3.87 \(\int f^{a+b x^2} x^2 \, dx\)

Optimal. Leaf size=59 \[ \frac{x f^{a+b x^2}}{2 b \log (f)}-\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{4 b^{3/2} \log ^{\frac{3}{2}}(f)} \]

[Out]

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

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Rubi [A]  time = 0.0596599, antiderivative size = 59, 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 \[ \frac{x f^{a+b x^2}}{2 b \log (f)}-\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{4 b^{3/2} \log ^{\frac{3}{2}}(f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^2)*x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.88825, size = 53, normalized size = 0.9 \[ \frac{f^{a + b x^{2}} x}{2 b \log{\left (f \right )}} - \frac{\sqrt{\pi } f^{a} \operatorname{erfi}{\left (\sqrt{b} x \sqrt{\log{\left (f \right )}} \right )}}{4 b^{\frac{3}{2}} \log{\left (f \right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**2+a)*x**2,x)

[Out]

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

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Mathematica [A]  time = 0.0338619, size = 59, normalized size = 1. \[ \frac{x f^{a+b x^2}}{2 b \log (f)}-\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{4 b^{3/2} \log ^{\frac{3}{2}}(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^2)*x^2,x]

[Out]

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

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Maple [A]  time = 0.026, size = 54, normalized size = 0.9 \[{\frac{{f}^{a}x{f}^{b{x}^{2}}}{2\,b\ln \left ( f \right ) }}-{\frac{{f}^{a}\sqrt{\pi }}{4\,b\ln \left ( f \right ) }{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^2+a)*x^2,x)

[Out]

1/2*f^a/ln(f)/b*x*f^(b*x^2)-1/4*f^a/ln(f)/b*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln
(f))^(1/2)*x)

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Maxima [A]  time = 0.767152, size = 72, normalized size = 1.22 \[ \frac{f^{b x^{2}} f^{a} x}{2 \, b \log \left (f\right )} - \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{4 \, \sqrt{-b \log \left (f\right )} b \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.276644, size = 73, normalized size = 1.24 \[ \frac{2 \, \sqrt{-b \log \left (f\right )} f^{b x^{2} + a} x - \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{4 \, \sqrt{-b \log \left (f\right )} b \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(2*sqrt(-b*log(f))*f^(b*x^2 + a)*x - sqrt(pi)*f^a*erf(sqrt(-b*log(f))*x))/(s
qrt(-b*log(f))*b*log(f))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(b*x**2+a)*x**2,x)

[Out]

Integral(f**(a + b*x**2)*x**2, x)

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GIAC/XCAS [A]  time = 0.229807, size = 80, normalized size = 1.36 \[ \frac{x e^{\left (b x^{2}{\rm ln}\left (f\right ) + a{\rm ln}\left (f\right )\right )}}{2 \, b{\rm ln}\left (f\right )} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b{\rm ln}\left (f\right )} x\right ) e^{\left (a{\rm ln}\left (f\right )\right )}}{4 \, \sqrt{-b{\rm ln}\left (f\right )} b{\rm ln}\left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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