3.2.65 \(\int \frac {f^{a+\frac {b}{x^3}}}{x^{16}} \, dx\) [165]

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

[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 [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 = {2249} \begin {gather*} -\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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)/x^16,x]

[Out]

-1/3*(f^(a + b/x^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))/(b^
5*x^12*Log[f]^5)

Rule 2249

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{p = Simplify
[(m + 1)/n]}, Simp[(-F^a)*((f/d)^m/(d*n*((-b)*Log[F])^p))*Simplify[FunctionExpand[Gamma[p, (-b)*(c + d*x)^n*Lo
g[F]]]], x] /; IGtQ[p, 0]] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] &&  !TrueQ[$UseGamma]

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] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.00, size = 24, normalized size = 0.35 \begin {gather*} -\frac {f^a \Gamma \left (5,-\frac {b \log (f)}{x^3}\right )}{3 b^5 \log ^5(f)} \end {gather*}

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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Maple [A]
time = 0.04, size = 71, normalized size = 1.03

method	result	size
meijerg	\(\frac {f^{a} \left (24-\frac {\left (\frac {5 b^{4} \ln \left (f \right )^{4}}{x^{12}}-\frac {20 b^{3} \ln \left (f \right )^{3}}{x^{9}}+\frac {60 b^{2} \ln \left (f \right )^{2}}{x^{6}}-\frac {120 b \ln \left (f \right )}{x^{3}}+120\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{3}}}}{5}\right )}{3 b^{5} \ln \left (f \right )^{5}}\)	\(71\)
risch	\(-\frac {\left (24 x^{12}-24 b \,x^{9} \ln \left (f \right )+12 b^{2} x^{6} \ln \left (f \right )^{2}-4 b^{3} x^{3} \ln \left (f \right )^{3}+b^{4} \ln \left (f \right )^{4}\right ) f^{\frac {a \,x^{3}+b}{x^{3}}}}{3 b^{5} \ln \left (f \right )^{5} x^{12}}\)	\(72\)
norman	\(\frac {-\frac {x^{3} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{3 \ln \left (f \right ) b}+\frac {4 x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{3 b^{2} \ln \left (f \right )^{2}}-\frac {4 x^{9} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{b^{3} \ln \left (f \right )^{3}}+\frac {8 x^{12} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{b^{4} \ln \left (f \right )^{4}}-\frac {8 x^{15} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \left (f \right )}}{b^{5} \ln \left (f \right )^{5}}}{x^{15}}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*f^a/b^5/ln(f)^5*(24-1/5*(5*b^4*ln(f)^4/x^12-20*b^3*ln(f)^3/x^9+60*b^2*ln(f)^2/x^6-120*b*ln(f)/x^3+120)*exp
(b*ln(f)/x^3))

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.32, size = 22, normalized size = 0.32 \begin {gather*} -\frac {f^{a} \Gamma \left (5, -\frac {b \log \left (f\right )}{x^{3}}\right )}{3 \, b^{5} \log \left (f\right )^{5}} \end {gather*}

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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Fricas [A]
time = 0.37, size = 71, normalized size = 1.03 \begin {gather*} -\frac {{\left (24 \, x^{12} - 24 \, b x^{9} \log \left (f\right ) + 12 \, b^{2} x^{6} \log \left (f\right )^{2} - 4 \, b^{3} x^{3} \log \left (f\right )^{3} + b^{4} \log \left (f\right )^{4}\right )} f^{\frac {a x^{3} + b}{x^{3}}}}{3 \, b^{5} x^{12} \log \left (f\right )^{5}} \end {gather*}

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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Sympy [A]
time = 0.06, size = 71, normalized size = 1.03 \begin {gather*} \frac {f^{a + \frac {b}{x^{3}}} \left (- b^{4} \log {\left (f \right )}^{4} + 4 b^{3} x^{3} \log {\left (f \right )}^{3} - 12 b^{2} x^{6} \log {\left (f \right )}^{2} + 24 b x^{9} \log {\left (f \right )} - 24 x^{12}\right )}{3 b^{5} x^{12} \log {\left (f \right )}^{5}} \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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