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

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

[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

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Rubi [C]  time = 0.02, 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*}

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Mathematica [C]  time = 0.00, size = 21, normalized size = 0.27 \[ \frac {f^a \Gamma \left (6,-\frac {b \log (f)}{x}\right )}{b^6 \log ^6(f)} \]

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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fricas [A]  time = 0.41, size = 79, normalized size = 1.03 \[ -\frac {{\left (b^{5} \log \relax (f)^{5} - 5 \, b^{4} x \log \relax (f)^{4} + 20 \, b^{3} x^{2} \log \relax (f)^{3} - 60 \, b^{2} x^{3} \log \relax (f)^{2} + 120 \, b x^{4} \log \relax (f) - 120 \, x^{5}\right )} f^{\frac {a x + b}{x}}}{b^{6} x^{5} \log \relax (f)^{6}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x}}}{x^{7}}\,{d x} \]

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)

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maple [A]  time = 0.03, size = 142, normalized size = 1.84 \[ \frac {-\frac {x \,{\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b \ln \relax (f )}+\frac {5 x^{2} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{2} \ln \relax (f )^{2}}-\frac {20 x^{3} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}+\frac {60 x^{4} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{4} \ln \relax (f )^{4}}-\frac {120 x^{5} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{5} \ln \relax (f )^{5}}+\frac {120 x^{6} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{6} \ln \relax (f )^{6}}}{x^{6}} \]

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*x^5*exp((a+b/x)*ln(f))/ln(f)^5+60/b^4*x^4*exp((a+b/x)*ln(f))/l
n(f)^4-20/b^3*x^3*exp((a+b/x)*ln(f))/ln(f)^3+5/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^6

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maxima [C]  time = 1.29, size = 21, normalized size = 0.27 \[ \frac {f^{a} \Gamma \left (6, -\frac {b \log \relax (f)}{x}\right )}{b^{6} \log \relax (f)^{6}} \]

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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mupad [B]  time = 3.65, size = 81, normalized size = 1.05 \[ -\frac {f^{a+\frac {b}{x}}\,\left (\frac {1}{b\,\ln \relax (f)}+\frac {20\,x^2}{b^3\,{\ln \relax (f)}^3}-\frac {60\,x^3}{b^4\,{\ln \relax (f)}^4}+\frac {120\,x^4}{b^5\,{\ln \relax (f)}^5}-\frac {120\,x^5}{b^6\,{\ln \relax (f)}^6}-\frac {5\,x}{b^2\,{\ln \relax (f)}^2}\right )}{x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.17, size = 80, normalized size = 1.04 \[ \frac {f^{a + \frac {b}{x}} \left (- b^{5} \log {\relax (f )}^{5} + 5 b^{4} x \log {\relax (f )}^{4} - 20 b^{3} x^{2} \log {\relax (f )}^{3} + 60 b^{2} x^{3} \log {\relax (f )}^{2} - 120 b x^{4} \log {\relax (f )} + 120 x^{5}\right )}{b^{6} x^{5} \log {\relax (f )}^{6}} \]

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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