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

Optimal. Leaf size=85 \[ \frac{3 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{4 b^{5/2} \log ^{\frac{5}{2}}(F)}-\frac{3 \sqrt{x} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac{x^{3/2} F^{a+b x}}{b \log (F)} \]

[Out]

(3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(4*b^(5/2)*Log[F]^(5/2)) - (
3*F^(a + b*x)*Sqrt[x])/(2*b^2*Log[F]^2) + (F^(a + b*x)*x^(3/2))/(b*Log[F])

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Rubi [A]  time = 0.10993, antiderivative size = 85, 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 \[ \frac{3 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{4 b^{5/2} \log ^{\frac{5}{2}}(F)}-\frac{3 \sqrt{x} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac{x^{3/2} F^{a+b x}}{b \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*x)*x^(3/2),x]

[Out]

(3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(4*b^(5/2)*Log[F]^(5/2)) - (
3*F^(a + b*x)*Sqrt[x])/(2*b^2*Log[F]^2) + (F^(a + b*x)*x^(3/2))/(b*Log[F])

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Rubi in Sympy [A]  time = 13.6344, size = 82, normalized size = 0.96 \[ \frac{3 \sqrt{\pi } F^{a} \operatorname{erfi}{\left (\sqrt{b} \sqrt{x} \sqrt{\log{\left (F \right )}} \right )}}{4 b^{\frac{5}{2}} \log{\left (F \right )}^{\frac{5}{2}}} + \frac{F^{a + b x} x^{\frac{3}{2}}}{b \log{\left (F \right )}} - \frac{3 F^{a + b x} \sqrt{x}}{2 b^{2} \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/(4*b**(5/2)*log(F)**(5/2)) +
F**(a + b*x)*x**(3/2)/(b*log(F)) - 3*F**(a + b*x)*sqrt(x)/(2*b**2*log(F)**2)

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Mathematica [A]  time = 0.0424413, size = 75, normalized size = 0.88 \[ \frac{F^a \left (3 \sqrt{\pi } \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )+2 \sqrt{b} \sqrt{x} \sqrt{\log (F)} F^{b x} (2 b x \log (F)-3)\right )}{4 b^{5/2} \log ^{\frac{5}{2}}(F)} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*(3*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]] + 2*Sqrt[b]*F^(b*x)*Sqrt[x]*
Sqrt[Log[F]]*(-3 + 2*b*x*Log[F])))/(4*b^(5/2)*Log[F]^(5/2))

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Maple [A]  time = 0.016, size = 75, normalized size = 0.9 \[ -{\frac{{F}^{a}}{b} \left ( -{\frac{ \left ( -10\,b\ln \left ( F \right ) x+15 \right ){{\rm e}^{b\ln \left ( F \right ) x}}}{10\,{b}^{2}}\sqrt{x} \left ( -b \right ) ^{{\frac{5}{2}}}\sqrt{\ln \left ( F \right ) }}+{\frac{3\,\sqrt{\pi }}{4} \left ( -b \right ) ^{{\frac{5}{2}}}{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ){b}^{-{\frac{5}{2}}}} \right ) \left ( -b \right ) ^{-{\frac{3}{2}}} \left ( \ln \left ( F \right ) \right ) ^{-{\frac{5}{2}}}} \]

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)^(5/2)/b*(-1/10*x^(1/2)*(-b)^(5/2)*ln(F)^(1/2)*(-10*b*ln(F)
*x+15)/b^2*exp(b*ln(F)*x)+3/4*(-b)^(5/2)/b^(5/2)*Pi^(1/2)*erfi(b^(1/2)*x^(1/2)*l
n(F)^(1/2)))

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Maxima [A]  time = 0.847287, size = 88, normalized size = 1.04 \[ \frac{1}{4} \, F^{a}{\left (\frac{2 \,{\left (2 \, b x^{\frac{3}{2}} \log \left (F\right ) - 3 \, \sqrt{x}\right )} F^{b x}}{b^{2} \log \left (F\right )^{2}} + \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{\sqrt{-b \log \left (F\right )} b^{2} \log \left (F\right )^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*F^a*(2*(2*b*x^(3/2)*log(F) - 3*sqrt(x))*F^(b*x)/(b^2*log(F)^2) + 3*sqrt(pi)*
erf(sqrt(-b*log(F))*sqrt(x))/(sqrt(-b*log(F))*b^2*log(F)^2))

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Fricas [A]  time = 0.279634, size = 86, normalized size = 1.01 \[ \frac{2 \,{\left (2 \, b x \log \left (F\right ) - 3\right )} \sqrt{-b \log \left (F\right )} F^{b x + a} \sqrt{x} + 3 \, \sqrt{\pi } F^{a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{4 \, \sqrt{-b \log \left (F\right )} b^{2} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(2*(2*b*x*log(F) - 3)*sqrt(-b*log(F))*F^(b*x + a)*sqrt(x) + 3*sqrt(pi)*F^a*e
rf(sqrt(-b*log(F))*sqrt(x)))/(sqrt(-b*log(F))*b^2*log(F)^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**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.254147, size = 97, normalized size = 1.14 \[ \frac{{\left (2 \, b x^{\frac{3}{2}}{\rm ln}\left (F\right ) - 3 \, \sqrt{x}\right )} e^{\left (b x{\rm ln}\left (F\right ) + a{\rm ln}\left (F\right )\right )}}{2 \, b^{2}{\rm ln}\left (F\right )^{2}} - \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (F\right )} \sqrt{x}\right ) e^{\left (a{\rm ln}\left (F\right )\right )}}{4 \, \sqrt{-b{\rm ln}\left (F\right )} b^{2}{\rm ln}\left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*b*x^(3/2)*ln(F) - 3*sqrt(x))*e^(b*x*ln(F) + a*ln(F))/(b^2*ln(F)^2) - 3/4*
sqrt(pi)*erf(-sqrt(-b*ln(F))*sqrt(x))*e^(a*ln(F))/(sqrt(-b*ln(F))*b^2*ln(F)^2)