3.586 \(\int F^{a+b \log ^2\left (c x^n\right )} x^2 \, dx\)

Optimal. Leaf size=89 \[ \frac{\sqrt{\pi } x^3 F^a \left (c x^n\right )^{-3/n} e^{-\frac{9}{4 b n^2 \log (F)}} \text{Erfi}\left (\frac{2 b n \log (F) \log \left (c x^n\right )+3}{2 \sqrt{b} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} n \sqrt{\log (F)}} \]

[Out]

(F^a*Sqrt[Pi]*x^3*Erfi[(3 + 2*b*n*Log[F]*Log[c*x^n])/(2*Sqrt[b]*n*Sqrt[Log[F]])]
)/(2*Sqrt[b]*E^(9/(4*b*n^2*Log[F]))*n*(c*x^n)^(3/n)*Sqrt[Log[F]])

_______________________________________________________________________________________

Rubi [A]  time = 0.15163, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{\pi } x^3 F^a \left (c x^n\right )^{-3/n} e^{-\frac{9}{4 b n^2 \log (F)}} \text{Erfi}\left (\frac{2 b n \log (F) \log \left (c x^n\right )+3}{2 \sqrt{b} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*Sqrt[Pi]*x^3*Erfi[(3 + 2*b*n*Log[F]*Log[c*x^n])/(2*Sqrt[b]*n*Sqrt[Log[F]])]
)/(2*Sqrt[b]*E^(9/(4*b*n^2*Log[F]))*n*(c*x^n)^(3/n)*Sqrt[Log[F]])

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int F^{a + b \log{\left (c x^{n} \right )}^{2}} x^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*ln(c*x**n)**2)*x**2,x)

[Out]

Integral(F**(a + b*log(c*x**n)**2)*x**2, x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.123557, size = 89, normalized size = 1. \[ \frac{\sqrt{\pi } x^3 F^a \left (c x^n\right )^{-3/n} e^{-\frac{9}{4 b n^2 \log (F)}} \text{Erfi}\left (\frac{2 b n \log (F) \log \left (c x^n\right )+3}{2 \sqrt{b} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*Sqrt[Pi]*x^3*Erfi[(3 + 2*b*n*Log[F]*Log[c*x^n])/(2*Sqrt[b]*n*Sqrt[Log[F]])]
)/(2*Sqrt[b]*E^(9/(4*b*n^2*Log[F]))*n*(c*x^n)^(3/n)*Sqrt[Log[F]])

_______________________________________________________________________________________

Maple [F]  time = 0.303, size = 0, normalized size = 0. \[ \int{F}^{a+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}}{x}^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*ln(c*x^n)^2)*x^2,x)

[Out]

int(F^(a+b*ln(c*x^n)^2)*x^2,x)

_______________________________________________________________________________________

Maxima [A]  time = 0.911358, size = 115, normalized size = 1.29 \[ \frac{\sqrt{\pi } F^{b \log \left (c\right )^{2} + a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} n \log \left (x\right ) - \frac{2 \, b n \log \left (F\right ) \log \left (c\right ) + 3}{2 \, \sqrt{-b \log \left (F\right )} n}\right ) e^{\left (-\frac{{\left (2 \, b n \log \left (F\right ) \log \left (c\right ) + 3\right )}^{2}}{4 \, b n^{2} \log \left (F\right )}\right )}}{2 \, \sqrt{-b \log \left (F\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.420169, size = 130, normalized size = 1.46 \[ \frac{\sqrt{\pi } b n \operatorname{erf}\left (\frac{{\left (2 \, b n^{2} \log \left (F\right ) \log \left (x\right ) + 2 \, b n \log \left (F\right ) \log \left (c\right ) + 3\right )} \sqrt{-b n^{2} \log \left (F\right )}}{2 \, b n^{2} \log \left (F\right )}\right ) e^{\left (\frac{4 \, a b n^{2} \log \left (F\right )^{2} - 12 \, b n \log \left (F\right ) \log \left (c\right ) - 9}{4 \, b n^{2} \log \left (F\right )}\right )} \log \left (F\right )}{2 \, \sqrt{-b n^{2} \log \left (F\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(pi)*b*n*erf(1/2*(2*b*n^2*log(F)*log(x) + 2*b*n*log(F)*log(c) + 3)*sqrt(
-b*n^2*log(F))/(b*n^2*log(F)))*e^(1/4*(4*a*b*n^2*log(F)^2 - 12*b*n*log(F)*log(c)
- 9)/(b*n^2*log(F)))*log(F)/sqrt(-b*n^2*log(F))

_______________________________________________________________________________________

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*ln(c*x**n)**2)*x**2,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(F^(b*log(c*x^n)^2 + a)*x^2, x)