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

Optimal. Leaf size=34 \[ \frac {f^a \Gamma \left (\frac {2}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^2 \left (-\frac {b \log (f)}{x^3}\right )^{2/3}} \]

[Out]

1/3*f^a*GAMMA(2/3,-b*ln(f)/x^3)/x^2/(-b*ln(f)/x^3)^(2/3)

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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} \[ \frac {f^a \text {Gamma}\left (\frac {2}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^2 \left (-\frac {b \log (f)}{x^3}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)/x^3,x]

[Out]

(f^a*Gamma[2/3, -((b*Log[f])/x^3)])/(3*x^2*(-((b*Log[f])/x^3))^(2/3))

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^3} \, dx &=\frac {f^a \Gamma \left (\frac {2}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^2 \left (-\frac {b \log (f)}{x^3}\right )^{2/3}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.00 \[ \frac {f^a \Gamma \left (\frac {2}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^2 \left (-\frac {b \log (f)}{x^3}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)/x^3,x]

[Out]

(f^a*Gamma[2/3, -((b*Log[f])/x^3)])/(3*x^2*(-((b*Log[f])/x^3))^(2/3))

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fricas [A]  time = 0.43, size = 29, normalized size = 0.85 \[ -\frac {\left (-b \log \relax (f)\right )^{\frac {1}{3}} f^{a} \Gamma \left (\frac {2}{3}, -\frac {b \log \relax (f)}{x^{3}}\right )}{3 \, b \log \relax (f)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x^3)/x^3, x)

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maple [B]  time = 0.06, size = 78, normalized size = 2.29 \[ \frac {\left (-b \right )^{\frac {1}{3}} \left (-\frac {\left (-b \right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}, -\frac {b \ln \relax (f )}{x^{3}}\right ) \ln \relax (f )^{\frac {2}{3}}}{\left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {2}{3}} x^{2}}+\frac {\left (-b \right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right ) \ln \relax (f )^{\frac {2}{3}}}{\left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {2}{3}} x^{2}}\right ) f^{a}}{3 b \ln \relax (f )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*f^a/b/ln(f)^(2/3)*(-b)^(1/3)*(1/x^2*(-b)^(2/3)*ln(f)^(2/3)*GAMMA(2/3)/(-b/x^3*ln(f))^(2/3)-1/x^2*(-b)^(2/3
)*ln(f)^(2/3)/(-b/x^3*ln(f))^(2/3)*GAMMA(2/3,-b/x^3*ln(f)))

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maxima [A]  time = 1.40, size = 28, normalized size = 0.82 \[ \frac {f^{a} \Gamma \left (\frac {2}{3}, -\frac {b \log \relax (f)}{x^{3}}\right )}{3 \, x^{2} \left (-\frac {b \log \relax (f)}{x^{3}}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.56, size = 33, normalized size = 0.97 \[ -\frac {f^a\,\left (\Gamma \left (\frac {2}{3}\right )-\Gamma \left (\frac {2}{3},-\frac {b\,\ln \relax (f)}{x^3}\right )\right )}{3\,x^2\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{2/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^3)/x^3,x)

[Out]

-(f^a*(gamma(2/3) - igamma(2/3, -(b*log(f))/x^3)))/(3*x^2*(-(b*log(f))/x^3)^(2/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b/x**3)/x**3, x)

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