Optimal. Leaf size=21 \[ \frac{2}{x+1}-\frac{1}{2 (x+1)^2}+\log (x+1) \]
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Rubi [A] time = 0.0072975, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{2}{x+1}-\frac{1}{2 (x+1)^2}+\log (x+1) \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{(1+x)^3} \, dx &=\int \left (\frac{1}{(1+x)^3}-\frac{2}{(1+x)^2}+\frac{1}{1+x}\right ) \, dx\\ &=-\frac{1}{2 (1+x)^2}+\frac{2}{1+x}+\log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0083873, size = 21, normalized size = 1. \[ \frac{2}{x+1}-\frac{1}{2 (x+1)^2}+\log (x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 20, normalized size = 1. \begin{align*} -{\frac{1}{2\, \left ( 1+x \right ) ^{2}}}+2\, \left ( 1+x \right ) ^{-1}+\ln \left ( 1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.922966, size = 30, normalized size = 1.43 \begin{align*} \frac{4 \, x + 3}{2 \,{\left (x^{2} + 2 \, x + 1\right )}} + \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80456, size = 84, normalized size = 4. \begin{align*} \frac{2 \,{\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) + 4 \, x + 3}{2 \,{\left (x^{2} + 2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.086027, size = 19, normalized size = 0.9 \begin{align*} \frac{4 x + 3}{2 x^{2} + 4 x + 2} + \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05038, size = 24, normalized size = 1.14 \begin{align*} \frac{4 \, x + 3}{2 \,{\left (x + 1\right )}^{2}} + \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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