\(\int f^{a+b x^2} x^2 \, dx\) [87]

Optimal result

Integrand size = 13, antiderivative size = 59 \[ \int f^{a+b x^2} x^2 \, dx=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {f^{a+b x^2} x}{2 b \log (f)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2243, 2235} \[ \int f^{a+b x^2} x^2 \, dx=\frac {x f^{a+b x^2}}{2 b \log (f)}-\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)} \]

[In]

Int[f^(a + b*x^2)*x^2,x]

[Out]

-1/4*(f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]])/(b^(3/2)*Log[f]^(3/2)) + (f^(a + b*x^2)*x)/(2*b*Log[f])

Rule 2235

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]

Rule 2243

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])

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int f^{a+b x^2} x^2 \, dx=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {f^{a+b x^2} x}{2 b \log (f)} \]

[In]

Integrate[f^(a + b*x^2)*x^2,x]

[Out]

-1/4*(f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]])/(b^(3/2)*Log[f]^(3/2)) + (f^(a + b*x^2)*x)/(2*b*Log[f])

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92

method	result	size
risch	\(\frac {f^{a} x \,f^{b \,x^{2}}}{2 \ln \left (f \right ) b}-\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x \right )}{4 \ln \left (f \right ) b \sqrt {-b \ln \left (f \right )}}\)	\(54\)
meijerg	\(-\frac {f^{a} \left (\frac {x \left (-b \right )^{\frac {3}{2}} \sqrt {\ln \left (f \right )}\, {\mathrm e}^{b \,x^{2} \ln \left (f \right )}}{b}-\frac {\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (x \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{2 b^{\frac {3}{2}}}\right )}{2 b \ln \left (f \right )^{\frac {3}{2}} \sqrt {-b}}\)	\(64\)

[In]

int(f^(b*x^2+a)*x^2,x,method=_RETURNVERBOSE)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int f^{a+b x^2} x^2 \, dx=\frac {2 \, b f^{b x^{2} + a} x \log \left (f\right ) + \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x\right )}{4 \, b^{2} \log \left (f\right )^{2}} \]

[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)

Sympy [F]

\[ \int f^{a+b x^2} x^2 \, dx=\int f^{a + b x^{2}} x^{2}\, dx \]

[In]

integrate(f**(b*x**2+a)*x**2,x)

[Out]

Integral(f**(a + b*x**2)*x**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int f^{a+b x^2} x^2 \, dx=\frac {f^{b x^{2}} f^{a} x}{2 \, b \log \left (f\right )} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x\right )}{4 \, \sqrt {-b \log \left (f\right )} b \log \left (f\right )} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int f^{a+b x^2} x^2 \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (f\right )} x\right )}{4 \, \sqrt {-b \log \left (f\right )} b \log \left (f\right )} + \frac {x e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{2 \, b \log \left (f\right )} \]

[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))

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int f^{a+b x^2} x^2 \, dx=\frac {f^a\,f^{b\,x^2}\,x}{2\,b\,\ln \left (f\right )}-\frac {f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \left (f\right )}{\sqrt {b\,\ln \left (f\right )}}\right )}{4\,b\,\ln \left (f\right )\,\sqrt {b\,\ln \left (f\right )}} \]

[In]

int(f^(a + b*x^2)*x^2,x)

[Out]

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