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

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

[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

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Rubi [C]  time = 0.02, 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*}

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Mathematica [C]  time = 0.00, size = 24, normalized size = 0.29 \[ \frac {f^a \Gamma \left (6,-\frac {b \log (f)}{x^2}\right )}{2 b^6 \log ^6(f)} \]

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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fricas [A]  time = 0.42, size = 84, normalized size = 1.02 \[ \frac {{\left (120 \, x^{10} - 120 \, b x^{8} \log \relax (f) + 60 \, b^{2} x^{6} \log \relax (f)^{2} - 20 \, b^{3} x^{4} \log \relax (f)^{3} + 5 \, b^{4} x^{2} \log \relax (f)^{4} - b^{5} \log \relax (f)^{5}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{2 \, b^{6} x^{10} \log \relax (f)^{6}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x^{2}}}}{x^{13}}\,{d x} \]

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)

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maple [A]  time = 0.04, size = 144, normalized size = 1.76 \[ \frac {-\frac {x^{2} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{2 b \ln \relax (f )}+\frac {5 x^{4} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{2 b^{2} \ln \relax (f )^{2}}-\frac {10 x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}+\frac {30 x^{8} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{4} \ln \relax (f )^{4}}-\frac {60 x^{10} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{5} \ln \relax (f )^{5}}+\frac {60 x^{12} {\mathrm e}^{\left (a +\frac {b}{x^{2}}\right ) \ln \relax (f )}}{b^{6} \ln \relax (f )^{6}}}{x^{12}} \]

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

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maxima [C]  time = 1.28, size = 22, normalized size = 0.27 \[ \frac {f^{a} \Gamma \left (6, -\frac {b \log \relax (f)}{x^{2}}\right )}{2 \, b^{6} \log \relax (f)^{6}} \]

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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mupad [B]  time = 3.61, size = 84, normalized size = 1.02 \[ -\frac {f^{a+\frac {b}{x^2}}\,\left (\frac {1}{2\,b\,\ln \relax (f)}-\frac {5\,x^2}{2\,b^2\,{\ln \relax (f)}^2}+\frac {10\,x^4}{b^3\,{\ln \relax (f)}^3}-\frac {30\,x^6}{b^4\,{\ln \relax (f)}^4}+\frac {60\,x^8}{b^5\,{\ln \relax (f)}^5}-\frac {60\,x^{10}}{b^6\,{\ln \relax (f)}^6}\right )}{x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.18, size = 85, normalized size = 1.04 \[ \frac {f^{a + \frac {b}{x^{2}}} \left (- b^{5} \log {\relax (f )}^{5} + 5 b^{4} x^{2} \log {\relax (f )}^{4} - 20 b^{3} x^{4} \log {\relax (f )}^{3} + 60 b^{2} x^{6} \log {\relax (f )}^{2} - 120 b x^{8} \log {\relax (f )} + 120 x^{10}\right )}{2 b^{6} x^{10} \log {\relax (f )}^{6}} \]

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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