Optimal. Leaf size=22 \[ (d-e) \log (x+1)-(d-2 e) \log (x+2) \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 31
Rule 632
Rule 1586
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.83, size = 22, normalized size = 1.00 \[ -{\left (d - 2 \, e\right )} \log \left (x + 2\right ) + {\left (d - e\right )} \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 26, normalized size = 1.18 \[ -{\left (d - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + {\left (d - e\right )} \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 1.32 \[ -d \ln \left (x +2\right )+d \ln \left (x +1\right )+2 e \ln \left (x +2\right )-e \ln \left (x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 22, normalized size = 1.00 \[ -{\left (d - 2 \, e\right )} \log \left (x + 2\right ) + {\left (d - e\right )} \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 22, normalized size = 1.00 \[ \ln \left (x+1\right )\,\left (d-e\right )-\ln \left (x+2\right )\,\left (d-2\,e\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 29, normalized size = 1.32 \[ \left (- d + 2 e\right ) \log {\left (x + \frac {4 d - 6 e}{2 d - 3 e} \right )} + \left (d - e\right ) \log {\left (x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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