3.34 \(\int \frac{F^{a+b x}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=38 \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{\sqrt{b} \sqrt{\log (F)}} \]

[Out]

(F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(Sqrt[b]*Sqrt[Log[F]])

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Rubi [A]  time = 0.0436502, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{\sqrt{b} \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*x)/Sqrt[x],x]

[Out]

(F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(Sqrt[b]*Sqrt[Log[F]])

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Rubi in Sympy [A]  time = 5.11173, size = 37, normalized size = 0.97 \[ \frac{\sqrt{\pi } F^{a} \operatorname{erfi}{\left (\sqrt{b} \sqrt{x} \sqrt{\log{\left (F \right )}} \right )}}{\sqrt{b} \sqrt{\log{\left (F \right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(b*x+a)/x**(1/2),x)

[Out]

sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/(sqrt(b)*sqrt(log(F)))

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Mathematica [A]  time = 0.00747896, size = 38, normalized size = 1. \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{\sqrt{b} \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*x)/Sqrt[x],x]

[Out]

(F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(Sqrt[b]*Sqrt[Log[F]])

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Maple [A]  time = 0.019, size = 27, normalized size = 0.7 \[{{F}^{a}\sqrt{\pi }{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ){\frac{1}{\sqrt{b}}}{\frac{1}{\sqrt{\ln \left ( F \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(b*x+a)/x^(1/2),x)

[Out]

F^a*erfi(b^(1/2)*x^(1/2)*ln(F)^(1/2))*Pi^(1/2)/b^(1/2)/ln(F)^(1/2)

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Maxima [A]  time = 0.796128, size = 35, normalized size = 0.92 \[ \frac{\sqrt{\pi } F^{a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{\sqrt{-b \log \left (F\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(pi)*F^a*erf(sqrt(-b*log(F))*sqrt(x))/sqrt(-b*log(F))

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Fricas [A]  time = 0.269375, size = 35, normalized size = 0.92 \[ \frac{\sqrt{\pi } F^{a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{\sqrt{-b \log \left (F\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(pi)*F^a*erf(sqrt(-b*log(F))*sqrt(x))/sqrt(-b*log(F))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + b x}}{\sqrt{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(b*x+a)/x**(1/2),x)

[Out]

Integral(F**(a + b*x)/sqrt(x), x)

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GIAC/XCAS [A]  time = 0.257126, size = 41, normalized size = 1.08 \[ -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} \sqrt{x}\right ) e^{\left (a{\rm ln}\left (F\right )\right )}}{\sqrt{-b{\rm ln}\left (F\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(pi)*erf(-sqrt(-b*ln(F))*sqrt(x))*e^(a*ln(F))/sqrt(-b*ln(F))