Optimal. Leaf size=47 \[ \log (x+1) (d-e+f-g)-\log (x+2) (d-2 e+4 f-8 g)+x (f-3 g)+\frac{g x^2}{2} \]
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Rubi [A] time = 0.0677106, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1586, 1657, 632, 31} \[ \log (x+1) (d-e+f-g)-\log (x+2) (d-2 e+4 f-8 g)+x (f-3 g)+\frac{g x^2}{2} \]
Antiderivative was successfully verified.
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Rule 1586
Rule 1657
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (2-3 x+x^2\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+3 x+x^2} \, dx\\ &=\int \left (f-3 g+g x+\frac{d-2 f+6 g+(e-3 f+7 g) x}{2+3 x+x^2}\right ) \, dx\\ &=(f-3 g) x+\frac{g x^2}{2}+\int \frac{d-2 f+6 g+(e-3 f+7 g) x}{2+3 x+x^2} \, dx\\ &=(f-3 g) x+\frac{g x^2}{2}-(d-2 e+4 f-8 g) \int \frac{1}{2+x} \, dx+(d-e+f-g) \int \frac{1}{1+x} \, dx\\ &=(f-3 g) x+\frac{g x^2}{2}+(d-e+f-g) \log (1+x)-(d-2 e+4 f-8 g) \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.020238, size = 44, normalized size = 0.94 \[ \log (x+1) (d-e+f-g)-\log (x+2) (d-2 e+4 f-8 g)+f x+\frac{1}{2} g (x-6) x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 69, normalized size = 1.5 \begin{align*}{\frac{g{x}^{2}}{2}}+fx-3\,gx-\ln \left ( 2+x \right ) d+2\,\ln \left ( 2+x \right ) e-4\,\ln \left ( 2+x \right ) f+8\,\ln \left ( 2+x \right ) g+\ln \left ( 1+x \right ) d-\ln \left ( 1+x \right ) e+\ln \left ( 1+x \right ) f-\ln \left ( 1+x \right ) g \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962906, size = 61, normalized size = 1.3 \begin{align*} \frac{1}{2} \, g x^{2} +{\left (f - 3 \, g\right )} x -{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) +{\left (d - e + f - g\right )} \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50892, size = 120, normalized size = 2.55 \begin{align*} \frac{1}{2} \, g x^{2} +{\left (f - 3 \, g\right )} x -{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) +{\left (d - e + f - g\right )} \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.885272, size = 66, normalized size = 1.4 \begin{align*} \frac{g x^{2}}{2} + x \left (f - 3 g\right ) + \left (- d + 2 e - 4 f + 8 g\right ) \log{\left (x + \frac{4 d - 6 e + 10 f - 18 g}{2 d - 3 e + 5 f - 9 g} \right )} + \left (d - e + f - g\right ) \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11015, size = 66, normalized size = 1.4 \begin{align*} \frac{1}{2} \, g x^{2} + f x - 3 \, g x -{\left (d + 4 \, f - 8 \, g - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) +{\left (d + f - g - e\right )} \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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