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 = 0.0344495, 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, Rules used = {2212, 2204} \[ \frac{x f^{a+b x^2}}{2 b \log (f)}-\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{4 b^{3/2} \log ^{\frac{3}{2}}(f)} \]

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])

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int f^{a+b x^2} x^2 \, dx &=\frac{f^{a+b x^2} x}{2 b \log (f)}-\frac{\int f^{a+b x^2} \, dx}{2 b \log (f)}\\ &=-\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{4 b^{3/2} \log ^{\frac{3}{2}}(f)}+\frac{f^{a+b x^2} x}{2 b \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0225355, 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 = 0.022, size = 54, normalized size = 0.9 \begin{align*}{\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 ) }}}} \end{align*}

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 = 1.0562, size = 72, normalized size = 1.22 \begin{align*} \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 )} \end{align*}

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 = 1.83004, size = 139, normalized size = 2.36 \begin{align*} \frac{2 \, b f^{b x^{2} + a} x \log \left (f\right ) + \sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{4 \, b^{2} \log \left (f\right )^{2}} \end{align*}

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

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

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 [A]  time = 1.25691, size = 77, normalized size = 1.31 \begin{align*} \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 )} + \frac{x e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{2 \, b \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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