Integrand size = 31, antiderivative size = 29 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2\right )}{4-5 x^2+x^4} \, dx=f x+(d-e+f) \log (1+x)-(d-2 e+4 f) \log (2+x) \]
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Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1600, 1671, 646, 31} \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2\right )}{4-5 x^2+x^4} \, dx=\log (x+1) (d-e+f)-\log (x+2) (d-2 e+4 f)+f x \]
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Rule 31
Rule 646
Rule 1600
Rule 1671
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2\right )}{4-5 x^2+x^4} \, dx=f x+(d-e+f) \log (1+x)+(-d+2 e-4 f) \log (2+x) \]
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Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07
method | result | size |
default | \(f x +\left (-d +2 e -4 f \right ) \ln \left (x +2\right )+\left (d -e +f \right ) \ln \left (x +1\right )\) | \(31\) |
norman | \(f x +\left (-d +2 e -4 f \right ) \ln \left (x +2\right )+\left (d -e +f \right ) \ln \left (x +1\right )\) | \(31\) |
parallelrisch | \(\ln \left (x +1\right ) d -\ln \left (x +1\right ) e +\ln \left (x +1\right ) f -\ln \left (x +2\right ) d +2 \ln \left (x +2\right ) e -4 \ln \left (x +2\right ) f +f x\) | \(45\) |
risch | \(f x +\ln \left (-x -1\right ) d -\ln \left (-x -1\right ) e +\ln \left (-x -1\right ) f -\ln \left (x +2\right ) d +2 \ln \left (x +2\right ) e -4 \ln \left (x +2\right ) f\) | \(51\) |
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Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2\right )}{4-5 x^2+x^4} \, dx=f x - {\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) + {\left (d - e + f\right )} \log \left (x + 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2\right )}{4-5 x^2+x^4} \, dx=f x + \left (- d + 2 e - 4 f\right ) \log {\left (x + \frac {4 d - 6 e + 10 f}{2 d - 3 e + 5 f} \right )} + \left (d - e + f\right ) \log {\left (x + 1 \right )} \]
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Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2\right )}{4-5 x^2+x^4} \, dx=f x - {\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) + {\left (d - e + f\right )} \log \left (x + 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2\right )}{4-5 x^2+x^4} \, dx=f x - {\left (d - 2 \, e + 4 \, f\right )} \log \left ({\left | x + 2 \right |}\right ) + {\left (d - e + f\right )} \log \left ({\left | x + 1 \right |}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2\right )}{4-5 x^2+x^4} \, dx=f\,x+\ln \left (x+1\right )\,\left (d-e+f\right )-\ln \left (x+2\right )\,\left (d-2\,e+4\,f\right ) \]
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