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

Optimal. Leaf size=73 \[ -\frac{2}{3} \sqrt{\pi } b^{3/2} f^a \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )+\frac{1}{3} x^3 f^{a+\frac{b}{x^2}}+\frac{2}{3} b x \log (f) f^{a+\frac{b}{x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0608214, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2214, 2206, 2211, 2204} \[ -\frac{2}{3} \sqrt{\pi } b^{3/2} f^a \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )+\frac{1}{3} x^3 f^{a+\frac{b}{x^2}}+\frac{2}{3} b x \log (f) f^{a+\frac{b}{x^2}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)*x^2,x]

[Out]

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

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), 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[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2206

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

Rule 2211

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

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+\frac{b}{x^2}} x^2 \, dx &=\frac{1}{3} f^{a+\frac{b}{x^2}} x^3+\frac{1}{3} (2 b \log (f)) \int f^{a+\frac{b}{x^2}} \, dx\\ &=\frac{1}{3} f^{a+\frac{b}{x^2}} x^3+\frac{2}{3} b f^{a+\frac{b}{x^2}} x \log (f)+\frac{1}{3} \left (4 b^2 \log ^2(f)\right ) \int \frac{f^{a+\frac{b}{x^2}}}{x^2} \, dx\\ &=\frac{1}{3} f^{a+\frac{b}{x^2}} x^3+\frac{2}{3} b f^{a+\frac{b}{x^2}} x \log (f)-\frac{1}{3} \left (4 b^2 \log ^2(f)\right ) \operatorname{Subst}\left (\int f^{a+b x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} f^{a+\frac{b}{x^2}} x^3+\frac{2}{3} b f^{a+\frac{b}{x^2}} x \log (f)-\frac{2}{3} b^{3/2} f^a \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right ) \log ^{\frac{3}{2}}(f)\\ \end{align*}

Mathematica [A]  time = 0.0243976, size = 60, normalized size = 0.82 \[ \frac{1}{3} f^a \left (x f^{\frac{b}{x^2}} \left (2 b \log (f)+x^2\right )-2 \sqrt{\pi } b^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)*x^2,x]

[Out]

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

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Maple [A]  time = 0.028, size = 67, normalized size = 0.9 \begin{align*}{\frac{{f}^{a}{x}^{3}}{3}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{2\,{f}^{a}\ln \left ( f \right ) bx}{3}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{2\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}\sqrt{\pi }}{3}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^2)*x^2,x)

[Out]

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

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Maxima [A]  time = 1.16611, size = 38, normalized size = 0.52 \begin{align*} \frac{1}{2} \, f^{a} x^{3} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{3}{2}} \Gamma \left (-\frac{3}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*f^a*x^3*(-b*log(f)/x^2)^(3/2)*gamma(-3/2, -b*log(f)/x^2)

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Fricas [A]  time = 1.80747, size = 153, normalized size = 2.1 \begin{align*} \frac{2}{3} \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} b f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) \log \left (f\right ) + \frac{1}{3} \,{\left (x^{3} + 2 \, b x \log \left (f\right )\right )} f^{\frac{a x^{2} + b}{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b/x**2)*x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x^2)*x^2, x)