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

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

[Out]

1/3*f^(a+b/x^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
)/b^6/x^15/ln(f)^6

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Rubi [C]  time = 0.02, 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*}

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Mathematica [C]  time = 0.00, size = 24, normalized size = 0.29 \[ \frac {f^a \Gamma \left (6,-\frac {b \log (f)}{x^3}\right )}{3 b^6 \log ^6(f)} \]

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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fricas [A]  time = 0.41, size = 84, normalized size = 1.02 \[ \frac {{\left (120 \, x^{15} - 120 \, b x^{12} \log \relax (f) + 60 \, b^{2} x^{9} \log \relax (f)^{2} - 20 \, b^{3} x^{6} \log \relax (f)^{3} + 5 \, b^{4} x^{3} \log \relax (f)^{4} - b^{5} \log \relax (f)^{5}\right )} f^{\frac {a x^{3} + b}{x^{3}}}}{3 \, b^{6} x^{15} \log \relax (f)^{6}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x^{3}}}}{x^{19}}\,{d x} \]

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)

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maple [A]  time = 0.03, size = 84, normalized size = 1.02 \[ -\frac {\left (-120 x^{15}+120 b \,x^{12} \ln \relax (f )-60 b^{2} x^{9} \ln \relax (f )^{2}+20 b^{3} x^{6} \ln \relax (f )^{3}-5 b^{4} x^{3} \ln \relax (f )^{4}+b^{5} \ln \relax (f )^{5}\right ) f^{\frac {a \,x^{3}+b}{x^{3}}}}{3 b^{6} x^{15} \ln \relax (f )^{6}} \]

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 = 1.13, size = 22, normalized size = 0.27 \[ \frac {f^{a} \Gamma \left (6, -\frac {b \log \relax (f)}{x^{3}}\right )}{3 \, b^{6} \log \relax (f)^{6}} \]

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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mupad [B]  time = 3.59, size = 84, normalized size = 1.02 \[ -\frac {f^{a+\frac {b}{x^3}}\,\left (\frac {1}{3\,b\,\ln \relax (f)}-\frac {5\,x^3}{3\,b^2\,{\ln \relax (f)}^2}+\frac {20\,x^6}{3\,b^3\,{\ln \relax (f)}^3}-\frac {20\,x^9}{b^4\,{\ln \relax (f)}^4}+\frac {40\,x^{12}}{b^5\,{\ln \relax (f)}^5}-\frac {40\,x^{15}}{b^6\,{\ln \relax (f)}^6}\right )}{x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^3)/x^19,x)

[Out]

-(f^(a + b/x^3)*(1/(3*b*log(f)) - (5*x^3)/(3*b^2*log(f)^2) + (20*x^6)/(3*b^3*log(f)^3) - (20*x^9)/(b^4*log(f)^
4) + (40*x^12)/(b^5*log(f)^5) - (40*x^15)/(b^6*log(f)^6)))/x^15

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sympy [A]  time = 0.18, size = 85, normalized size = 1.04 \[ \frac {f^{a + \frac {b}{x^{3}}} \left (- b^{5} \log {\relax (f )}^{5} + 5 b^{4} x^{3} \log {\relax (f )}^{4} - 20 b^{3} x^{6} \log {\relax (f )}^{3} + 60 b^{2} x^{9} \log {\relax (f )}^{2} - 120 b x^{12} \log {\relax (f )} + 120 x^{15}\right )}{3 b^{6} x^{15} \log {\relax (f )}^{6}} \]

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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