\(\int \frac {f^{a+\frac {b}{x^3}}}{x^{10}} \, dx\) [163]

Optimal result

Integrand size = 13, antiderivative size = 67 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 f^{a+\frac {b}{x^3}}}{3 b^3 \log ^3(f)}+\frac {2 f^{a+\frac {b}{x^3}}}{3 b^2 x^3 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^6 \log (f)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2243, 2240} \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 f^{a+\frac {b}{x^3}}}{3 b^3 \log ^3(f)}+\frac {2 f^{a+\frac {b}{x^3}}}{3 b^2 x^3 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^6 \log (f)} \]

[In]

Int[f^(a + b/x^3)/x^10,x]

[Out]

(-2*f^(a + b/x^3))/(3*b^3*Log[f]^3) + (2*f^(a + b/x^3))/(3*b^2*x^3*Log[f]^2) - f^(a + b/x^3)/(3*b*x^6*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^6 \log (f)}-\frac {2 \int \frac {f^{a+\frac {b}{x^3}}}{x^7} \, dx}{b \log (f)} \\ & = \frac {2 f^{a+\frac {b}{x^3}}}{3 b^2 x^3 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^6 \log (f)}+\frac {2 \int \frac {f^{a+\frac {b}{x^3}}}{x^4} \, dx}{b^2 \log ^2(f)} \\ & = -\frac {2 f^{a+\frac {b}{x^3}}}{3 b^3 \log ^3(f)}+\frac {2 f^{a+\frac {b}{x^3}}}{3 b^2 x^3 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^6 \log (f)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.67 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {f^{a+\frac {b}{x^3}} \left (2 x^6-2 b x^3 \log (f)+b^2 \log ^2(f)\right )}{3 b^3 x^6 \log ^3(f)} \]

[In]

Integrate[f^(a + b/x^3)/x^10,x]

[Out]

-1/3*(f^(a + b/x^3)*(2*x^6 - 2*b*x^3*Log[f] + b^2*Log[f]^2))/(b^3*x^6*Log[f]^3)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70

method	result	size
meijerg	\(\frac {f^{a} \left (2-\frac {\left (\frac {3 b^{2} \ln \left (f \right )^{2}}{x^{6}}-\frac {6 b \ln \left (f \right )}{x^{3}}+6\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{3}}}}{3}\right )}{3 b^{3} \ln \left (f \right )^{3}}\)	\(47\)
risch	\(-\frac {\left (2 x^{6}-2 b \,x^{3} \ln \left (f \right )+\ln \left (f \right )^{2} b^{2}\right ) f^{\frac {a \,x^{3}+b}{x^{3}}}}{3 \ln \left (f \right )^{3} b^{3} x^{6}}\)	\(48\)
parallelrisch	\(\frac {-2 f^{a +\frac {b}{x^{3}}} x^{6}+2 b \,f^{a +\frac {b}{x^{3}}} x^{3} \ln \left (f \right )-f^{a +\frac {b}{x^{3}}} \ln \left (f \right )^{2} b^{2}}{3 x^{6} \ln \left (f \right )^{3} b^{3}}\)	\(63\)
norman	\(\frac {-\frac {x^{3} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{3 b \ln \left (f \right )}+\frac {2 x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{3 b^{2} \ln \left (f \right )^{2}}-\frac {2 x^{9} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{3 b^{3} \ln \left (f \right )^{3}}}{x^{9}}\)	\(75\)

[In]

int(f^(a+b/x^3)/x^10,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {{\left (2 \, x^{6} - 2 \, b x^{3} \log \left (f\right ) + b^{2} \log \left (f\right )^{2}\right )} f^{\frac {a x^{3} + b}{x^{3}}}}{3 \, b^{3} x^{6} \log \left (f\right )^{3}} \]

[In]

integrate(f^(a+b/x^3)/x^10,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.66 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{10}} \, dx=\frac {f^{a + \frac {b}{x^{3}}} \left (- b^{2} \log {\left (f \right )}^{2} + 2 b x^{3} \log {\left (f \right )} - 2 x^{6}\right )}{3 b^{3} x^{6} \log {\left (f \right )}^{3}} \]

[In]

integrate(f**(a+b/x**3)/x**10,x)

[Out]

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

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 = 0.33 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {f^{a} \Gamma \left (3, -\frac {b \log \left (f\right )}{x^{3}}\right )}{3 \, b^{3} \log \left (f\right )^{3}} \]

[In]

integrate(f^(a+b/x^3)/x^10,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{10}} \, dx=\int { \frac {f^{a + \frac {b}{x^{3}}}}{x^{10}} \,d x } \]

[In]

integrate(f^(a+b/x^3)/x^10,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)/x^10, x)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {f^{a+\frac {b}{x^3}}\,\left (\frac {1}{3\,b\,\ln \left (f\right )}-\frac {2\,x^3}{3\,b^2\,{\ln \left (f\right )}^2}+\frac {2\,x^6}{3\,b^3\,{\ln \left (f\right )}^3}\right )}{x^6} \]

[In]

int(f^(a + b/x^3)/x^10,x)

[Out]

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