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

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

[Out]

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

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Rubi [C]  time = 0.0226648, 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*}

Mathematica [C]  time = 0.0027375, size = 24, normalized size = 0.35 \[ -\frac{f^a \text{Gamma}\left (5,-\frac{b \log (f)}{x^3}\right )}{3 b^5 \log ^5(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)/x^16,x]

[Out]

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

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Maple [A]  time = 0.028, size = 121, normalized size = 1.8 \begin{align*}{\frac{1}{{x}^{15}} \left ( -8\,{\frac{{x}^{15}}{{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}+8\,{\frac{{x}^{12}}{{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}-4\,{\frac{{x}^{9}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}+{\frac{4\,{x}^{6}}{3\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}-{\frac{{x}^{3}}{3\,b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}} \right ) } \end{align*}

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

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Maxima [C]  time = 1.30783, size = 30, normalized size = 0.43 \begin{align*} -\frac{f^{a} \Gamma \left (5, -\frac{b \log \left (f\right )}{x^{3}}\right )}{3 \, 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^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 = 1.65528, size = 178, normalized size = 2.58 \begin{align*} -\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{align*}

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 = 0.152325, size = 71, normalized size = 1.03 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{3}}}}{x^{16}}\,{d x} \end{align*}

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)