Optimal. Leaf size=14 \[ (d-2 e) \log (x+2)+e x \]
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Rubi [A] time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1586, 43} \begin {gather*} (d-2 e) \log (x+2)+e x \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 1586
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{4-5 x^2+x^4} \, dx &=\int \frac {d+e x}{2+x} \, dx\\ &=\int \left (e+\frac {d-2 e}{2+x}\right ) \, dx\\ &=e x+(d-2 e) \log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.14 \begin {gather*} (d-2 e) \log (x+2)+e (x+2) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{4-5 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.38, size = 14, normalized size = 1.00 \begin {gather*} e x + {\left (d - 2 \, e\right )} \log \left (x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 17, normalized size = 1.21 \begin {gather*} x e + {\left (d - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 1.29 \begin {gather*} d \ln \left (x +2\right )+e x -2 e \ln \left (x +2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 14, normalized size = 1.00 \begin {gather*} e x + {\left (d - 2 \, e\right )} \log \left (x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 14, normalized size = 1.00 \begin {gather*} \ln \left (x+2\right )\,\left (d-2\,e\right )+e\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 12, normalized size = 0.86 \begin {gather*} e x + \left (d - 2 e\right ) \log {\left (x + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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