3.139 \(\int \frac {f^{a+\frac {b}{x^2}}}{x^{11}} \, dx\)

Optimal. Leaf size=69 \[ -\frac {f^{a+\frac {b}{x^2}} \left (b^4 \log ^4(f)-4 b^3 x^2 \log ^3(f)+12 b^2 x^4 \log ^2(f)-24 b x^6 \log (f)+24 x^8\right )}{2 b^5 x^8 \log ^5(f)} \]

[Out]

-1/2*f^(a+b/x^2)*(24*x^8-24*b*x^6*ln(f)+12*b^2*x^4*ln(f)^2-4*b^3*x^2*ln(f)^3+b^4*ln(f)^4)/b^5/x^8/ln(f)^5

________________________________________________________________________________________

Rubi [C]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 0.35, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2218} \[ -\frac {f^a \text {Gamma}\left (5,-\frac {b \log (f)}{x^2}\right )}{2 b^5 \log ^5(f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)/x^11,x]

[Out]

-(f^a*Gamma[5, -((b*Log[f])/x^2)])/(2*b^5*Log[f]^5)

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin {align*} \int \frac {f^{a+\frac {b}{x^2}}}{x^{11}} \, dx &=-\frac {f^a \Gamma \left (5,-\frac {b \log (f)}{x^2}\right )}{2 b^5 \log ^5(f)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.00, size = 24, normalized size = 0.35 \[ -\frac {f^a \Gamma \left (5,-\frac {b \log (f)}{x^2}\right )}{2 b^5 \log ^5(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)/x^11,x]

[Out]

-1/2*(f^a*Gamma[5, -((b*Log[f])/x^2)])/(b^5*Log[f]^5)

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 71, normalized size = 1.03 \[ -\frac {{\left (24 \, x^{8} - 24 \, b x^{6} \log \relax (f) + 12 \, b^{2} x^{4} \log \relax (f)^{2} - 4 \, b^{3} x^{2} \log \relax (f)^{3} + b^{4} \log \relax (f)^{4}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{2 \, b^{5} x^{8} \log \relax (f)^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^11,x, algorithm="fricas")

[Out]

-1/2*(24*x^8 - 24*b*x^6*log(f) + 12*b^2*x^4*log(f)^2 - 4*b^3*x^2*log(f)^3 + b^4*log(f)^4)*f^((a*x^2 + b)/x^2)/
(b^5*x^8*log(f)^5)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x^{2}}}}{x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^11,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)/x^11, x)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 121, normalized size = 1.75 \[ \frac {-\frac {x^{2} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{2 b \ln \relax (f )}+\frac {2 x^{4} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{2} \ln \relax (f )^{2}}-\frac {6 x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}+\frac {12 x^{8} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{4} \ln \relax (f )^{4}}-\frac {12 x^{10} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{5} \ln \relax (f )^{5}}}{x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^2)/x^11,x)

[Out]

(-12/b^5/ln(f)^5*x^10*exp((a+b/x^2)*ln(f))+12/b^4*x^8*exp((a+b/x^2)*ln(f))/ln(f)^4-6/b^3*x^6*exp((a+b/x^2)*ln(
f))/ln(f)^3+2/b^2*x^4*exp((a+b/x^2)*ln(f))/ln(f)^2-1/2/b*x^2*exp((a+b/x^2)*ln(f))/ln(f))/x^10

________________________________________________________________________________________

maxima [C]  time = 1.27, size = 22, normalized size = 0.32 \[ -\frac {f^{a} \Gamma \left (5, -\frac {b \log \relax (f)}{x^{2}}\right )}{2 \, b^{5} \log \relax (f)^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^11,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 3.60, size = 72, normalized size = 1.04 \[ -\frac {f^{a+\frac {b}{x^2}}\,\left (\frac {1}{2\,b\,\ln \relax (f)}-\frac {2\,x^2}{b^2\,{\ln \relax (f)}^2}+\frac {6\,x^4}{b^3\,{\ln \relax (f)}^3}-\frac {12\,x^6}{b^4\,{\ln \relax (f)}^4}+\frac {12\,x^8}{b^5\,{\ln \relax (f)}^5}\right )}{x^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^2)/x^11,x)

[Out]

-(f^(a + b/x^2)*(1/(2*b*log(f)) - (2*x^2)/(b^2*log(f)^2) + (6*x^4)/(b^3*log(f)^3) - (12*x^6)/(b^4*log(f)^4) +
(12*x^8)/(b^5*log(f)^5)))/x^8

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 71, normalized size = 1.03 \[ \frac {f^{a + \frac {b}{x^{2}}} \left (- b^{4} \log {\relax (f )}^{4} + 4 b^{3} x^{2} \log {\relax (f )}^{3} - 12 b^{2} x^{4} \log {\relax (f )}^{2} + 24 b x^{6} \log {\relax (f )} - 24 x^{8}\right )}{2 b^{5} x^{8} \log {\relax (f )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**2)/x**11,x)

[Out]

f**(a + b/x**2)*(-b**4*log(f)**4 + 4*b**3*x**2*log(f)**3 - 12*b**2*x**4*log(f)**2 + 24*b*x**6*log(f) - 24*x**8
)/(2*b**5*x**8*log(f)**5)

________________________________________________________________________________________