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

Optimal. Leaf size=46 \[ -\frac{\sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{2 \sqrt{b} d \sqrt{\log (F)}} \]

[Out]

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

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Rubi [A]  time = 0.0915427, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{\sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{2 \sqrt{b} d \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.39149, size = 42, normalized size = 0.91 \[ - \frac{\sqrt{\pi } F^{a} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (F \right )}}}{c + d x} \right )}}{2 \sqrt{b} d \sqrt{\log{\left (F \right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(log(F))/(c + d*x))/(2*sqrt(b)*d*sqrt(log(F)))

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Mathematica [A]  time = 0.0130025, size = 46, normalized size = 1. \[ -\frac{\sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{2 \sqrt{b} d \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.037, size = 35, normalized size = 0.8 \[ -{\frac{\sqrt{\pi }{F}^{a}}{2\,d}{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-b\ln \left ( F \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.258082, size = 55, normalized size = 1.2 \[ -\frac{\sqrt{\pi } F^{a} \operatorname{erf}\left (\frac{d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{d x + c}\right )}{2 \, d^{2} \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(pi)*F^a*erf(d*sqrt(-b*log(F)/d^2)/(d*x + c))/(d^2*sqrt(-b*log(F)/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(F**(a+b/(d*x+c)**2)/(d*x+c)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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