3.165 \(\int \frac {f^{a+\frac {b}{x^3}}}{x^{16}} \, dx\)

Optimal. Leaf size=69 \[ -\frac {f^{a+\frac {b}{x^3}} \left (b^4 \log ^4(f)-4 b^3 x^3 \log ^3(f)+12 b^2 x^6 \log ^2(f)-24 b x^9 \log (f)+24 x^{12}\right )}{3 b^5 x^{12} \log ^5(f)} \]

[Out]

-1/3*f^(a+b/x^3)*(24*x^12-24*b*x^9*ln(f)+12*b^2*x^6*ln(f)^2-4*b^3*x^3*ln(f)^3+b^4*ln(f)^4)/b^5/x^12/ln(f)^5

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Rubi [C]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 0.35, 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} \[ -\frac {f^a \text {Gamma}\left (5,-\frac {b \log (f)}{x^3}\right )}{3 b^5 \log ^5(f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)/x^16,x]

[Out]

-(f^a*Gamma[5, -((b*Log[f])/x^3)])/(3*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^3}}}{x^{16}} \, dx &=-\frac {f^a \Gamma \left (5,-\frac {b \log (f)}{x^3}\right )}{3 b^5 \log ^5(f)}\\ \end {align*}

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Mathematica [C]  time = 0.00, size = 24, normalized size = 0.35 \[ -\frac {f^a \Gamma \left (5,-\frac {b \log (f)}{x^3}\right )}{3 b^5 \log ^5(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)/x^16,x]

[Out]

-1/3*(f^a*Gamma[5, -((b*Log[f])/x^3)])/(b^5*Log[f]^5)

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fricas [A]  time = 0.42, size = 71, normalized size = 1.03 \[ -\frac {{\left (24 \, x^{12} - 24 \, b x^{9} \log \relax (f) + 12 \, b^{2} x^{6} \log \relax (f)^{2} - 4 \, b^{3} x^{3} \log \relax (f)^{3} + b^{4} \log \relax (f)^{4}\right )} f^{\frac {a x^{3} + b}{x^{3}}}}{3 \, b^{5} x^{12} \log \relax (f)^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^16,x, algorithm="fricas")

[Out]

-1/3*(24*x^12 - 24*b*x^9*log(f) + 12*b^2*x^6*log(f)^2 - 4*b^3*x^3*log(f)^3 + b^4*log(f)^4)*f^((a*x^3 + b)/x^3)
/(b^5*x^12*log(f)^5)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x^{3}}}}{x^{16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^16,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)/x^16, x)

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maple [A]  time = 0.04, size = 121, normalized size = 1.75 \[ \frac {-\frac {8 x^{15} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{b^{5} \ln \relax (f )^{5}}+\frac {8 x^{12} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{b^{4} \ln \relax (f )^{4}}-\frac {4 x^{9} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}+\frac {4 x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{3 b^{2} \ln \relax (f )^{2}}-\frac {x^{3} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{3 b \ln \relax (f )}}{x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^3)/x^16,x)

[Out]

(-8/b^5/ln(f)^5*x^15*exp((a+b/x^3)*ln(f))+8/b^4*x^12*exp((a+b/x^3)*ln(f))/ln(f)^4-4/b^3*x^9*exp((a+b/x^3)*ln(f
))/ln(f)^3+4/3/b^2*x^6*exp((a+b/x^3)*ln(f))/ln(f)^2-1/3/b*x^3*exp((a+b/x^3)*ln(f))/ln(f))/x^15

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maxima [C]  time = 1.39, size = 22, normalized size = 0.32 \[ -\frac {f^{a} \Gamma \left (5, -\frac {b \log \relax (f)}{x^{3}}\right )}{3 \, b^{5} \log \relax (f)^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^16,x, algorithm="maxima")

[Out]

-1/3*f^a*gamma(5, -b*log(f)/x^3)/(b^5*log(f)^5)

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mupad [B]  time = 3.52, size = 72, normalized size = 1.04 \[ -\frac {f^{a+\frac {b}{x^3}}\,\left (\frac {1}{3\,b\,\ln \relax (f)}-\frac {4\,x^3}{3\,b^2\,{\ln \relax (f)}^2}+\frac {4\,x^6}{b^3\,{\ln \relax (f)}^3}-\frac {8\,x^9}{b^4\,{\ln \relax (f)}^4}+\frac {8\,x^{12}}{b^5\,{\ln \relax (f)}^5}\right )}{x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^3)/x^16,x)

[Out]

-(f^(a + b/x^3)*(1/(3*b*log(f)) - (4*x^3)/(3*b^2*log(f)^2) + (4*x^6)/(b^3*log(f)^3) - (8*x^9)/(b^4*log(f)^4) +
(8*x^12)/(b^5*log(f)^5)))/x^12

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sympy [A]  time = 0.17, size = 71, normalized size = 1.03 \[ \frac {f^{a + \frac {b}{x^{3}}} \left (- b^{4} \log {\relax (f )}^{4} + 4 b^{3} x^{3} \log {\relax (f )}^{3} - 12 b^{2} x^{6} \log {\relax (f )}^{2} + 24 b x^{9} \log {\relax (f )} - 24 x^{12}\right )}{3 b^{5} x^{12} \log {\relax (f )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**3)/x**16,x)

[Out]

f**(a + b/x**3)*(-b**4*log(f)**4 + 4*b**3*x**3*log(f)**3 - 12*b**2*x**6*log(f)**2 + 24*b*x**9*log(f) - 24*x**1
2)/(3*b**5*x**12*log(f)**5)

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