\(\int \frac {f^{a+\frac {b}{x^3}}}{x^{13}} \, dx\) [164]

Optimal result

Integrand size = 13, antiderivative size = 83 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{13}} \, dx=\frac {2 f^{a+\frac {b}{x^3}}}{b^4 \log ^4(f)}-\frac {2 f^{a+\frac {b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)} \]

[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] (verified)

Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2243, 2240} \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{13}} \, dx=\frac {2 f^{a+\frac {b}{x^3}}}{b^4 \log ^4(f)}-\frac {2 f^{a+\frac {b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)} \]

[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 2240

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 2243

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*} \text {integral}& = -\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] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.70 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{13}} \, dx=\frac {f^{a+\frac {b}{x^3}} \left (6 x^9-6 b x^6 \log (f)+3 b^2 x^3 \log ^2(f)-b^3 \log ^3(f)\right )}{3 b^4 x^9 \log ^4(f)} \]

[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)

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.71

method	result	size
meijerg	\(-\frac {f^{a} \left (6-\frac {\left (-\frac {4 b^{3} \ln \left (f \right )^{3}}{x^{9}}+\frac {12 b^{2} \ln \left (f \right )^{2}}{x^{6}}-\frac {24 b \ln \left (f \right )}{x^{3}}+24\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{3}}}}{4}\right )}{3 b^{4} \ln \left (f \right )^{4}}\)	\(59\)
risch	\(-\frac {\left (-6 x^{9}+6 b \,x^{6} \ln \left (f \right )-3 b^{2} x^{3} \ln \left (f \right )^{2}+\ln \left (f \right )^{3} b^{3}\right ) f^{\frac {a \,x^{3}+b}{x^{3}}}}{3 \ln \left (f \right )^{4} b^{4} x^{9}}\)	\(60\)
parallelrisch	\(\frac {6 f^{a +\frac {b}{x^{3}}} x^{9}-6 b \,f^{a +\frac {b}{x^{3}}} x^{6} \ln \left (f \right )+3 b^{2} f^{a +\frac {b}{x^{3}}} x^{3} \ln \left (f \right )^{2}-f^{a +\frac {b}{x^{3}}} \ln \left (f \right )^{3} b^{3}}{3 x^{9} \ln \left (f \right )^{4} b^{4}}\)	\(84\)
norman	\(\frac {\frac {x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{b^{2} \ln \left (f \right )^{2}}-\frac {x^{3} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{3 b \ln \left (f \right )}-\frac {2 x^{9} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{b^{3} \ln \left (f \right )^{3}}+\frac {2 x^{12} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{b^{4} \ln \left (f \right )^{4}}}{x^{12}}\)	\(97\)

[In]

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

[Out]

-1/3*f^a/b^4/ln(f)^4*(6-1/4*(-4*b^3*ln(f)^3/x^9+12*b^2*ln(f)^2/x^6-24*b*ln(f)/x^3+24)*exp(b*ln(f)/x^3))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.72 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{13}} \, dx=\frac {{\left (6 \, x^{9} - 6 \, b x^{6} \log \left (f\right ) + 3 \, b^{2} x^{3} \log \left (f\right )^{2} - b^{3} \log \left (f\right )^{3}\right )} f^{\frac {a x^{3} + b}{x^{3}}}}{3 \, b^{4} x^{9} \log \left (f\right )^{4}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.70 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{13}} \, dx=\frac {f^{a + \frac {b}{x^{3}}} \left (- b^{3} \log {\left (f \right )}^{3} + 3 b^{2} x^{3} \log {\left (f \right )}^{2} - 6 b x^{6} \log {\left (f \right )} + 6 x^{9}\right )}{3 b^{4} x^{9} \log {\left (f \right )}^{4}} \]

[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)

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 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{13}} \, dx=\frac {f^{a} \Gamma \left (4, -\frac {b \log \left (f\right )}{x^{3}}\right )}{3 \, b^{4} \log \left (f\right )^{4}} \]

[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)

Giac [F]

\[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{13}} \, dx=\int { \frac {f^{a + \frac {b}{x^{3}}}}{x^{13}} \,d x } \]

[In]

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

[Out]

integrate(f^(a + b/x^3)/x^13, x)

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.72 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{13}} \, dx=-\frac {f^{a+\frac {b}{x^3}}\,\left (\frac {1}{3\,b\,\ln \left (f\right )}-\frac {x^3}{b^2\,{\ln \left (f\right )}^2}+\frac {2\,x^6}{b^3\,{\ln \left (f\right )}^3}-\frac {2\,x^9}{b^4\,{\ln \left (f\right )}^4}\right )}{x^9} \]

[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