Optimal. Leaf size=29 \[ \log (x+1) (d-e+f)-\log (x+2) (d-2 e+4 f)+f x \]
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Rubi [A] time = 0.0500694, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {1586, 1657, 632, 31} \[ \log (x+1) (d-e+f)-\log (x+2) (d-2 e+4 f)+f x \]
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\right )}{4-5 x^2+x^4} \, dx &=\int \frac{d+e x+f x^2}{2+3 x+x^2} \, dx\\ &=\int \left (f+\frac{d-2 f+(e-3 f) x}{2+3 x+x^2}\right ) \, dx\\ &=f x+\int \frac{d-2 f+(e-3 f) x}{2+3 x+x^2} \, dx\\ &=f x+(d-e+f) \int \frac{1}{1+x} \, dx-(d-2 e+4 f) \int \frac{1}{2+x} \, dx\\ &=f x+(d-e+f) \log (1+x)-(d-2 e+4 f) \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.013349, size = 30, normalized size = 1.03 \[ \log (x+1) (d-e+f)+\log (x+2) (-d+2 e-4 f)+f x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 45, normalized size = 1.6 \begin{align*} fx-\ln \left ( 2+x \right ) d+2\,\ln \left ( 2+x \right ) e-4\,\ln \left ( 2+x \right ) f+\ln \left ( 1+x \right ) d-\ln \left ( 1+x \right ) e+\ln \left ( 1+x \right ) f \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968562, size = 39, normalized size = 1.34 \begin{align*} f x -{\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) +{\left (d - e + f\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 = 1.48516, size = 80, normalized size = 2.76 \begin{align*} f x -{\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) +{\left (d - e + f\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 = 0.594252, size = 44, normalized size = 1.52 \begin{align*} f x + \left (- d + 2 e - 4 f\right ) \log{\left (x + \frac{4 d - 6 e + 10 f}{2 d - 3 e + 5 f} \right )} + \left (d - e + f\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 = 1.06533, size = 45, normalized size = 1.55 \begin{align*} f x -{\left (d + 4 \, f - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) +{\left (d + f - 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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