3.164 \(\int \frac {f^{a+\frac {b}{x^3}}}{x^{13}} \, dx\)

Optimal. Leaf size=83 \[ \frac {2 f^{a+\frac {b}{x^3}}}{b^4 \log ^4(f)}-\frac {2 f^{a+\frac {b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)} \]

[Out]

2*f^(a+b/x^3)/b^4/ln(f)^4-2*f^(a+b/x^3)/b^3/x^3/ln(f)^3+f^(a+b/x^3)/b^2/x^6/ln(f)^2-1/3*f^(a+b/x^3)/b/x^9/ln(f
)

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ \frac {f^{a+\frac {b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac {2 f^{a+\frac {b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac {2 f^{a+\frac {b}{x^3}}}{b^4 \log ^4(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)/x^13,x]

[Out]

(2*f^(a + b/x^3))/(b^4*Log[f]^4) - (2*f^(a + b/x^3))/(b^3*x^3*Log[f]^3) + f^(a + b/x^3)/(b^2*x^6*Log[f]^2) - f
^(a + b/x^3)/(3*b*x^9*Log[f])

Rule 2209

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

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

\begin {align*} \int \frac {f^{a+\frac {b}{x^3}}}{x^{13}} \, dx &=-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)}-\frac {3 \int \frac {f^{a+\frac {b}{x^3}}}{x^{10}} \, dx}{b \log (f)}\\ &=\frac {f^{a+\frac {b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)}+\frac {6 \int \frac {f^{a+\frac {b}{x^3}}}{x^7} \, dx}{b^2 \log ^2(f)}\\ &=-\frac {2 f^{a+\frac {b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)}-\frac {6 \int \frac {f^{a+\frac {b}{x^3}}}{x^4} \, dx}{b^3 \log ^3(f)}\\ &=\frac {2 f^{a+\frac {b}{x^3}}}{b^4 \log ^4(f)}-\frac {2 f^{a+\frac {b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 58, normalized size = 0.70 \[ \frac {f^{a+\frac {b}{x^3}} \left (-b^3 \log ^3(f)+3 b^2 x^3 \log ^2(f)-6 b x^6 \log (f)+6 x^9\right )}{3 b^4 x^9 \log ^4(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)/x^13,x]

[Out]

(f^(a + b/x^3)*(6*x^9 - 6*b*x^6*Log[f] + 3*b^2*x^3*Log[f]^2 - b^3*Log[f]^3))/(3*b^4*x^9*Log[f]^4)

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 60, normalized size = 0.72 \[ \frac {{\left (6 \, x^{9} - 6 \, b x^{6} \log \relax (f) + 3 \, b^{2} x^{3} \log \relax (f)^{2} - b^{3} \log \relax (f)^{3}\right )} f^{\frac {a x^{3} + b}{x^{3}}}}{3 \, b^{4} x^{9} \log \relax (f)^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^13,x, algorithm="fricas")

[Out]

1/3*(6*x^9 - 6*b*x^6*log(f) + 3*b^2*x^3*log(f)^2 - b^3*log(f)^3)*f^((a*x^3 + b)/x^3)/(b^4*x^9*log(f)^4)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x^{3}}}}{x^{13}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^13,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)/x^13, x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 97, normalized size = 1.17 \[ \frac {\frac {2 x^{12} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{b^{4} \ln \relax (f )^{4}}-\frac {2 x^{9} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}+\frac {x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{b^{2} \ln \relax (f )^{2}}-\frac {x^{3} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{3 b \ln \relax (f )}}{x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^3)/x^13,x)

[Out]

(1/b^2*x^6*exp((a+b/x^3)*ln(f))/ln(f)^2+2/b^4/ln(f)^4*x^12*exp((a+b/x^3)*ln(f))-2/b^3*x^9*exp((a+b/x^3)*ln(f))
/ln(f)^3-1/3/b*x^3*exp((a+b/x^3)*ln(f))/ln(f))/x^12

________________________________________________________________________________________

maxima [C]  time = 1.28, size = 22, normalized size = 0.27 \[ \frac {f^{a} \Gamma \left (4, -\frac {b \log \relax (f)}{x^{3}}\right )}{3 \, b^{4} \log \relax (f)^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^13,x, algorithm="maxima")

[Out]

1/3*f^a*gamma(4, -b*log(f)/x^3)/(b^4*log(f)^4)

________________________________________________________________________________________

mupad [B]  time = 3.54, size = 60, normalized size = 0.72 \[ -\frac {f^{a+\frac {b}{x^3}}\,\left (\frac {1}{3\,b\,\ln \relax (f)}-\frac {x^3}{b^2\,{\ln \relax (f)}^2}+\frac {2\,x^6}{b^3\,{\ln \relax (f)}^3}-\frac {2\,x^9}{b^4\,{\ln \relax (f)}^4}\right )}{x^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^3)/x^13,x)

[Out]

-(f^(a + b/x^3)*(1/(3*b*log(f)) - x^3/(b^2*log(f)^2) + (2*x^6)/(b^3*log(f)^3) - (2*x^9)/(b^4*log(f)^4)))/x^9

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 58, normalized size = 0.70 \[ \frac {f^{a + \frac {b}{x^{3}}} \left (- b^{3} \log {\relax (f )}^{3} + 3 b^{2} x^{3} \log {\relax (f )}^{2} - 6 b x^{6} \log {\relax (f )} + 6 x^{9}\right )}{3 b^{4} x^{9} \log {\relax (f )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**3)/x**13,x)

[Out]

f**(a + b/x**3)*(-b**3*log(f)**3 + 3*b**2*x**3*log(f)**2 - 6*b*x**6*log(f) + 6*x**9)/(3*b**4*x**9*log(f)**4)

________________________________________________________________________________________