3.35 \(\int \frac{F^{a+b x}}{x^{3/2}} \, dx\)

Optimal. Leaf size=54 \[ 2 \sqrt{\pi } \sqrt{b} F^a \sqrt{\log (F)} \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{2 F^{a+b x}}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0766526, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ 2 \sqrt{\pi } \sqrt{b} F^a \sqrt{\log (F)} \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{2 F^{a+b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*x)/x^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.2138, size = 53, normalized size = 0.98 \[ 2 \sqrt{\pi } F^{a} \sqrt{b} \sqrt{\log{\left (F \right )}} \operatorname{erfi}{\left (\sqrt{b} \sqrt{x} \sqrt{\log{\left (F \right )}} \right )} - \frac{2 F^{a + b x}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(b*x+a)/x**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0324719, size = 54, normalized size = 1. \[ 2 \sqrt{\pi } \sqrt{b} F^a \sqrt{\log (F)} \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{2 F^{a+b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*x)/x^(3/2),x]

[Out]

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

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Maple [A]  time = 0.017, size = 64, normalized size = 1.2 \[ -{\frac{{F}^{a}}{b} \left ( -b \right ) ^{{\frac{3}{2}}}\sqrt{\ln \left ( F \right ) } \left ( -2\,{\frac{{{\rm e}^{b\ln \left ( F \right ) x}}}{\sqrt{x}\sqrt{-b}\sqrt{\ln \left ( F \right ) }}}+2\,{\frac{\sqrt{b}\sqrt{\pi }{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ) }{\sqrt{-b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(b*x+a)/x^(3/2),x)

[Out]

-F^a*(-b)^(3/2)*ln(F)^(1/2)/b*(-2/x^(1/2)/(-b)^(1/2)/ln(F)^(1/2)*exp(b*ln(F)*x)+
2/(-b)^(1/2)*b^(1/2)*Pi^(1/2)*erfi(b^(1/2)*x^(1/2)*ln(F)^(1/2)))

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Maxima [A]  time = 0.851139, size = 32, normalized size = 0.59 \[ -\frac{\sqrt{-b x \log \left (F\right )} F^{a} \Gamma \left (-\frac{1}{2}, -b x \log \left (F\right )\right )}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*x + a)/x^(3/2),x, algorithm="maxima")

[Out]

-sqrt(-b*x*log(F))*F^a*gamma(-1/2, -b*x*log(F))/sqrt(x)

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Fricas [A]  time = 0.2682, size = 73, normalized size = 1.35 \[ \frac{2 \,{\left (\sqrt{\pi } F^{a} b \sqrt{x} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right ) \log \left (F\right ) - \sqrt{-b \log \left (F\right )} F^{b x + a}\right )}}{\sqrt{-b \log \left (F\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*x + a)/x^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + b x}}{x^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(b*x+a)/x**(3/2),x)

[Out]

Integral(F**(a + b*x)/x**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{b x + a}}{x^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*x + a)/x^(3/2),x, algorithm="giac")

[Out]

integrate(F^(b*x + a)/x^(3/2), x)