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

Optimal. Leaf size=77 \[ \frac{f^{a+\frac{b}{x}} \left (60 b^2 x^3 \log ^2(f)-20 b^3 x^2 \log ^3(f)+5 b^4 x \log ^4(f)-b^5 \log ^5(f)-120 b x^4 \log (f)+120 x^5\right )}{b^6 x^5 \log ^6(f)} \]

[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)

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Rubi [C]  time = 0.0190488, antiderivative size = 21, normalized size of antiderivative = 0.27, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2218} \[ \frac{f^a \text{Gamma}\left (6,-\frac{b \log (f)}{x}\right )}{b^6 \log ^6(f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x)/x^7,x]

[Out]

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

Rule 2218

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

Rubi steps

\begin{align*} \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)}\\ \end{align*}

Mathematica [C]  time = 0.002575, size = 21, normalized size = 0.27 \[ \frac{f^a \text{Gamma}\left (6,-\frac{b \log (f)}{x}\right )}{b^6 \log ^6(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x)/x^7,x]

[Out]

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

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Maple [A]  time = 0.016, size = 142, normalized size = 1.8 \begin{align*}{\frac{1}{{x}^{6}} \left ( 120\,{\frac{{x}^{6}}{{b}^{6} \left ( \ln \left ( f \right ) \right ) ^{6}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}-120\,{\frac{{x}^{5}}{{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}+60\,{\frac{{x}^{4}}{{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}-20\,{\frac{{x}^{3}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}+5\,{\frac{{x}^{2}}{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}-{\frac{x}{b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.16134, size = 28, normalized size = 0.36 \begin{align*} \frac{f^{a} \Gamma \left (6, -\frac{b \log \left (f\right )}{x}\right )}{b^{6} \log \left (f\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.93082, size = 194, normalized size = 2.52 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.162109, size = 80, normalized size = 1.04 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x}}}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x)/x^7, x)