\(\int \frac {f^{a+\frac {b}{x^2}}}{x^{13}} \, dx\) [140]

Optimal result

Integrand size = 13, antiderivative size = 82 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{13}} \, dx=\frac {f^{a+\frac {b}{x^2}} \left (120 x^{10}-120 b x^8 \log (f)+60 b^2 x^6 \log ^2(f)-20 b^3 x^4 \log ^3(f)+5 b^4 x^2 \log ^4(f)-b^5 \log ^5(f)\right )}{2 b^6 x^{10} \log ^6(f)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2249} \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{13}} \, dx=\frac {f^{a+\frac {b}{x^2}} \left (-b^5 \log ^5(f)+5 b^4 x^2 \log ^4(f)-20 b^3 x^4 \log ^3(f)+60 b^2 x^6 \log ^2(f)-120 b x^8 \log (f)+120 x^{10}\right )}{2 b^6 x^{10} \log ^6(f)} \]

[In]

Int[f^(a + b/x^2)/x^13,x]

[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)

Rule 2249

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{p = Simplify
[(m + 1)/n]}, Simp[(-F^a)*((f/d)^m/(d*n*((-b)*Log[F])^p))*Simplify[FunctionExpand[Gamma[p, (-b)*(c + d*x)^n*Lo
g[F]]]], x] /; IGtQ[p, 0]] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = \frac {f^{a+\frac {b}{x^2}} \left (120 x^{10}-120 b x^8 \log (f)+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)\right )}{2 b^6 x^{10} \log ^6(f)} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.29 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{13}} \, dx=\frac {f^a \Gamma \left (6,-\frac {b \log (f)}{x^2}\right )}{2 b^6 \log ^6(f)} \]

[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)

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.01

method	result	size
meijerg	\(-\frac {f^{a} \left (120-\frac {\left (-\frac {6 b^{5} \ln \left (f \right )^{5}}{x^{10}}+\frac {30 b^{4} \ln \left (f \right )^{4}}{x^{8}}-\frac {120 b^{3} \ln \left (f \right )^{3}}{x^{6}}+\frac {360 b^{2} \ln \left (f \right )^{2}}{x^{4}}-\frac {720 b \ln \left (f \right )}{x^{2}}+720\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{6}\right )}{2 b^{6} \ln \left (f \right )^{6}}\)	\(83\)
risch	\(-\frac {\left (b^{5} \ln \left (f \right )^{5}-5 b^{4} x^{2} \ln \left (f \right )^{4}+20 b^{3} x^{4} \ln \left (f \right )^{3}-60 b^{2} x^{6} \ln \left (f \right )^{2}+120 b \,x^{8} \ln \left (f \right )-120 x^{10}\right ) f^{\frac {a \,x^{2}+b}{x^{2}}}}{2 b^{6} \ln \left (f \right )^{6} x^{10}}\)	\(84\)
parallelrisch	\(\frac {-f^{a +\frac {b}{x^{2}}} b^{5} \ln \left (f \right )^{5}+5 f^{a +\frac {b}{x^{2}}} x^{2} b^{4} \ln \left (f \right )^{4}-20 f^{a +\frac {b}{x^{2}}} x^{4} b^{3} \ln \left (f \right )^{3}+60 f^{a +\frac {b}{x^{2}}} x^{6} b^{2} \ln \left (f \right )^{2}-120 f^{a +\frac {b}{x^{2}}} x^{8} b \ln \left (f \right )+120 f^{a +\frac {b}{x^{2}}} x^{10}}{2 x^{10} b^{6} \ln \left (f \right )^{6}}\)	\(126\)
norman	\(\frac {\frac {60 x^{12} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{b^{6} \ln \left (f \right )^{6}}-\frac {x^{2} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{2 b \ln \left (f \right )}+\frac {5 x^{4} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{2 b^{2} \ln \left (f \right )^{2}}-\frac {10 x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{b^{3} \ln \left (f \right )^{3}}+\frac {30 x^{8} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{b^{4} \ln \left (f \right )^{4}}-\frac {60 x^{10} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \left (f \right )}}{b^{5} \ln \left (f \right )^{5}}}{x^{12}}\)	\(144\)

[In]

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

[Out]

-1/2*f^a/b^6/ln(f)^6*(120-1/6*(-6*b^5*ln(f)^5/x^10+30*b^4*ln(f)^4/x^8-120*b^3*ln(f)^3/x^6+360*b^2*ln(f)^2/x^4-
720*b*ln(f)/x^2+720)*exp(b*ln(f)/x^2))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.02 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{13}} \, dx=\frac {{\left (120 \, x^{10} - 120 \, b x^{8} \log \left (f\right ) + 60 \, b^{2} x^{6} \log \left (f\right )^{2} - 20 \, b^{3} x^{4} \log \left (f\right )^{3} + 5 \, b^{4} x^{2} \log \left (f\right )^{4} - b^{5} \log \left (f\right )^{5}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{2 \, b^{6} x^{10} \log \left (f\right )^{6}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{13}} \, dx=\frac {f^{a + \frac {b}{x^{2}}} \left (- b^{5} \log {\left (f \right )}^{5} + 5 b^{4} x^{2} \log {\left (f \right )}^{4} - 20 b^{3} x^{4} \log {\left (f \right )}^{3} + 60 b^{2} x^{6} \log {\left (f \right )}^{2} - 120 b x^{8} \log {\left (f \right )} + 120 x^{10}\right )}{2 b^{6} x^{10} \log {\left (f \right )}^{6}} \]

[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)

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.27 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{13}} \, dx=\frac {f^{a} \Gamma \left (6, -\frac {b \log \left (f\right )}{x^{2}}\right )}{2 \, b^{6} \log \left (f\right )^{6}} \]

[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)

Giac [F]

\[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{13}} \, dx=\int { \frac {f^{a + \frac {b}{x^{2}}}}{x^{13}} \,d x } \]

[In]

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

[Out]

integrate(f^(a + b/x^2)/x^13, x)

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.02 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{13}} \, dx=-\frac {f^{a+\frac {b}{x^2}}\,\left (\frac {1}{2\,b\,\ln \left (f\right )}-\frac {5\,x^2}{2\,b^2\,{\ln \left (f\right )}^2}+\frac {10\,x^4}{b^3\,{\ln \left (f\right )}^3}-\frac {30\,x^6}{b^4\,{\ln \left (f\right )}^4}+\frac {60\,x^8}{b^5\,{\ln \left (f\right )}^5}-\frac {60\,x^{10}}{b^6\,{\ln \left (f\right )}^6}\right )}{x^{10}} \]

[In]

int(f^(a + b/x^2)/x^13,x)

[Out]

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