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

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

[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

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Rubi [C]  time = 0.02, 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*}

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Mathematica [C]  time = 0.00, size = 22, normalized size = 0.34 \[ -\frac {f^a \Gamma \left (5,-\frac {b \log (f)}{x}\right )}{b^5 \log ^5(f)} \]

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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fricas [A]  time = 0.42, size = 67, normalized size = 1.03 \[ -\frac {{\left (b^{4} \log \relax (f)^{4} - 4 \, b^{3} x \log \relax (f)^{3} + 12 \, b^{2} x^{2} \log \relax (f)^{2} - 24 \, b x^{3} \log \relax (f) + 24 \, x^{4}\right )} f^{\frac {a x + b}{x}}}{b^{5} x^{4} \log \relax (f)^{5}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x}}}{x^{6}}\,{d x} \]

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)

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maple [A]  time = 0.03, size = 119, normalized size = 1.83 \[ \frac {-\frac {x \,{\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b \ln \relax (f )}+\frac {4 x^{2} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{2} \ln \relax (f )^{2}}-\frac {12 x^{3} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}+\frac {24 x^{4} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{4} \ln \relax (f )^{4}}-\frac {24 x^{5} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{5} \ln \relax (f )^{5}}}{x^{5}} \]

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

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maxima [C]  time = 1.24, size = 22, normalized size = 0.34 \[ -\frac {f^{a} \Gamma \left (5, -\frac {b \log \relax (f)}{x}\right )}{b^{5} \log \relax (f)^{5}} \]

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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mupad [B]  time = 3.60, size = 69, normalized size = 1.06 \[ -\frac {f^{a+\frac {b}{x}}\,\left (\frac {1}{b\,\ln \relax (f)}+\frac {12\,x^2}{b^3\,{\ln \relax (f)}^3}-\frac {24\,x^3}{b^4\,{\ln \relax (f)}^4}+\frac {24\,x^4}{b^5\,{\ln \relax (f)}^5}-\frac {4\,x}{b^2\,{\ln \relax (f)}^2}\right )}{x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.16, size = 66, normalized size = 1.02 \[ \frac {f^{a + \frac {b}{x}} \left (- b^{4} \log {\relax (f )}^{4} + 4 b^{3} x \log {\relax (f )}^{3} - 12 b^{2} x^{2} \log {\relax (f )}^{2} + 24 b x^{3} \log {\relax (f )} - 24 x^{4}\right )}{b^{5} x^{4} \log {\relax (f )}^{5}} \]

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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