3.173 \(\int \frac {f^{a+\frac {b}{x^3}}}{x^5} \, dx\)

Optimal. Leaf size=34 \[ \frac {f^a \Gamma \left (\frac {4}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^4 \left (-\frac {b \log (f)}{x^3}\right )^{4/3}} \]

[Out]

1/3*f^a*GAMMA(4/3,-b*ln(f)/x^3)/x^4/(-b*ln(f)/x^3)^(4/3)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 34, 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 = {2218} \[ \frac {f^a \text {Gamma}\left (\frac {4}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^4 \left (-\frac {b \log (f)}{x^3}\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)/x^5,x]

[Out]

(f^a*Gamma[4/3, -((b*Log[f])/x^3)])/(3*x^4*(-((b*Log[f])/x^3))^(4/3))

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^5} \, dx &=\frac {f^a \Gamma \left (\frac {4}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^4 \left (-\frac {b \log (f)}{x^3}\right )^{4/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 34, normalized size = 1.00 \[ \frac {f^a \Gamma \left (\frac {4}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^4 \left (-\frac {b \log (f)}{x^3}\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)/x^5,x]

[Out]

(f^a*Gamma[4/3, -((b*Log[f])/x^3)])/(3*x^4*(-((b*Log[f])/x^3))^(4/3))

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 53, normalized size = 1.56 \[ \frac {\left (-b \log \relax (f)\right )^{\frac {2}{3}} f^{a} x \Gamma \left (\frac {1}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) - 3 \, b f^{\frac {a x^{3} + b}{x^{3}}} \log \relax (f)}{9 \, b^{2} x \log \relax (f)^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^5,x, algorithm="fricas")

[Out]

1/9*((-b*log(f))^(2/3)*f^a*x*gamma(1/3, -b*log(f)/x^3) - 3*b*f^((a*x^3 + b)/x^3)*log(f))/(b^2*x*log(f)^2)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x^{3}}}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^5,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)/x^5, x)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 112, normalized size = 3.29 \[ -\frac {\left (\frac {\left (-b \right )^{\frac {4}{3}} \Gamma \left (\frac {1}{3}, -\frac {b \ln \relax (f )}{x^{3}}\right ) \ln \relax (f )^{\frac {1}{3}}}{3 \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {1}{3}} b x}+\frac {\left (-b \right )^{\frac {4}{3}} {\mathrm e}^{\frac {b \ln \relax (f )}{x^{3}}} \ln \relax (f )^{\frac {1}{3}}}{b x}-\frac {2 \left (-b \right )^{\frac {4}{3}} \pi \sqrt {3}\, \ln \relax (f )^{\frac {1}{3}}}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {1}{3}} b x}\right ) f^{a}}{3 \left (-b \right )^{\frac {4}{3}} \ln \relax (f )^{\frac {4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^3)/x^5,x)

[Out]

-1/3*f^a/(-b)^(4/3)/ln(f)^(4/3)*(-2/9/x*(-b)^(4/3)*ln(f)^(1/3)/b*Pi*3^(1/2)/GAMMA(2/3)/(-b/x^3*ln(f))^(1/3)+1/
x*(-b)^(4/3)*ln(f)^(1/3)/b*exp(b/x^3*ln(f))+1/3/x*(-b)^(4/3)*ln(f)^(1/3)/b/(-b/x^3*ln(f))^(1/3)*GAMMA(1/3,-b/x
^3*ln(f)))

________________________________________________________________________________________

maxima [A]  time = 1.04, size = 28, normalized size = 0.82 \[ \frac {f^{a} \Gamma \left (\frac {4}{3}, -\frac {b \log \relax (f)}{x^{3}}\right )}{3 \, x^{4} \left (-\frac {b \log \relax (f)}{x^{3}}\right )^{\frac {4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^5,x, algorithm="maxima")

[Out]

1/3*f^a*gamma(4/3, -b*log(f)/x^3)/(x^4*(-b*log(f)/x^3)^(4/3))

________________________________________________________________________________________

mupad [B]  time = 3.54, size = 77, normalized size = 2.26 \[ \frac {f^a\,\Gamma \left (\frac {1}{3},-\frac {b\,\ln \relax (f)}{x^3}\right )}{9\,x^4\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{4/3}}-\frac {f^a\,f^{\frac {b}{x^3}}}{3\,b\,x\,\ln \relax (f)}-\frac {2\,\pi \,\sqrt {3}\,f^a}{27\,x^4\,\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{4/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^3)/x^5,x)

[Out]

(f^a*igamma(1/3, -(b*log(f))/x^3))/(9*x^4*(-(b*log(f))/x^3)^(4/3)) - (f^a*f^(b/x^3))/(3*b*x*log(f)) - (2*3^(1/
2)*f^a*pi)/(27*x^4*gamma(2/3)*(-(b*log(f))/x^3)^(4/3))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**3)/x**5,x)

[Out]

Timed out

________________________________________________________________________________________