\(\int \frac {f^{a+\frac {b}{x}}}{x^7} \, dx\) [127]

Optimal result

Integrand size = 13, antiderivative size = 77 \[ \int \frac {f^{a+\frac {b}{x}}}{x^7} \, dx=\frac {f^{a+\frac {b}{x}} \left (120 x^5-120 b x^4 \log (f)+60 b^2 x^3 \log ^2(f)-20 b^3 x^2 \log ^3(f)+5 b^4 x \log ^4(f)-b^5 \log ^5(f)\right )}{b^6 x^5 \log ^6(f)} \]

[Out]

f^(a+b/x)*(120*x^5-120*b*x^4*ln(f)+60*b^2*x^3*ln(f)^2-20*b^3*x^2*ln(f)^3+5*b^4*x*ln(f)^4-b^5*ln(f)^5)/b^6/x^5/
ln(f)^6

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 77, 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} \[ \int \frac {f^{a+\frac {b}{x}}}{x^7} \, dx=\frac {f^{a+\frac {b}{x}} \left (-b^5 \log ^5(f)+5 b^4 x \log ^4(f)-20 b^3 x^2 \log ^3(f)+60 b^2 x^3 \log ^2(f)-120 b x^4 \log (f)+120 x^5\right )}{b^6 x^5 \log ^6(f)} \]

[In]

Int[f^(a + b/x)/x^7,x]

[Out]

(f^(a + b/x)*(120*x^5 - 120*b*x^4*Log[f] + 60*b^2*x^3*Log[f]^2 - 20*b^3*x^2*Log[f]^3 + 5*b^4*x*Log[f]^4 - b^5*
Log[f]^5))/(b^6*x^5*Log[f]^6)

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 (120 x^5-120 b x^4 \log (f)+60 b^2 x^3 \log ^2(f)-20 b^3 x^2 \log ^3(f)+5 b^4 x \log ^4(f)-b^5 \log ^5(f)\right )}{b^6 x^5 \log ^6(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 = 21, normalized size of antiderivative = 0.27 \[ \int \frac {f^{a+\frac {b}{x}}}{x^7} \, dx=\frac {f^a \Gamma \left (6,-\frac {b \log (f)}{x}\right )}{b^6 \log ^6(f)} \]

[In]

Integrate[f^(a + b/x)/x^7,x]

[Out]

(f^a*Gamma[6, -((b*Log[f])/x)])/(b^6*Log[f]^6)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {\left (b^{5} \ln \left (f \right )^{5}-5 b^{4} x \ln \left (f \right )^{4}+20 b^{3} x^{2} \ln \left (f \right )^{3}-60 b^{2} x^{3} \ln \left (f \right )^{2}+120 b \,x^{4} \ln \left (f \right )-120 x^{5}\right ) f^{\frac {a x +b}{x}}}{b^{6} \ln \left (f \right )^{6} x^{5}}\)	\(80\)
meijerg	\(-\frac {f^{a} \left (120-\frac {\left (-\frac {6 b^{5} \ln \left (f \right )^{5}}{x^{5}}+\frac {30 b^{4} \ln \left (f \right )^{4}}{x^{4}}-\frac {120 b^{3} \ln \left (f \right )^{3}}{x^{3}}+\frac {360 b^{2} \ln \left (f \right )^{2}}{x^{2}}-\frac {720 b \ln \left (f \right )}{x}+720\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x}}}{6}\right )}{b^{6} \ln \left (f \right )^{6}}\)	\(83\)
parallelrisch	\(\frac {-\ln \left (f \right )^{5} f^{a +\frac {b}{x}} b^{5}+5 \ln \left (f \right )^{4} x \,f^{a +\frac {b}{x}} b^{4}-20 \ln \left (f \right )^{3} x^{2} f^{a +\frac {b}{x}} b^{3}+60 \ln \left (f \right )^{2} x^{3} f^{a +\frac {b}{x}} b^{2}-120 \ln \left (f \right ) x^{4} f^{a +\frac {b}{x}} b +120 f^{a +\frac {b}{x}} x^{5}}{x^{5} b^{6} \ln \left (f \right )^{6}}\)	\(123\)
norman	\(\frac {\frac {120 x^{6} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{6} \ln \left (f \right )^{6}}-\frac {x \,{\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b \ln \left (f \right )}+\frac {5 x^{2} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{2} \ln \left (f \right )^{2}}-\frac {20 x^{3} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{3} \ln \left (f \right )^{3}}+\frac {60 x^{4} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{4} \ln \left (f \right )^{4}}-\frac {120 x^{5} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \left (f \right )}}{b^{5} \ln \left (f \right )^{5}}}{x^{6}}\)	\(142\)

[In]

int(f^(a+b/x)/x^7,x,method=_RETURNVERBOSE)

[Out]

-(b^5*ln(f)^5-5*b^4*x*ln(f)^4+20*b^3*x^2*ln(f)^3-60*b^2*x^3*ln(f)^2+120*b*x^4*ln(f)-120*x^5)/b^6/ln(f)^6/x^5*f
^((a*x+b)/x)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03 \[ \int \frac {f^{a+\frac {b}{x}}}{x^7} \, dx=-\frac {{\left (b^{5} \log \left (f\right )^{5} - 5 \, b^{4} x \log \left (f\right )^{4} + 20 \, b^{3} x^{2} \log \left (f\right )^{3} - 60 \, b^{2} x^{3} \log \left (f\right )^{2} + 120 \, b x^{4} \log \left (f\right ) - 120 \, x^{5}\right )} f^{\frac {a x + b}{x}}}{b^{6} x^{5} \log \left (f\right )^{6}} \]

[In]

integrate(f^(a+b/x)/x^7,x, algorithm="fricas")

[Out]

-(b^5*log(f)^5 - 5*b^4*x*log(f)^4 + 20*b^3*x^2*log(f)^3 - 60*b^2*x^3*log(f)^2 + 120*b*x^4*log(f) - 120*x^5)*f^
((a*x + b)/x)/(b^6*x^5*log(f)^6)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.04 \[ \int \frac {f^{a+\frac {b}{x}}}{x^7} \, dx=\frac {f^{a + \frac {b}{x}} \left (- b^{5} \log {\left (f \right )}^{5} + 5 b^{4} x \log {\left (f \right )}^{4} - 20 b^{3} x^{2} \log {\left (f \right )}^{3} + 60 b^{2} x^{3} \log {\left (f \right )}^{2} - 120 b x^{4} \log {\left (f \right )} + 120 x^{5}\right )}{b^{6} x^{5} \log {\left (f \right )}^{6}} \]

[In]

integrate(f**(a+b/x)/x**7,x)

[Out]

f**(a + b/x)*(-b**5*log(f)**5 + 5*b**4*x*log(f)**4 - 20*b**3*x**2*log(f)**3 + 60*b**2*x**3*log(f)**2 - 120*b*x
**4*log(f) + 120*x**5)/(b**6*x**5*log(f)**6)

Maxima [C] (verification not implemented)

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

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.27 \[ \int \frac {f^{a+\frac {b}{x}}}{x^7} \, dx=\frac {f^{a} \Gamma \left (6, -\frac {b \log \left (f\right )}{x}\right )}{b^{6} \log \left (f\right )^{6}} \]

[In]

integrate(f^(a+b/x)/x^7,x, algorithm="maxima")

[Out]

f^a*gamma(6, -b*log(f)/x)/(b^6*log(f)^6)

Giac [F]

\[ \int \frac {f^{a+\frac {b}{x}}}{x^7} \, dx=\int { \frac {f^{a + \frac {b}{x}}}{x^{7}} \,d x } \]

[In]

integrate(f^(a+b/x)/x^7,x, algorithm="giac")

[Out]

integrate(f^(a + b/x)/x^7, x)

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int \frac {f^{a+\frac {b}{x}}}{x^7} \, dx=-\frac {f^{a+\frac {b}{x}}\,\left (\frac {1}{b\,\ln \left (f\right )}+\frac {20\,x^2}{b^3\,{\ln \left (f\right )}^3}-\frac {60\,x^3}{b^4\,{\ln \left (f\right )}^4}+\frac {120\,x^4}{b^5\,{\ln \left (f\right )}^5}-\frac {120\,x^5}{b^6\,{\ln \left (f\right )}^6}-\frac {5\,x}{b^2\,{\ln \left (f\right )}^2}\right )}{x^5} \]

[In]

int(f^(a + b/x)/x^7,x)

[Out]

-(f^(a + b/x)*(1/(b*log(f)) + (20*x^2)/(b^3*log(f)^3) - (60*x^3)/(b^4*log(f)^4) + (120*x^4)/(b^5*log(f)^5) - (
120*x^5)/(b^6*log(f)^6) - (5*x)/(b^2*log(f)^2)))/x^5