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

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

[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

_______________________________________________________________________________________

Rubi [A]  time = 0.104748, 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 \[ -\frac{2}{3} \sqrt{\pi } b^{3/2} f^a \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )+\frac{2}{3} b x \log (f) f^{a+\frac{b}{x^2}}+\frac{1}{3} x^3 f^{a+\frac{b}{x^2}} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.83937, size = 70, normalized size = 0.96 \[ - \frac{2 \sqrt{\pi } b^{\frac{3}{2}} f^{a} \log{\left (f \right )}^{\frac{3}{2}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (f \right )}}}{x} \right )}}{3} + \frac{2 b f^{a + \frac{b}{x^{2}}} x \log{\left (f \right )}}{3} + \frac{f^{a + \frac{b}{x^{2}}} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0336177, 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

_______________________________________________________________________________________

Maple [A]  time = 0.029, size = 67, normalized size = 0.9 \[{\frac{{f}^{a}{x}^{3}}{3}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{2\,{f}^{a}\ln \left ( f \right ) bx}{3}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{2\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}\sqrt{\pi }}{3}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.880215, size = 92, normalized size = 1.26 \[ -\frac{2 \, \sqrt{\pi } b^{2} f^{a}{\left (\operatorname{erf}\left (\sqrt{-\frac{b \log \left (f\right )}{x^{2}}}\right ) - 1\right )} \log \left (f\right )^{2}}{3 \, x \sqrt{-\frac{b \log \left (f\right )}{x^{2}}}} + \frac{1}{3} \,{\left (f^{a} x^{3} + 2 \, b f^{a} x \log \left (f\right )\right )} f^{\frac{b}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.318096, size = 93, normalized size = 1.27 \[ -\frac{2 \, \sqrt{\pi } b^{2} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{2} -{\left (x^{3} + 2 \, b x \log \left (f\right )\right )} \sqrt{-b \log \left (f\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{3 \, \sqrt{-b \log \left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x^{2}}} x^{2}\,{d x} \]

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)