Optimal. Leaf size=14 \[ \frac{1}{x}+2 \log (x)+3 \log (x+2) \]
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Rubi [A] time = 0.0233708, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1593, 893} \[ \frac{1}{x}+2 \log (x)+3 \log (x+2) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 893
Rubi steps
\begin{align*} \int \frac{-2+3 x+5 x^2}{2 x^2+x^3} \, dx &=\int \frac{-2+3 x+5 x^2}{x^2 (2+x)} \, dx\\ &=\int \left (-\frac{1}{x^2}+\frac{2}{x}+\frac{3}{2+x}\right ) \, dx\\ &=\frac{1}{x}+2 \log (x)+3 \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0036714, size = 14, normalized size = 1. \[ \frac{1}{x}+2 \log (x)+3 \log (x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 15, normalized size = 1.1 \begin{align*}{x}^{-1}+2\,\ln \left ( x \right ) +3\,\ln \left ( 2+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.927164, size = 19, normalized size = 1.36 \begin{align*} \frac{1}{x} + 3 \, \log \left (x + 2\right ) + 2 \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74643, size = 50, normalized size = 3.57 \begin{align*} \frac{3 \, x \log \left (x + 2\right ) + 2 \, x \log \left (x\right ) + 1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.100243, size = 14, normalized size = 1. \begin{align*} 2 \log{\left (x \right )} + 3 \log{\left (x + 2 \right )} + \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05714, size = 22, normalized size = 1.57 \begin{align*} \frac{1}{x} + 3 \, \log \left ({\left | x + 2 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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