3.171 \(\int \frac{f^{a+\frac{b}{x^3}}}{x^2} \, dx\)

Optimal. Leaf size=34 \[ \frac{f^a \text{Gamma}\left (\frac{1}{3},-\frac{b \log (f)}{x^3}\right )}{3 x \sqrt [3]{-\frac{b \log (f)}{x^3}}} \]

[Out]

(f^a*Gamma[1/3, -((b*Log[f])/x^3)])/(3*x*(-((b*Log[f])/x^3))^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 0.0236808, 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} \[ \frac{f^a \text{Gamma}\left (\frac{1}{3},-\frac{b \log (f)}{x^3}\right )}{3 x \sqrt [3]{-\frac{b \log (f)}{x^3}}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)/x^2,x]

[Out]

(f^a*Gamma[1/3, -((b*Log[f])/x^3)])/(3*x*(-((b*Log[f])/x^3))^(1/3))

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^2} \, dx &=\frac{f^a \Gamma \left (\frac{1}{3},-\frac{b \log (f)}{x^3}\right )}{3 x \sqrt [3]{-\frac{b \log (f)}{x^3}}}\\ \end{align*}

Mathematica [A]  time = 0.0052051, size = 34, normalized size = 1. \[ \frac{f^a \text{Gamma}\left (\frac{1}{3},-\frac{b \log (f)}{x^3}\right )}{3 x \sqrt [3]{-\frac{b \log (f)}{x^3}}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)/x^2,x]

[Out]

(f^a*Gamma[1/3, -((b*Log[f])/x^3)])/(3*x*(-((b*Log[f])/x^3))^(1/3))

________________________________________________________________________________________

Maple [B]  time = 0.03, size = 82, normalized size = 2.4 \begin{align*} -{\frac{{f}^{a}}{3} \left ({\frac{2\,\pi \,\sqrt{3}}{3\,x\Gamma \left ( 2/3 \right ) }\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) }{\frac{1}{\sqrt [3]{-{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}}-{\frac{1}{x}\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) }\Gamma \left ({\frac{1}{3}},-{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ){\frac{1}{\sqrt [3]{-{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}} \right ){\frac{1}{\sqrt [3]{-b}}}{\frac{1}{\sqrt [3]{\ln \left ( f \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^3)/x^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.20205, size = 38, normalized size = 1.12 \begin{align*} \frac{f^{a} \Gamma \left (\frac{1}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right )}{3 \, x \left (-\frac{b \log \left (f\right )}{x^{3}}\right )^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.59362, size = 88, normalized size = 2.59 \begin{align*} -\frac{\left (-b \log \left (f\right )\right )^{\frac{2}{3}} f^{a} \Gamma \left (\frac{1}{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^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{3}}}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**3)/x**2,x)

[Out]

Integral(f**(a + b/x**3)/x**2, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{3}}}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^2,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)/x^2, x)