Optimal. Leaf size=22 \[ (d-e) \log (x+1)-(d-2 e) \log (x+2) \]
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Rubi [A] time = 0.0206828, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1586, 632, 31} \[ (d-e) \log (x+1)-(d-2 e) \log (x+2) \]
Antiderivative was successfully verified.
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Rule 1586
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (2-3 x+x^2\right )}{4-5 x^2+x^4} \, dx &=\int \frac{d+e x}{2+3 x+x^2} \, dx\\ &=-\left ((d-2 e) \int \frac{1}{2+x} \, dx\right )+(d-e) \int \frac{1}{1+x} \, dx\\ &=(d-e) \log (1+x)-(d-2 e) \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0073826, size = 23, normalized size = 1.05 \[ (d-e) \log (x+1)+(2 e-d) \log (x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 29, normalized size = 1.3 \begin{align*} -\ln \left ( 2+x \right ) d+2\,\ln \left ( 2+x \right ) e+\ln \left ( 1+x \right ) d-\ln \left ( 1+x \right ) e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.9715, size = 30, normalized size = 1.36 \begin{align*} -{\left (d - 2 \, e\right )} \log \left (x + 2\right ) +{\left (d - e\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 = 1.52891, size = 59, normalized size = 2.68 \begin{align*} -{\left (d - 2 \, e\right )} \log \left (x + 2\right ) +{\left (d - e\right )} \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.252077, size = 29, normalized size = 1.32 \begin{align*} \left (- d + 2 e\right ) \log{\left (x + \frac{4 d - 6 e}{2 d - 3 e} \right )} + \left (d - e\right ) \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06985, size = 35, normalized size = 1.59 \begin{align*} -{\left (d - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) +{\left (d - e\right )} \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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