3.149 \(\int \frac{f^{a+\frac{b}{x^2}}}{x^6} \, dx\)

Optimal. Leaf size=86 \[ -\frac{3 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{8 b^{5/2} \log ^{\frac{5}{2}}(f)}+\frac{3 f^{a+\frac{b}{x^2}}}{4 b^2 x \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^3 \log (f)} \]

[Out]

(-3*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(8*b^(5/2)*Log[f]^(5/2)) + (3*f
^(a + b/x^2))/(4*b^2*x*Log[f]^2) - f^(a + b/x^2)/(2*b*x^3*Log[f])

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Rubi [A]  time = 0.132087, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{3 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{8 b^{5/2} \log ^{\frac{5}{2}}(f)}+\frac{3 f^{a+\frac{b}{x^2}}}{4 b^2 x \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^3 \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x^2)/x^6,x]

[Out]

(-3*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(8*b^(5/2)*Log[f]^(5/2)) + (3*f
^(a + b/x^2))/(4*b^2*x*Log[f]^2) - f^(a + b/x^2)/(2*b*x^3*Log[f])

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Rubi in Sympy [A]  time = 12.6779, size = 78, normalized size = 0.91 \[ - \frac{f^{a + \frac{b}{x^{2}}}}{2 b x^{3} \log{\left (f \right )}} + \frac{3 f^{a + \frac{b}{x^{2}}}}{4 b^{2} x \log{\left (f \right )}^{2}} - \frac{3 \sqrt{\pi } f^{a} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (f \right )}}}{x} \right )}}{8 b^{\frac{5}{2}} \log{\left (f \right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x**2)/x**6,x)

[Out]

-f**(a + b/x**2)/(2*b*x**3*log(f)) + 3*f**(a + b/x**2)/(4*b**2*x*log(f)**2) - 3*
sqrt(pi)*f**a*erfi(sqrt(b)*sqrt(log(f))/x)/(8*b**(5/2)*log(f)**(5/2))

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Mathematica [A]  time = 0.0795551, size = 74, normalized size = 0.86 \[ \frac{f^{a+\frac{b}{x^2}} \left (3 x^2-2 b \log (f)\right )}{4 b^2 x^3 \log ^2(f)}-\frac{3 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{8 b^{5/2} \log ^{\frac{5}{2}}(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^2)/x^6,x]

[Out]

(-3*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(8*b^(5/2)*Log[f]^(5/2)) + (f^(
a + b/x^2)*(3*x^2 - 2*b*Log[f]))/(4*b^2*x^3*Log[f]^2)

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Maple [A]  time = 0.043, size = 80, normalized size = 0.9 \[ -{\frac{{f}^{a}}{2\,b{x}^{3}\ln \left ( f \right ) }{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{3\,{f}^{a}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}x}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{3\,{f}^{a}\sqrt{\pi }}{8\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x^2)/x^6,x)

[Out]

-1/2*f^a/ln(f)/b/x^3*f^(b/x^2)+3/4*f^a/ln(f)^2/b^2/x*f^(b/x^2)-3/8*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 = 0.901625, size = 38, normalized size = 0.44 \[ \frac{f^{a} \Gamma \left (\frac{5}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{5} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)/x^6,x, algorithm="maxima")

[Out]

1/2*f^a*gamma(5/2, -b*log(f)/x^2)/(x^5*(-b*log(f)/x^2)^(5/2))

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Fricas [A]  time = 0.265458, size = 103, normalized size = 1.2 \[ -\frac{3 \, \sqrt{\pi } f^{a} x^{3} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) - 2 \,{\left (3 \, x^{2} - 2 \, b \log \left (f\right )\right )} \sqrt{-b \log \left (f\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{8 \, \sqrt{-b \log \left (f\right )} b^{2} x^{3} \log \left (f\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)/x^6,x, algorithm="fricas")

[Out]

-1/8*(3*sqrt(pi)*f^a*x^3*erf(sqrt(-b*log(f))/x) - 2*(3*x^2 - 2*b*log(f))*sqrt(-b
*log(f))*f^((a*x^2 + b)/x^2))/(sqrt(-b*log(f))*b^2*x^3*log(f)^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x**2)/x**6,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{a + \frac{b}{x^{2}}}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)/x^6,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)/x^6, x)