3.36 \(\int \frac{F^{a+b x}}{x^{5/2}} \, dx\)

Optimal. Leaf size=77 \[ \frac{4}{3} \sqrt{\pi } b^{3/2} F^a \log ^{\frac{3}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{2 F^{a+b x}}{3 x^{3/2}}-\frac{4 b \log (F) F^{a+b x}}{3 \sqrt{x}} \]

[Out]

(-2*F^(a + b*x))/(3*x^(3/2)) - (4*b*F^(a + b*x)*Log[F])/(3*Sqrt[x]) + (4*b^(3/2)
*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Log[F]^(3/2))/3

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Rubi [A]  time = 0.106497, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{4}{3} \sqrt{\pi } b^{3/2} F^a \log ^{\frac{3}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{2 F^{a+b x}}{3 x^{3/2}}-\frac{4 b \log (F) F^{a+b x}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*x)/x^(5/2),x]

[Out]

(-2*F^(a + b*x))/(3*x^(3/2)) - (4*b*F^(a + b*x)*Log[F])/(3*Sqrt[x]) + (4*b^(3/2)
*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Log[F]^(3/2))/3

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Rubi in Sympy [A]  time = 11.7905, size = 76, normalized size = 0.99 \[ \frac{4 \sqrt{\pi } F^{a} b^{\frac{3}{2}} \log{\left (F \right )}^{\frac{3}{2}} \operatorname{erfi}{\left (\sqrt{b} \sqrt{x} \sqrt{\log{\left (F \right )}} \right )}}{3} - \frac{4 F^{a + b x} b \log{\left (F \right )}}{3 \sqrt{x}} - \frac{2 F^{a + b x}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(b*x+a)/x**(5/2),x)

[Out]

4*sqrt(pi)*F**a*b**(3/2)*log(F)**(3/2)*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/3 - 4*
F**(a + b*x)*b*log(F)/(3*sqrt(x)) - 2*F**(a + b*x)/(3*x**(3/2))

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Mathematica [A]  time = 0.0651847, size = 64, normalized size = 0.83 \[ \frac{2}{3} F^a \left (2 \sqrt{\pi } b^{3/2} \log ^{\frac{3}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{F^{b x} (2 b x \log (F)+1)}{x^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*x)/x^(5/2),x]

[Out]

(2*F^a*(2*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Log[F]^(3/2) - (F^
(b*x)*(1 + 2*b*x*Log[F]))/x^(3/2)))/3

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Maple [A]  time = 0.019, size = 72, normalized size = 0.9 \[ -{\frac{{F}^{a}}{b} \left ( -b \right ) ^{{\frac{5}{2}}} \left ( \ln \left ( F \right ) \right ) ^{{\frac{3}{2}}} \left ( -{\frac{ \left ( 4\,b\ln \left ( F \right ) x+2 \right ){{\rm e}^{b\ln \left ( F \right ) x}}}{3}{x}^{-{\frac{3}{2}}} \left ( -b \right ) ^{-{\frac{3}{2}}} \left ( \ln \left ( F \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{4\,\sqrt{\pi }}{3}{b}^{{\frac{3}{2}}}{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ) \left ( -b \right ) ^{-{\frac{3}{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(b*x+a)/x^(5/2),x)

[Out]

-F^a*(-b)^(5/2)*ln(F)^(3/2)/b*(-2/3/x^(3/2)/(-b)^(3/2)/ln(F)^(3/2)*(2*b*ln(F)*x+
1)*exp(b*ln(F)*x)+4/3/(-b)^(3/2)*b^(3/2)*Pi^(1/2)*erfi(b^(1/2)*x^(1/2)*ln(F)^(1/
2)))

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Maxima [A]  time = 0.836279, size = 32, normalized size = 0.42 \[ -\frac{\left (-b x \log \left (F\right )\right )^{\frac{3}{2}} F^{a} \Gamma \left (-\frac{3}{2}, -b x \log \left (F\right )\right )}{x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*x + a)/x^(5/2),x, algorithm="maxima")

[Out]

-(-b*x*log(F))^(3/2)*F^a*gamma(-3/2, -b*x*log(F))/x^(3/2)

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Fricas [A]  time = 0.276743, size = 90, normalized size = 1.17 \[ \frac{2 \,{\left (2 \, \sqrt{\pi } F^{a} b^{2} x^{\frac{3}{2}} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right ) \log \left (F\right )^{2} -{\left (2 \, b x \log \left (F\right ) + 1\right )} \sqrt{-b \log \left (F\right )} F^{b x + a}\right )}}{3 \, \sqrt{-b \log \left (F\right )} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*x + a)/x^(5/2),x, algorithm="fricas")

[Out]

2/3*(2*sqrt(pi)*F^a*b^2*x^(3/2)*erf(sqrt(-b*log(F))*sqrt(x))*log(F)^2 - (2*b*x*l
og(F) + 1)*sqrt(-b*log(F))*F^(b*x + a))/(sqrt(-b*log(F))*x^(3/2))

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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**(b*x+a)/x**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{b x + a}}{x^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*x + a)/x^(5/2),x, algorithm="giac")

[Out]

integrate(F^(b*x + a)/x^(5/2), x)