3.151 \(\int \frac{f^{a+\frac{b}{x^2}}}{x^{10}} \, dx\)

Optimal. Leaf size=132 \[ -\frac{105 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{32 b^{9/2} \log ^{\frac{9}{2}}(f)}+\frac{105 f^{a+\frac{b}{x^2}}}{16 b^4 x \log ^4(f)}-\frac{35 f^{a+\frac{b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac{7 f^{a+\frac{b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^7 \log (f)} \]

[Out]

(-105*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(32*b^(9/2)*Log[f]^(9/2)) + (
105*f^(a + b/x^2))/(16*b^4*x*Log[f]^4) - (35*f^(a + b/x^2))/(8*b^3*x^3*Log[f]^3)
+ (7*f^(a + b/x^2))/(4*b^2*x^5*Log[f]^2) - f^(a + b/x^2)/(2*b*x^7*Log[f])

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Rubi [A]  time = 0.225015, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{105 \sqrt{\pi } f^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )}{32 b^{9/2} \log ^{\frac{9}{2}}(f)}+\frac{105 f^{a+\frac{b}{x^2}}}{16 b^4 x \log ^4(f)}-\frac{35 f^{a+\frac{b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac{7 f^{a+\frac{b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac{f^{a+\frac{b}{x^2}}}{2 b x^7 \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x^2)/x^10,x]

[Out]

(-105*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(32*b^(9/2)*Log[f]^(9/2)) + (
105*f^(a + b/x^2))/(16*b^4*x*Log[f]^4) - (35*f^(a + b/x^2))/(8*b^3*x^3*Log[f]^3)
+ (7*f^(a + b/x^2))/(4*b^2*x^5*Log[f]^2) - f^(a + b/x^2)/(2*b*x^7*Log[f])

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Rubi in Sympy [A]  time = 25.2915, size = 126, normalized size = 0.95 \[ - \frac{f^{a + \frac{b}{x^{2}}}}{2 b x^{7} \log{\left (f \right )}} + \frac{7 f^{a + \frac{b}{x^{2}}}}{4 b^{2} x^{5} \log{\left (f \right )}^{2}} - \frac{35 f^{a + \frac{b}{x^{2}}}}{8 b^{3} x^{3} \log{\left (f \right )}^{3}} + \frac{105 f^{a + \frac{b}{x^{2}}}}{16 b^{4} x \log{\left (f \right )}^{4}} - \frac{105 \sqrt{\pi } f^{a} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (f \right )}}}{x} \right )}}{32 b^{\frac{9}{2}} \log{\left (f \right )}^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x**2)/x**10,x)

[Out]

-f**(a + b/x**2)/(2*b*x**7*log(f)) + 7*f**(a + b/x**2)/(4*b**2*x**5*log(f)**2) -
35*f**(a + b/x**2)/(8*b**3*x**3*log(f)**3) + 105*f**(a + b/x**2)/(16*b**4*x*log
(f)**4) - 105*sqrt(pi)*f**a*erfi(sqrt(b)*sqrt(log(f))/x)/(32*b**(9/2)*log(f)**(9
/2))

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Mathematica [A]  time = 0.132737, size = 100, normalized size = 0.76 \[ \frac{f^a \left (\frac{2 \sqrt{b} \sqrt{\log (f)} f^{\frac{b}{x^2}} \left (-8 b^3 \log ^3(f)+28 b^2 x^2 \log ^2(f)-70 b x^4 \log (f)+105 x^6\right )}{x^7}-105 \sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )\right )}{32 b^{9/2} \log ^{\frac{9}{2}}(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^2)/x^10,x]

[Out]

(f^a*(-105*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x] + (2*Sqrt[b]*f^(b/x^2)*Sqrt[L
og[f]]*(105*x^6 - 70*b*x^4*Log[f] + 28*b^2*x^2*Log[f]^2 - 8*b^3*Log[f]^3))/x^7))
/(32*b^(9/2)*Log[f]^(9/2))

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Maple [A]  time = 0.063, size = 124, normalized size = 0.9 \[ -{\frac{{f}^{a}}{2\,b\ln \left ( f \right ){x}^{7}}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{7\,{f}^{a}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{5}}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{35\,{f}^{a}}{8\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{x}^{3}}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{105\,{f}^{a}}{16\, \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}x}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{105\,{f}^{a}\sqrt{\pi }}{32\, \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x^2)/x^10,x)

[Out]

-1/2*f^a/ln(f)/b/x^7*f^(b/x^2)+7/4*f^a/ln(f)^2/b^2/x^5*f^(b/x^2)-35/8*f^a/ln(f)^
3/b^3/x^3*f^(b/x^2)+105/16*f^a/ln(f)^4/b^4/x*f^(b/x^2)-105/32*f^a/ln(f)^4/b^4*Pi
^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2)/x)

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Maxima [A]  time = 0.846342, size = 38, normalized size = 0.29 \[ \frac{f^{a} \Gamma \left (\frac{9}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{9} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)/x^10,x, algorithm="maxima")

[Out]

1/2*f^a*gamma(9/2, -b*log(f)/x^2)/(x^9*(-b*log(f)/x^2)^(9/2))

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Fricas [A]  time = 0.256445, size = 135, normalized size = 1.02 \[ -\frac{105 \, \sqrt{\pi } f^{a} x^{7} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) - 2 \,{\left (105 \, x^{6} - 70 \, b x^{4} \log \left (f\right ) + 28 \, b^{2} x^{2} \log \left (f\right )^{2} - 8 \, b^{3} \log \left (f\right )^{3}\right )} \sqrt{-b \log \left (f\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{32 \, \sqrt{-b \log \left (f\right )} b^{4} x^{7} \log \left (f\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)/x^10,x, algorithm="fricas")

[Out]

-1/32*(105*sqrt(pi)*f^a*x^7*erf(sqrt(-b*log(f))/x) - 2*(105*x^6 - 70*b*x^4*log(f
) + 28*b^2*x^2*log(f)^2 - 8*b^3*log(f)^3)*sqrt(-b*log(f))*f^((a*x^2 + b)/x^2))/(
sqrt(-b*log(f))*b^4*x^7*log(f)^4)

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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**10,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{a + \frac{b}{x^{2}}}}{x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)/x^10,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)/x^10, x)