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

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

[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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Rubi [A]  time = 0.0222683, antiderivative size = 34, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0046829, size = 34, normalized size = 1. \[ \frac{f^a \text{Gamma}\left (\frac{2}{3},-\frac{b \log (f)}{x^3}\right )}{3 x^2 \left (-\frac{b \log (f)}{x^3}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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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Maple [B]  time = 0.03, size = 78, normalized size = 2.3 \begin{align*}{\frac{{f}^{a}}{3\,b}\sqrt [3]{-b} \left ({\frac{\Gamma \left ({\frac{2}{3}} \right ) }{{x}^{2}} \left ( -b \right ) ^{{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}} \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{{x}^{2}} \left ( -b \right ) ^{{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}}\Gamma \left ({\frac{2}{3}},-{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) ^{-{\frac{2}{3}}}} \right ) \left ( \ln \left ( f \right ) \right ) ^{-{\frac{2}{3}}}} \end{align*}

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

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Maxima [A]  time = 1.2261, size = 38, normalized size = 1.12 \begin{align*} \frac{f^{a} \Gamma \left (\frac{2}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right )}{3 \, x^{2} \left (-\frac{b \log \left (f\right )}{x^{3}}\right )^{\frac{2}{3}}} \end{align*}

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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Fricas [A]  time = 1.59512, size = 88, normalized size = 2.59 \begin{align*} -\frac{\left (-b \log \left (f\right )\right )^{\frac{1}{3}} f^{a} \Gamma \left (\frac{2}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right )}{3 \, b \log \left (f\right )} \end{align*}

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

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

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)