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

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

[Out]

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

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Rubi [A]  time = 0.0230954, 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} \[ \frac{f^a \text{Gamma}\left (\frac{4}{3},-\frac{b \log (f)}{x^3}\right )}{3 x^4 \left (-\frac{b \log (f)}{x^3}\right )^{4/3}} \]

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 = 0.005421, size = 34, normalized size = 1. \[ \frac{f^a \text{Gamma}\left (\frac{4}{3},-\frac{b \log (f)}{x^3}\right )}{3 x^4 \left (-\frac{b \log (f)}{x^3}\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

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))

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Maple [B]  time = 0.037, size = 112, normalized size = 3.3 \begin{align*} -{\frac{{f}^{a}}{3} \left ( -{\frac{2\,\pi \,\sqrt{3}}{9\,bx\Gamma \left ( 2/3 \right ) } \left ( -b \right ) ^{{\frac{4}{3}}}\sqrt [3]{\ln \left ( f \right ) }{\frac{1}{\sqrt [3]{-{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}}+{\frac{1}{bx} \left ( -b \right ) ^{{\frac{4}{3}}}\sqrt [3]{\ln \left ( f \right ) }{{\rm e}^{{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}+{\frac{1}{3\,bx} \left ( -b \right ) ^{{\frac{4}{3}}}\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 ) \left ( -b \right ) ^{-{\frac{4}{3}}} \left ( \ln \left ( f \right ) \right ) ^{-{\frac{4}{3}}}} \end{align*}

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*ln(f)/x^3)^(1/3)+1/
x*(-b)^(4/3)*ln(f)^(1/3)/b*exp(b*ln(f)/x^3)+1/3/x*(-b)^(4/3)*ln(f)^(1/3)/b/(-b*ln(f)/x^3)^(1/3)*GAMMA(1/3,-b*l
n(f)/x^3))

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Maxima [A]  time = 1.22927, size = 38, normalized size = 1.12 \begin{align*} \frac{f^{a} \Gamma \left (\frac{4}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right )}{3 \, x^{4} \left (-\frac{b \log \left (f\right )}{x^{3}}\right )^{\frac{4}{3}}} \end{align*}

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))

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Fricas [A]  time = 1.63414, size = 144, normalized size = 4.24 \begin{align*} \frac{\left (-b \log \left (f\right )\right )^{\frac{2}{3}} f^{a} x \Gamma \left (\frac{1}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right ) - 3 \, b f^{\frac{a x^{3} + b}{x^{3}}} \log \left (f\right )}{9 \, b^{2} x \log \left (f\right )^{2}} \end{align*}

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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{3}}}}{x^{5}}\,{d x} \end{align*}

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)