3.320 \(\int \frac{F^{a+\frac{b}{(c+d x)^2}}}{c+d x} \, dx\)

Optimal. Leaf size=22 \[ -\frac{F^a \text{ExpIntegralEi}\left (\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

[Out]

-(F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)^2])/(2*d)

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Rubi [A]  time = 0.0713568, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{F^a \text{ExpIntegralEi}\left (\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

-(F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)^2])/(2*d)

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Rubi in Sympy [A]  time = 4.86458, size = 20, normalized size = 0.91 \[ - \frac{F^{a} \operatorname{Ei}{\left (\frac{b \log{\left (F \right )}}{\left (c + d x\right )^{2}} \right )}}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-F**a*Ei(b*log(F)/(c + d*x)**2)/(2*d)

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Mathematica [A]  time = 0.00934062, size = 22, normalized size = 1. \[ -\frac{F^a \text{ExpIntegralEi}\left (\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

-(F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)^2])/(2*d)

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Maple [A]  time = 0.029, size = 23, normalized size = 1.1 \[{\frac{{F}^{a}}{2\,d}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{ \left ( dx+c \right ) ^{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/d*F^a*Ei(1,-b*ln(F)/(d*x+c)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c), x)

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Fricas [A]  time = 0.26512, size = 42, normalized size = 1.91 \[ -\frac{F^{a}{\rm Ei}\left (\frac{b \log \left (F\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*F^a*Ei(b*log(F)/(d^2*x^2 + 2*c*d*x + c^2))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{c + d x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b/(d*x+c)**2)/(d*x+c),x)

[Out]

Integral(F**(a + b/(c + d*x)**2)/(c + d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c), x)