Optimal. Leaf size=31 \[ (e-4 f) x+\frac {1}{2} f (2+x)^2+(d-2 e+4 f) \log (2+x) \]
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Rubi [A]
time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1600, 712}
\begin {gather*} \log (x+2) (d-2 e+4 f)+x (e-4 f)+\frac {1}{2} f (x+2)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rule 1600
Rubi steps
\begin {align*} \int \frac {\left (d+e x+f x^2\right ) \left (2-x-2 x^2+x^3\right )}{4-5 x^2+x^4} \, dx &=\int \frac {d+e x+f x^2}{2+x} \, dx\\ &=\int \left (e-4 f+\frac {d-2 e+4 f}{2+x}+f (2+x)\right ) \, dx\\ &=(e-4 f) x+\frac {1}{2} f (2+x)^2+(d-2 e+4 f) \log (2+x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 0.97 \begin {gather*} \frac {1}{2} (2 e+f (-6+x)) (2+x)+(d-2 e+4 f) \log (2+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 28, normalized size = 0.90
method | result | size |
default | \(\frac {f \,x^{2}}{2}+e x -2 f x +\left (d -2 e +4 f \right ) \ln \left (x +2\right )\) | \(28\) |
norman | \(\left (e -2 f \right ) x +\frac {f \,x^{2}}{2}+\left (d -2 e +4 f \right ) \ln \left (x +2\right )\) | \(28\) |
risch | \(\frac {f \,x^{2}}{2}+e x -2 f x +\ln \left (x +2\right ) d -2 \ln \left (x +2\right ) e +4 \ln \left (x +2\right ) f\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 32, normalized size = 1.03 \begin {gather*} \frac {1}{2} \, f x^{2} - {\left (2 \, f - e\right )} x + {\left (d + 4 \, f - 2 \, e\right )} \log \left (x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, f x^{2} + {\left (e - 2 \, f\right )} x + {\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 26, normalized size = 0.84 \begin {gather*} \frac {f x^{2}}{2} + x \left (e - 2 f\right ) + \left (d - 2 e + 4 f\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 = 4.53, size = 30, normalized size = 0.97 \begin {gather*} \frac {1}{2} \, f x^{2} - 2 \, f x + x e + {\left (d + 4 \, f - 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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Mupad [B]
time = 0.04, size = 27, normalized size = 0.87 \begin {gather*} x\,\left (e-2\,f\right )+\frac {f\,x^2}{2}+\ln \left (x+2\right )\,\left (d-2\,e+4\,f\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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