Optimal. Leaf size=13 \[ -\frac{3}{16} \left (\frac{1}{x^2}+1\right )^{8/3} \]
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Rubi [A] time = 0.0035764, antiderivative size = 13, 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 = {261} \[ -\frac{3}{16} \left (\frac{1}{x^2}+1\right )^{8/3} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{\left (1+\frac{1}{x^2}\right )^{5/3}}{x^3} \, dx &=-\frac{3}{16} \left (1+\frac{1}{x^2}\right )^{8/3}\\ \end{align*}
Mathematica [A] time = 0.0072235, size = 21, normalized size = 1.62 \[ -\frac{3 \left (\frac{1}{x^2}+1\right )^{5/3} \left (x^2+1\right )}{16 x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 22, normalized size = 1.7 \begin{align*} -{\frac{3\,{x}^{2}+3}{16\,{x}^{2}} \left ({\frac{{x}^{2}+1}{{x}^{2}}} \right ) ^{{\frac{5}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960308, size = 12, normalized size = 0.92 \begin{align*} -\frac{3}{16} \,{\left (\frac{1}{x^{2}} + 1\right )}^{\frac{8}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52766, size = 69, normalized size = 5.31 \begin{align*} -\frac{3 \,{\left (x^{4} + 2 \, x^{2} + 1\right )} \left (\frac{x^{2} + 1}{x^{2}}\right )^{\frac{2}{3}}}{16 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.08207, size = 48, normalized size = 3.69 \begin{align*} - \frac{3 \left (1 + \frac{1}{x^{2}}\right )^{\frac{2}{3}}}{16} - \frac{3 \left (1 + \frac{1}{x^{2}}\right )^{\frac{2}{3}}}{8 x^{2}} - \frac{3 \left (1 + \frac{1}{x^{2}}\right )^{\frac{2}{3}}}{16 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (\frac{1}{x^{2}} + 1\right )}^{\frac{5}{3}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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