{a \log (c+d x)}{d}+\frac{b \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d} \]

[Out]

(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 + (a*Log[c + d*x])/d

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Rubi [A]  time = 0.225586, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{a \log (c+d x)}{d}+\frac{b \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d} \]

Antiderivative was successfully verified.

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

[Out]

(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 + (a*Log[c + d*x])/d

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Rubi in Sympy [A]  time = 15.4127, size = 65, normalized size = 0.96 \[ \frac{F^{g n \left (- e - f x\right )} F^{- \frac{g n \left (c f - d e\right )}{d}} b \left (F^{g \left (e + f x\right )}\right )^{n} \operatorname{Ei}{\left (\frac{f g n \left (c + d x\right ) \log{\left (F \right )}}{d} \right )}}{d} + \frac{a \log{\left (c + d x \right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0988741, size = 56, normalized size = 0.82 \[ \frac{a \log (c+d x)+b \left (F^{g (e+f x)}\right )^n F^{-\frac{f g n (c+d x)}{d}} \text{ExpIntegralEi}\left (\frac{f g n \log (F) (c+d x)}{d}\right )}{d} \]

Antiderivative was successfully verified.

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

[Out]

((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) + a*Log[c + d*x])/d

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{\frac{a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n}}{dx+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.283091, size = 68, normalized size = 1. \[ \frac{F^{\frac{{\left (d e - c f\right )} g n}{d}} b{\rm Ei}\left (\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + a \log \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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