3.152 \(\int \frac{f^{a+\frac{b}{x^2}}}{x^{12}} \, dx\)

Optimal. Leaf size=34 \[ \frac{f^a \text{Gamma}\left (\frac{11}{2},-\frac{b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac{b \log (f)}{x^2}\right )^{11/2}} \]

[Out]

(f^a*Gamma[11/2, -((b*Log[f])/x^2)])/(2*x^11*(-((b*Log[f])/x^2))^(11/2))

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Rubi [A]  time = 0.0378495, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{f^a \text{Gamma}\left (\frac{11}{2},-\frac{b \log (f)}{x^2}\right )}{2 x^{11} \left (-\frac{b \log (f)}{x^2}\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x^2)/x^12,x]

[Out]

(f^a*Gamma[11/2, -((b*Log[f])/x^2)])/(2*x^11*(-((b*Log[f])/x^2))^(11/2))

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Rubi in Sympy [A]  time = 3.18876, size = 34, normalized size = 1. \[ \frac{f^{a} \Gamma{\left (\frac{11}{2},- \frac{b \log{\left (f \right )}}{x^{2}} \right )}}{2 x^{11} \left (- \frac{b \log{\left (f \right )}}{x^{2}}\right )^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x**2)/x**12,x)

[Out]

f**a*Gamma(11/2, -b*log(f)/x**2)/(2*x**11*(-b*log(f)/x**2)**(11/2))

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Mathematica [B]  time = 0.100471, size = 112, normalized size = 3.29 \[ \frac{f^a \left (945 \sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )-\frac{2 \sqrt{b} \sqrt{\log (f)} f^{\frac{b}{x^2}} \left (16 b^4 \log ^4(f)-72 b^3 x^2 \log ^3(f)+252 b^2 x^4 \log ^2(f)-630 b x^6 \log (f)+945 x^8\right )}{x^9}\right )}{64 b^{11/2} \log ^{\frac{11}{2}}(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^2)/x^12,x]

[Out]

(f^a*(945*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x] - (2*Sqrt[b]*f^(b/x^2)*Sqrt[Lo
g[f]]*(945*x^8 - 630*b*x^6*Log[f] + 252*b^2*x^4*Log[f]^2 - 72*b^3*x^2*Log[f]^3 +
16*b^4*Log[f]^4))/x^9))/(64*b^(11/2)*Log[f]^(11/2))

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Maple [A]  time = 0.073, size = 146, normalized size = 4.3 \[ -{\frac{{f}^{a}}{2\,b\ln \left ( f \right ){x}^{9}}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{9\,{f}^{a}}{4\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{7}}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{63\,{f}^{a}}{8\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{x}^{5}}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{315\,{f}^{a}}{16\, \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}{x}^{3}}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{945\,{f}^{a}}{32\, \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}x}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{945\,{f}^{a}\sqrt{\pi }}{64\, \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x^2)/x^12,x)

[Out]

-1/2*f^a/ln(f)/b/x^9*f^(b/x^2)+9/4*f^a/ln(f)^2/b^2/x^7*f^(b/x^2)-63/8*f^a/ln(f)^
3/b^3/x^5*f^(b/x^2)+315/16*f^a/ln(f)^4/b^4/x^3*f^(b/x^2)-945/32*f^a/ln(f)^5/b^5/
x*f^(b/x^2)+945/64*f^a/ln(f)^5/b^5*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2
)/x)

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Maxima [A]  time = 0.911784, size = 38, normalized size = 1.12 \[ \frac{f^{a} \Gamma \left (\frac{11}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{11} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)/x^12,x, algorithm="maxima")

[Out]

1/2*f^a*gamma(11/2, -b*log(f)/x^2)/(x^11*(-b*log(f)/x^2)^(11/2))

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Fricas [A]  time = 0.269456, size = 151, normalized size = 4.44 \[ \frac{945 \, \sqrt{\pi } f^{a} x^{9} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) - 2 \,{\left (945 \, x^{8} - 630 \, b x^{6} \log \left (f\right ) + 252 \, b^{2} x^{4} \log \left (f\right )^{2} - 72 \, b^{3} x^{2} \log \left (f\right )^{3} + 16 \, b^{4} \log \left (f\right )^{4}\right )} \sqrt{-b \log \left (f\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{64 \, \sqrt{-b \log \left (f\right )} b^{5} x^{9} \log \left (f\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)/x^12,x, algorithm="fricas")

[Out]

1/64*(945*sqrt(pi)*f^a*x^9*erf(sqrt(-b*log(f))/x) - 2*(945*x^8 - 630*b*x^6*log(f
) + 252*b^2*x^4*log(f)^2 - 72*b^3*x^2*log(f)^3 + 16*b^4*log(f)^4)*sqrt(-b*log(f)
)*f^((a*x^2 + b)/x^2))/(sqrt(-b*log(f))*b^5*x^9*log(f)^5)

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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**12,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{a + \frac{b}{x^{2}}}}{x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)/x^12,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)/x^12, x)