Optimal. Leaf size=21 \[ \frac {2}{x+1}-\frac {1}{2 (x+1)^2}+\log (x+1) \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ \frac {2}{x+1}-\frac {1}{2 (x+1)^2}+\log (x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 31, normalized size = 1.48 \[ \frac {2 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) + 4 \, x + 3}{2 \, {\left (x^{2} + 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 18, normalized size = 0.86 \[ \frac {4 \, x + 3}{2 \, {\left (x + 1\right )}^{2}} + \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 0.95 \[ \ln \left (x +1\right )-\frac {1}{2 \left (x +1\right )^{2}}+\frac {2}{x +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 22, normalized size = 1.05 \[ \frac {4 \, x + 3}{2 \, {\left (x^{2} + 2 \, x + 1\right )}} + \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 21, normalized size = 1.00 \[ \ln \left (x+1\right )+\frac {2\,x+\frac {3}{2}}{x^2+2\,x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 19, normalized size = 0.90 \[ \frac {4 x + 3}{2 x^{2} + 4 x + 2} + \log {\left (x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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