Optimal. Leaf size=42 \[ -\frac{1}{2} (d+e) \log (1-x)+\frac{1}{3} (d+2 e) \log (2-x)+\frac{1}{6} (d-e) \log (x+1) \]
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Rubi [A] time = 0.0518755, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {1586, 2074} \[ -\frac{1}{2} (d+e) \log (1-x)+\frac{1}{3} (d+2 e) \log (2-x)+\frac{1}{6} (d-e) \log (x+1) \]
Antiderivative was successfully verified.
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Rule 1586
Rule 2074
Rubi steps
\begin{align*} \int \frac{(2+x) (d+e x)}{4-5 x^2+x^4} \, dx &=\int \frac{d+e x}{2-x-2 x^2+x^3} \, dx\\ &=\int \left (\frac{d+2 e}{3 (-2+x)}+\frac{-d-e}{2 (-1+x)}+\frac{d-e}{6 (1+x)}\right ) \, dx\\ &=-\frac{1}{2} (d+e) \log (1-x)+\frac{1}{3} (d+2 e) \log (2-x)+\frac{1}{6} (d-e) \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0187446, size = 39, normalized size = 0.93 \[ \frac{1}{6} (-3 (d+e) \log (1-x)+2 (d+2 e) \log (2-x)+(d-e) \log (x+1)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 44, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}+{\frac{\ln \left ( x-2 \right ) d}{3}}+{\frac{2\,\ln \left ( x-2 \right ) e}{3}}-{\frac{\ln \left ( x-1 \right ) d}{2}}-{\frac{\ln \left ( x-1 \right ) e}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969047, size = 43, normalized size = 1.02 \begin{align*} \frac{1}{6} \,{\left (d - e\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (d + e\right )} \log \left (x - 1\right ) + \frac{1}{3} \,{\left (d + 2 \, e\right )} \log \left (x - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50864, size = 103, normalized size = 2.45 \begin{align*} \frac{1}{6} \,{\left (d - e\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (d + e\right )} \log \left (x - 1\right ) + \frac{1}{3} \,{\left (d + 2 \, e\right )} \log \left (x - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.14819, size = 304, normalized size = 7.24 \begin{align*} \frac{\left (d - e\right ) \log{\left (x + \frac{26 d^{3} + 66 d^{2} e - 9 d^{2} \left (d - e\right ) + 78 d e^{2} - 12 d e \left (d - e\right ) - 7 d \left (d - e\right )^{2} + 46 e^{3} + 3 e^{2} \left (d - e\right ) - 8 e \left (d - e\right )^{2}}{10 d^{3} + 69 d^{2} e + 102 d e^{2} + 35 e^{3}} \right )}}{6} - \frac{\left (d + e\right ) \log{\left (x + \frac{26 d^{3} + 66 d^{2} e + 27 d^{2} \left (d + e\right ) + 78 d e^{2} + 36 d e \left (d + e\right ) - 63 d \left (d + e\right )^{2} + 46 e^{3} - 9 e^{2} \left (d + e\right ) - 72 e \left (d + e\right )^{2}}{10 d^{3} + 69 d^{2} e + 102 d e^{2} + 35 e^{3}} \right )}}{2} + \frac{\left (d + 2 e\right ) \log{\left (x + \frac{26 d^{3} + 66 d^{2} e - 18 d^{2} \left (d + 2 e\right ) + 78 d e^{2} - 24 d e \left (d + 2 e\right ) - 28 d \left (d + 2 e\right )^{2} + 46 e^{3} + 6 e^{2} \left (d + 2 e\right ) - 32 e \left (d + 2 e\right )^{2}}{10 d^{3} + 69 d^{2} e + 102 d e^{2} + 35 e^{3}} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07426, size = 51, normalized size = 1.21 \begin{align*} \frac{1}{6} \,{\left (d - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\left (d + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{3} \,{\left (d + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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