3.166 \(\int \frac{f^{a+\frac{b}{x^3}}}{x^{19}} \, dx\)

Optimal. Leaf size=82 \[ \frac{f^{a+\frac{b}{x^3}} \left (60 b^2 x^9 \log ^2(f)-20 b^3 x^6 \log ^3(f)+5 b^4 x^3 \log ^4(f)-b^5 \log ^5(f)-120 b x^{12} \log (f)+120 x^{15}\right )}{3 b^6 x^{15} \log ^6(f)} \]

[Out]

(f^(a + b/x^3)*(120*x^15 - 120*b*x^12*Log[f] + 60*b^2*x^9*Log[f]^2 - 20*b^3*x^6*Log[f]^3 + 5*b^4*x^3*Log[f]^4
- b^5*Log[f]^5))/(3*b^6*x^15*Log[f]^6)

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Rubi [C]  time = 0.0228049, antiderivative size = 24, normalized size of antiderivative = 0.29, 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} \[ \frac{f^a \text{Gamma}\left (6,-\frac{b \log (f)}{x^3}\right )}{3 b^6 \log ^6(f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)/x^19,x]

[Out]

(f^a*Gamma[6, -((b*Log[f])/x^3)])/(3*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^3}}}{x^{19}} \, dx &=\frac{f^a \Gamma \left (6,-\frac{b \log (f)}{x^3}\right )}{3 b^6 \log ^6(f)}\\ \end{align*}

Mathematica [C]  time = 0.0029883, size = 24, normalized size = 0.29 \[ \frac{f^a \text{Gamma}\left (6,-\frac{b \log (f)}{x^3}\right )}{3 b^6 \log ^6(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)/x^19,x]

[Out]

(f^a*Gamma[6, -((b*Log[f])/x^3)])/(3*b^6*Log[f]^6)

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Maple [A]  time = 0.029, size = 84, normalized size = 1. \begin{align*} -{\frac{-120\,{x}^{15}+120\,b{x}^{12}\ln \left ( f \right ) -60\,{b}^{2}{x}^{9} \left ( \ln \left ( f \right ) \right ) ^{2}+20\,{b}^{3}{x}^{6} \left ( \ln \left ( f \right ) \right ) ^{3}-5\,{b}^{4}{x}^{3} \left ( \ln \left ( f \right ) \right ) ^{4}+{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}{3\, \left ( \ln \left ( f \right ) \right ) ^{6}{b}^{6}{x}^{15}}{f}^{{\frac{a{x}^{3}+b}{{x}^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^3)/x^19,x)

[Out]

-1/3*(-120*x^15+120*b*x^12*ln(f)-60*b^2*x^9*ln(f)^2+20*b^3*x^6*ln(f)^3-5*b^4*x^3*ln(f)^4+b^5*ln(f)^5)/ln(f)^6/
b^6/x^15*f^((a*x^3+b)/x^3)

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Maxima [C]  time = 1.18733, size = 30, normalized size = 0.37 \begin{align*} \frac{f^{a} \Gamma \left (6, -\frac{b \log \left (f\right )}{x^{3}}\right )}{3 \, b^{6} \log \left (f\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^19,x, algorithm="maxima")

[Out]

1/3*f^a*gamma(6, -b*log(f)/x^3)/(b^6*log(f)^6)

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Fricas [A]  time = 2.01496, size = 211, normalized size = 2.57 \begin{align*} \frac{{\left (120 \, x^{15} - 120 \, b x^{12} \log \left (f\right ) + 60 \, b^{2} x^{9} \log \left (f\right )^{2} - 20 \, b^{3} x^{6} \log \left (f\right )^{3} + 5 \, b^{4} x^{3} \log \left (f\right )^{4} - b^{5} \log \left (f\right )^{5}\right )} f^{\frac{a x^{3} + b}{x^{3}}}}{3 \, b^{6} x^{15} \log \left (f\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^19,x, algorithm="fricas")

[Out]

1/3*(120*x^15 - 120*b*x^12*log(f) + 60*b^2*x^9*log(f)^2 - 20*b^3*x^6*log(f)^3 + 5*b^4*x^3*log(f)^4 - b^5*log(f
)^5)*f^((a*x^3 + b)/x^3)/(b^6*x^15*log(f)^6)

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Sympy [A]  time = 0.161392, size = 85, normalized size = 1.04 \begin{align*} \frac{f^{a + \frac{b}{x^{3}}} \left (- b^{5} \log{\left (f \right )}^{5} + 5 b^{4} x^{3} \log{\left (f \right )}^{4} - 20 b^{3} x^{6} \log{\left (f \right )}^{3} + 60 b^{2} x^{9} \log{\left (f \right )}^{2} - 120 b x^{12} \log{\left (f \right )} + 120 x^{15}\right )}{3 b^{6} x^{15} \log{\left (f \right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**3)/x**19,x)

[Out]

f**(a + b/x**3)*(-b**5*log(f)**5 + 5*b**4*x**3*log(f)**4 - 20*b**3*x**6*log(f)**3 + 60*b**2*x**9*log(f)**2 - 1
20*b*x**12*log(f) + 120*x**15)/(3*b**6*x**15*log(f)**6)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{3}}}}{x^{19}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^19,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)/x^19, x)