3.397 \(\int \frac{F^{a+\frac{b}{c+d x}}}{e+f x} \, dx\)

Optimal. Leaf size=71 \[ \frac{F^{a-\frac{b f}{d e-c f}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )}{f}-\frac{F^a \text{ExpIntegralEi}\left (\frac{b \log (F)}{c+d x}\right )}{f} \]

[Out]

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

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Rubi [A]  time = 0.620657, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{F^{a-\frac{b f}{d e-c f}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )}{f}-\frac{F^a \text{ExpIntegralEi}\left (\frac{b \log (F)}{c+d x}\right )}{f} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.4203, size = 65, normalized size = 0.92 \[ - \frac{F^{a} \operatorname{Ei}{\left (\frac{b \log{\left (F \right )}}{c + d x} \right )}}{f} + \frac{F^{\frac{a \left (c f - d e\right ) + b f}{c f - d e}} \operatorname{Ei}{\left (- \frac{b d \left (e + f x\right ) \log{\left (F \right )}}{\left (c + d x\right ) \left (c f - d e\right )} \right )}}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0864028, size = 66, normalized size = 0.93 \[ \frac{F^a \left (F^{\frac{b f}{c f-d e}} \text{ExpIntegralEi}\left (\frac{b d \log (F) (e+f x)}{(c+d x) (d e-c f)}\right )-\text{ExpIntegralEi}\left (\frac{b \log (F)}{c+d x}\right )\right )}{f} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*(-ExpIntegralEi[(b*Log[F])/(c + d*x)] + F^((b*f)/(-(d*e) + c*f))*ExpIntegra
lEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]))/f

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Maple [A]  time = 0.043, size = 106, normalized size = 1.5 \[ -{\frac{1}{f}{F}^{{\frac{acf-ade+bf}{cf-ed}}}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}}-\ln \left ( F \right ) a-{\frac{-\ln \left ( F \right ) acf+\ln \left ( F \right ) ade-\ln \left ( F \right ) bf}{cf-ed}} \right ) }+{\frac{{F}^{a}}{f}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b/(d*x+c))/(f*x+e),x)

[Out]

-1/f*F^((a*c*f-a*d*e+b*f)/(c*f-d*e))*Ei(1,-b*ln(F)/(d*x+c)-ln(F)*a-(-ln(F)*a*c*f
+ln(F)*a*d*e-ln(F)*b*f)/(c*f-d*e))+1/f*F^a*Ei(1,-b*ln(F)/(d*x+c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + \frac{b}{d x + c}}}{f x + e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.282594, size = 120, normalized size = 1.69 \[ \frac{F^{\frac{a d e -{\left (a c + b\right )} f}{d e - c f}}{\rm Ei}\left (\frac{{\left (b d f x + b d e\right )} \log \left (F\right )}{c d e - c^{2} f +{\left (d^{2} e - c d f\right )} x}\right ) - F^{a}{\rm Ei}\left (\frac{b \log \left (F\right )}{d x + c}\right )}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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