Optimal. Leaf size=51 \[ \log (x+2) (d-2 e+4 f-8 g)+x (e-4 f+12 g)+\frac {1}{2} (x+2)^2 (f-6 g)+\frac {1}{3} g (x+2)^3 \]
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Rubi [A] time = 0.08, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {1586, 1850} \begin {gather*} \log (x+2) (d-2 e+4 f-8 g)+x (e-4 f+12 g)+\frac {1}{2} (x+2)^2 (f-6 g)+\frac {1}{3} g (x+2)^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 1586
Rule 1850
Rubi steps
\begin {align*} \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{4-5 x^2+x^4} \, dx &=\int \frac {d+e x+f x^2+g x^3}{2+x} \, dx\\ &=\int \left (e-4 f+12 g+\frac {d-2 e+4 f-8 g}{2+x}+(f-6 g) (2+x)+g (2+x)^2\right ) \, dx\\ &=(e-4 f+12 g) x+\frac {1}{2} (f-6 g) (2+x)^2+\frac {1}{3} g (2+x)^3+(d-2 e+4 f-8 g) \log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.88 \begin {gather*} \log (x+2) (d-2 e+4 f-8 g)+\frac {1}{6} (x+2) \left (6 e+3 f (x-6)+2 g \left (x^2-5 x+22\right )\right ) \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 {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g 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 = 43, normalized size = 0.84 \begin {gather*} \frac {1}{3} \, g x^{3} + \frac {1}{2} \, {\left (f - 2 \, g\right )} x^{2} + {\left (e - 2 \, f + 4 \, g\right )} x + {\left (d - 2 \, e + 4 \, f - 8 \, g\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 = 0.25, size = 49, normalized size = 0.96 \begin {gather*} \frac {1}{3} \, g x^{3} + \frac {1}{2} \, f x^{2} - g x^{2} - 2 \, f x + 4 \, g x + x e + {\left (d + 4 \, f - 8 \, g - 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 = 58, normalized size = 1.14 \begin {gather*} \frac {g \,x^{3}}{3}+\frac {f \,x^{2}}{2}-g \,x^{2}+d \ln \left (x +2\right )+e x -2 e \ln \left (x +2\right )-2 f x +4 f \ln \left (x +2\right )+4 g x -8 g \ln \left (x +2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 43, normalized size = 0.84 \begin {gather*} \frac {1}{3} \, g x^{3} + \frac {1}{2} \, {\left (f - 2 \, g\right )} x^{2} + {\left (e - 2 \, f + 4 \, g\right )} x + {\left (d - 2 \, e + 4 \, f - 8 \, g\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 = 0.04, size = 44, normalized size = 0.86 \begin {gather*} x^2\,\left (\frac {f}{2}-g\right )+x\,\left (e-2\,f+4\,g\right )+\frac {g\,x^3}{3}+\ln \left (x+2\right )\,\left (d-2\,e+4\,f-8\,g\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 41, normalized size = 0.80 \begin {gather*} \frac {g x^{3}}{3} + x^{2} \left (\frac {f}{2} - g\right ) + x \left (e - 2 f + 4 g\right ) + \left (d - 2 e + 4 f - 8 g\right ) \log {\left (x + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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