3.142 \(\int f^{a+\frac{b}{x^2}} x^8 \, dx\)

Optimal. Leaf size=34 \[ \frac{1}{2} x^9 f^a \left (-\frac{b \log (f)}{x^2}\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-\frac{b \log (f)}{x^2}\right ) \]

[Out]

(f^a*x^9*Gamma[-9/2, -((b*Log[f])/x^2)]*(-((b*Log[f])/x^2))^(9/2))/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0385288, 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 \[ \frac{1}{2} x^9 f^a \left (-\frac{b \log (f)}{x^2}\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-\frac{b \log (f)}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x^2)*x^8,x]

[Out]

(f^a*x^9*Gamma[-9/2, -((b*Log[f])/x^2)]*(-((b*Log[f])/x^2))^(9/2))/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.23892, size = 36, normalized size = 1.06 \[ \frac{f^{a} x^{9} \left (- \frac{b \log{\left (f \right )}}{x^{2}}\right )^{\frac{9}{2}} \Gamma{\left (- \frac{9}{2},- \frac{b \log{\left (f \right )}}{x^{2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x**2)*x**8,x)

[Out]

f**a*x**9*(-b*log(f)/x**2)**(9/2)*Gamma(-9/2, -b*log(f)/x**2)/2

_______________________________________________________________________________________

Mathematica [B]  time = 0.0643937, size = 98, normalized size = 2.88 \[ \frac{1}{945} f^a \left (x f^{\frac{b}{x^2}} \left (16 b^4 \log ^4(f)+8 b^3 x^2 \log ^3(f)+12 b^2 x^4 \log ^2(f)+30 b x^6 \log (f)+105 x^8\right )-16 \sqrt{\pi } b^{9/2} \log ^{\frac{9}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^2)*x^8,x]

[Out]

(f^a*(-16*b^(9/2)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Log[f]^(9/2) + f^(b/x^
2)*x*(105*x^8 + 30*b*x^6*Log[f] + 12*b^2*x^4*Log[f]^2 + 8*b^3*x^2*Log[f]^3 + 16*
b^4*Log[f]^4)))/945

_______________________________________________________________________________________

Maple [A]  time = 0.041, size = 133, normalized size = 3.9 \[{\frac{{f}^{a}{x}^{9}}{9}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{2\,{f}^{a}\ln \left ( f \right ) b{x}^{7}}{63}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{5}}{315}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{8\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}{x}^{3}}{945}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{16\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}x}{945}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{16\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{5}{b}^{5}\sqrt{\pi }}{945}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x^2)*x^8,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.968771, size = 154, normalized size = 4.53 \[ -\frac{16 \, \sqrt{\pi } b^{5} f^{a}{\left (\operatorname{erf}\left (\sqrt{-\frac{b \log \left (f\right )}{x^{2}}}\right ) - 1\right )} \log \left (f\right )^{5}}{945 \, x \sqrt{-\frac{b \log \left (f\right )}{x^{2}}}} + \frac{1}{945} \,{\left (105 \, f^{a} x^{9} + 30 \, b f^{a} x^{7} \log \left (f\right ) + 12 \, b^{2} f^{a} x^{5} \log \left (f\right )^{2} + 8 \, b^{3} f^{a} x^{3} \log \left (f\right )^{3} + 16 \, b^{4} f^{a} x \log \left (f\right )^{4}\right )} f^{\frac{b}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)*x^8,x, algorithm="maxima")

[Out]

-16/945*sqrt(pi)*b^5*f^a*(erf(sqrt(-b*log(f)/x^2)) - 1)*log(f)^5/(x*sqrt(-b*log(
f)/x^2)) + 1/945*(105*f^a*x^9 + 30*b*f^a*x^7*log(f) + 12*b^2*f^a*x^5*log(f)^2 +
8*b^3*f^a*x^3*log(f)^3 + 16*b^4*f^a*x*log(f)^4)*f^(b/x^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.284652, size = 144, normalized size = 4.24 \[ -\frac{16 \, \sqrt{\pi } b^{5} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{5} -{\left (105 \, x^{9} + 30 \, b x^{7} \log \left (f\right ) + 12 \, b^{2} x^{5} \log \left (f\right )^{2} + 8 \, b^{3} x^{3} \log \left (f\right )^{3} + 16 \, b^{4} x \log \left (f\right )^{4}\right )} \sqrt{-b \log \left (f\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{945 \, \sqrt{-b \log \left (f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)*x^8,x, algorithm="fricas")

[Out]

-1/945*(16*sqrt(pi)*b^5*f^a*erf(sqrt(-b*log(f))/x)*log(f)^5 - (105*x^9 + 30*b*x^
7*log(f) + 12*b^2*x^5*log(f)^2 + 8*b^3*x^3*log(f)^3 + 16*b^4*x*log(f)^4)*sqrt(-b
*log(f))*f^((a*x^2 + b)/x^2))/sqrt(-b*log(f))

_______________________________________________________________________________________

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**8,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x^{2}}} x^{8}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^2)*x^8,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)*x^8, x)