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

Optimal. Leaf size=65 \[ -\frac{f^{a+\frac{b}{x}} \left (12 b^2 x^2 \log ^2(f)-4 b^3 x \log ^3(f)+b^4 \log ^4(f)-24 b x^3 \log (f)+24 x^4\right )}{b^5 x^4 \log ^5(f)} \]

[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))

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Rubi [C]  time = 0.0183533, antiderivative size = 22, normalized size of antiderivative = 0.34, 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} \[ -\frac{f^a \text{Gamma}\left (5,-\frac{b \log (f)}{x}\right )}{b^5 \log ^5(f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x)/x^6,x]

[Out]

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

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^6} \, dx &=-\frac{f^a \Gamma \left (5,-\frac{b \log (f)}{x}\right )}{b^5 \log ^5(f)}\\ \end{align*}

Mathematica [C]  time = 0.0027207, size = 22, normalized size = 0.34 \[ -\frac{f^a \text{Gamma}\left (5,-\frac{b \log (f)}{x}\right )}{b^5 \log ^5(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x)/x^6,x]

[Out]

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

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Maple [A]  time = 0.013, size = 119, normalized size = 1.8 \begin{align*}{\frac{1}{{x}^{5}} \left ( -24\,{\frac{{x}^{5}}{{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}+24\,{\frac{{x}^{4}}{{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}-12\,{\frac{{x}^{3}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}+4\,{\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^6,x)

[Out]

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

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Maxima [C]  time = 1.28846, size = 30, normalized size = 0.46 \begin{align*} -\frac{f^{a} \Gamma \left (5, -\frac{b \log \left (f\right )}{x}\right )}{b^{5} \log \left (f\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.8285, size = 162, normalized size = 2.49 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.155604, size = 66, normalized size = 1.02 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x}}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x)/x^6, x)