3.140 \(\int \frac{f^{a+\frac{b}{x^2}}}{x^{13}} \, dx\)

Optimal. Leaf size=82 \[ \frac{f^{a+\frac{b}{x^2}} \left (60 b^2 x^6 \log ^2(f)-20 b^3 x^4 \log ^3(f)+5 b^4 x^2 \log ^4(f)-b^5 \log ^5(f)-120 b x^8 \log (f)+120 x^{10}\right )}{2 b^6 x^{10} \log ^6(f)} \]

[Out]

(f^(a + b/x^2)*(120*x^10 - 120*b*x^8*Log[f] + 60*b^2*x^6*Log[f]^2 - 20*b^3*x^4*Log[f]^3 + 5*b^4*x^2*Log[f]^4 -
b^5*Log[f]^5))/(2*b^6*x^10*Log[f]^6)

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Rubi [C]  time = 0.0240109, antiderivative size = 24, normalized size of antiderivative = 0.29, 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} \[ \frac{f^a \text{Gamma}\left (6,-\frac{b \log (f)}{x^2}\right )}{2 b^6 \log ^6(f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^2)/x^13,x]

[Out]

(f^a*Gamma[6, -((b*Log[f])/x^2)])/(2*b^6*Log[f]^6)

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^{13}} \, dx &=\frac{f^a \Gamma \left (6,-\frac{b \log (f)}{x^2}\right )}{2 b^6 \log ^6(f)}\\ \end{align*}

Mathematica [C]  time = 0.0027983, size = 24, normalized size = 0.29 \[ \frac{f^a \text{Gamma}\left (6,-\frac{b \log (f)}{x^2}\right )}{2 b^6 \log ^6(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^2)/x^13,x]

[Out]

(f^a*Gamma[6, -((b*Log[f])/x^2)])/(2*b^6*Log[f]^6)

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Maple [A]  time = 0.023, size = 144, normalized size = 1.8 \begin{align*}{\frac{1}{{x}^{12}} \left ( 60\,{\frac{{x}^{12}}{{b}^{6} \left ( \ln \left ( f \right ) \right ) ^{6}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{2}}} \right ) \ln \left ( f \right ) }}}-60\,{\frac{{x}^{10}}{{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{2}}} \right ) \ln \left ( f \right ) }}}+30\,{\frac{{x}^{8}}{{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{2}}} \right ) \ln \left ( f \right ) }}}-10\,{\frac{{x}^{6}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{2}}} \right ) \ln \left ( f \right ) }}}+{\frac{5\,{x}^{4}}{2\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{2}}} \right ) \ln \left ( f \right ) }}}-{\frac{{x}^{2}}{2\,b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{{x}^{2}}} \right ) \ln \left ( f \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^2)/x^13,x)

[Out]

(60/b^6/ln(f)^6*x^12*exp((a+b/x^2)*ln(f))-60/b^5/ln(f)^5*x^10*exp((a+b/x^2)*ln(f))+30/b^4/ln(f)^4*x^8*exp((a+b
/x^2)*ln(f))-10/b^3/ln(f)^3*x^6*exp((a+b/x^2)*ln(f))+5/2/b^2/ln(f)^2*x^4*exp((a+b/x^2)*ln(f))-1/2/b/ln(f)*x^2*
exp((a+b/x^2)*ln(f)))/x^12

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Maxima [C]  time = 1.18374, size = 30, normalized size = 0.37 \begin{align*} \frac{f^{a} \Gamma \left (6, -\frac{b \log \left (f\right )}{x^{2}}\right )}{2 \, b^{6} \log \left (f\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^13,x, algorithm="maxima")

[Out]

1/2*f^a*gamma(6, -b*log(f)/x^2)/(b^6*log(f)^6)

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Fricas [A]  time = 1.75167, size = 209, normalized size = 2.55 \begin{align*} \frac{{\left (120 \, x^{10} - 120 \, b x^{8} \log \left (f\right ) + 60 \, b^{2} x^{6} \log \left (f\right )^{2} - 20 \, b^{3} x^{4} \log \left (f\right )^{3} + 5 \, b^{4} x^{2} \log \left (f\right )^{4} - b^{5} \log \left (f\right )^{5}\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{2 \, b^{6} x^{10} \log \left (f\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^13,x, algorithm="fricas")

[Out]

1/2*(120*x^10 - 120*b*x^8*log(f) + 60*b^2*x^6*log(f)^2 - 20*b^3*x^4*log(f)^3 + 5*b^4*x^2*log(f)^4 - b^5*log(f)
^5)*f^((a*x^2 + b)/x^2)/(b^6*x^10*log(f)^6)

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Sympy [A]  time = 0.168652, size = 85, normalized size = 1.04 \begin{align*} \frac{f^{a + \frac{b}{x^{2}}} \left (- b^{5} \log{\left (f \right )}^{5} + 5 b^{4} x^{2} \log{\left (f \right )}^{4} - 20 b^{3} x^{4} \log{\left (f \right )}^{3} + 60 b^{2} x^{6} \log{\left (f \right )}^{2} - 120 b x^{8} \log{\left (f \right )} + 120 x^{10}\right )}{2 b^{6} x^{10} \log{\left (f \right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**2)/x**13,x)

[Out]

f**(a + b/x**2)*(-b**5*log(f)**5 + 5*b**4*x**2*log(f)**4 - 20*b**3*x**4*log(f)**3 + 60*b**2*x**6*log(f)**2 - 1
20*b*x**8*log(f) + 120*x**10)/(2*b**6*x**10*log(f)**6)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{2}}}}{x^{13}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^2)/x^13,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^2)/x^13, x)