\(\int \frac {f^{a+\frac {b}{x}}}{x^6} \, dx\) [126]

Optimal result

Integrand size = 13, antiderivative size = 65 \[ \int \frac {f^{a+\frac {b}{x}}}{x^6} \, dx=-\frac {f^{a+\frac {b}{x}} \left (24 x^4-24 b x^3 \log (f)+12 b^2 x^2 \log ^2(f)-4 b^3 x \log ^3(f)+b^4 \log ^4(f)\right )}{b^5 x^4 \log ^5(f)} \]

[Out]

-f^(a+b/x)*(24*x^4-24*b*x^3*ln(f)+12*b^2*x^2*ln(f)^2-4*b^3*x*ln(f)^3+b^4*ln(f)^4)/b^5/x^4/ln(f)^5

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, 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} \[ \int \frac {f^{a+\frac {b}{x}}}{x^6} \, dx=-\frac {f^{a+\frac {b}{x}} \left (b^4 \log ^4(f)-4 b^3 x \log ^3(f)+12 b^2 x^2 \log ^2(f)-24 b x^3 \log (f)+24 x^4\right )}{b^5 x^4 \log ^5(f)} \]

[In]

Int[f^(a + b/x)/x^6,x]

[Out]

-((f^(a + b/x)*(24*x^4 - 24*b*x^3*Log[f] + 12*b^2*x^2*Log[f]^2 - 4*b^3*x*Log[f]^3 + b^4*Log[f]^4))/(b^5*x^4*Lo
g[f]^5))

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}} \left (24 x^4-24 b x^3 \log (f)+12 b^2 x^2 \log ^2(f)-4 b^3 x \log ^3(f)+b^4 \log ^4(f)\right )}{b^5 x^4 \log ^5(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 = 22, normalized size of antiderivative = 0.34 \[ \int \frac {f^{a+\frac {b}{x}}}{x^6} \, dx=-\frac {f^a \Gamma \left (5,-\frac {b \log (f)}{x}\right )}{b^5 \log ^5(f)} \]

[In]

Integrate[f^(a + b/x)/x^6,x]

[Out]

-((f^a*Gamma[5, -((b*Log[f])/x)])/(b^5*Log[f]^5))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.05

method	result	size
risch	\(-\frac {\left (24 x^{4}-24 b \,x^{3} \ln \left (f \right )+12 b^{2} x^{2} \ln \left (f \right )^{2}-4 b^{3} x \ln \left (f \right )^{3}+b^{4} \ln \left (f \right )^{4}\right ) f^{\frac {a x +b}{x}}}{b^{5} \ln \left (f \right )^{5} x^{4}}\)	\(68\)
meijerg	\(\frac {f^{a} \left (24-\frac {\left (\frac {5 b^{4} \ln \left (f \right )^{4}}{x^{4}}-\frac {20 b^{3} \ln \left (f \right )^{3}}{x^{3}}+\frac {60 b^{2} \ln \left (f \right )^{2}}{x^{2}}-\frac {120 b \ln \left (f \right )}{x}+120\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x}}}{5}\right )}{b^{5} \ln \left (f \right )^{5}}\)	\(70\)
parallelrisch	\(\frac {-\ln \left (f \right )^{4} f^{a +\frac {b}{x}} b^{4}+4 \ln \left (f \right )^{3} x \,f^{a +\frac {b}{x}} b^{3}-12 \ln \left (f \right )^{2} x^{2} f^{a +\frac {b}{x}} b^{2}+24 \ln \left (f \right ) x^{3} f^{a +\frac {b}{x}} b -24 f^{a +\frac {b}{x}} x^{4}}{x^{4} b^{5} \ln \left (f \right )^{5}}\)	\(102\)
norman	\(\frac {-\frac {x \,{\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b \ln \left (f \right )}+\frac {4 x^{2} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{2} \ln \left (f \right )^{2}}-\frac {12 x^{3} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{3} \ln \left (f \right )^{3}}+\frac {24 x^{4} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{4} \ln \left (f \right )^{4}}-\frac {24 x^{5} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{5} \ln \left (f \right )^{5}}}{x^{5}}\)	\(119\)

[In]

int(f^(a+b/x)/x^6,x,method=_RETURNVERBOSE)

[Out]

-(24*x^4-24*b*x^3*ln(f)+12*b^2*x^2*ln(f)^2-4*b^3*x*ln(f)^3+b^4*ln(f)^4)/b^5/ln(f)^5/x^4*f^((a*x+b)/x)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03 \[ \int \frac {f^{a+\frac {b}{x}}}{x^6} \, dx=-\frac {{\left (b^{4} \log \left (f\right )^{4} - 4 \, b^{3} x \log \left (f\right )^{3} + 12 \, b^{2} x^{2} \log \left (f\right )^{2} - 24 \, b x^{3} \log \left (f\right ) + 24 \, x^{4}\right )} f^{\frac {a x + b}{x}}}{b^{5} x^{4} \log \left (f\right )^{5}} \]

[In]

integrate(f^(a+b/x)/x^6,x, algorithm="fricas")

[Out]

-(b^4*log(f)^4 - 4*b^3*x*log(f)^3 + 12*b^2*x^2*log(f)^2 - 24*b*x^3*log(f) + 24*x^4)*f^((a*x + b)/x)/(b^5*x^4*l
og(f)^5)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.02 \[ \int \frac {f^{a+\frac {b}{x}}}{x^6} \, dx=\frac {f^{a + \frac {b}{x}} \left (- b^{4} \log {\left (f \right )}^{4} + 4 b^{3} x \log {\left (f \right )}^{3} - 12 b^{2} x^{2} \log {\left (f \right )}^{2} + 24 b x^{3} \log {\left (f \right )} - 24 x^{4}\right )}{b^{5} x^{4} \log {\left (f \right )}^{5}} \]

[In]

integrate(f**(a+b/x)/x**6,x)

[Out]

f**(a + b/x)*(-b**4*log(f)**4 + 4*b**3*x*log(f)**3 - 12*b**2*x**2*log(f)**2 + 24*b*x**3*log(f) - 24*x**4)/(b**
5*x**4*log(f)**5)

Maxima [C] (verification not implemented)

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

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.34 \[ \int \frac {f^{a+\frac {b}{x}}}{x^6} \, dx=-\frac {f^{a} \Gamma \left (5, -\frac {b \log \left (f\right )}{x}\right )}{b^{5} \log \left (f\right )^{5}} \]

[In]

integrate(f^(a+b/x)/x^6,x, algorithm="maxima")

[Out]

-f^a*gamma(5, -b*log(f)/x)/(b^5*log(f)^5)

Giac [F]

\[ \int \frac {f^{a+\frac {b}{x}}}{x^6} \, dx=\int { \frac {f^{a + \frac {b}{x}}}{x^{6}} \,d x } \]

[In]

integrate(f^(a+b/x)/x^6,x, algorithm="giac")

[Out]

integrate(f^(a + b/x)/x^6, x)

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.06 \[ \int \frac {f^{a+\frac {b}{x}}}{x^6} \, dx=-\frac {f^{a+\frac {b}{x}}\,\left (\frac {1}{b\,\ln \left (f\right )}+\frac {12\,x^2}{b^3\,{\ln \left (f\right )}^3}-\frac {24\,x^3}{b^4\,{\ln \left (f\right )}^4}+\frac {24\,x^4}{b^5\,{\ln \left (f\right )}^5}-\frac {4\,x}{b^2\,{\ln \left (f\right )}^2}\right )}{x^4} \]

[In]

int(f^(a + b/x)/x^6,x)

[Out]

-(f^(a + b/x)*(1/(b*log(f)) + (12*x^2)/(b^3*log(f)^3) - (24*x^3)/(b^4*log(f)^4) + (24*x^4)/(b^5*log(f)^5) - (4
*x)/(b^2*log(f)^2)))/x^4