Integrand size = 20, antiderivative size = 14 \[ \int \frac {-16+9 x+3 x^2}{4 x+x^2} \, dx=5+3 x+\log \left (\frac {16 (4+x)}{x^4}\right ) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1607, 907} \[ \int \frac {-16+9 x+3 x^2}{4 x+x^2} \, dx=3 x-4 \log (x)+\log (x+4) \]
[In]
[Out]
Rule 907
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {-16+9 x+3 x^2}{x (4+x)} \, dx \\ & = \int \left (3-\frac {4}{x}+\frac {1}{4+x}\right ) \, dx \\ & = 3 x-4 \log (x)+\log (4+x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-16+9 x+3 x^2}{4 x+x^2} \, dx=3 x-4 \log (x)+\log (4+x) \]
[In]
[Out]
Time = 0.48 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(3 x -4 \ln \left (x \right )+\ln \left (4+x \right )\) | \(13\) |
norman | \(3 x -4 \ln \left (x \right )+\ln \left (4+x \right )\) | \(13\) |
risch | \(3 x -4 \ln \left (x \right )+\ln \left (4+x \right )\) | \(13\) |
parallelrisch | \(3 x -4 \ln \left (x \right )+\ln \left (4+x \right )\) | \(13\) |
meijerg | \(-4 \ln \left (x \right )+8 \ln \left (2\right )+\ln \left (1+\frac {x}{4}\right )+3 x\) | \(19\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-16+9 x+3 x^2}{4 x+x^2} \, dx=3 \, x + \log \left (x + 4\right ) - 4 \, \log \left (x\right ) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-16+9 x+3 x^2}{4 x+x^2} \, dx=3 x - 4 \log {\left (x \right )} + \log {\left (x + 4 \right )} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-16+9 x+3 x^2}{4 x+x^2} \, dx=3 \, x + \log \left (x + 4\right ) - 4 \, \log \left (x\right ) \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-16+9 x+3 x^2}{4 x+x^2} \, dx=3 \, x + \log \left ({\left | x + 4 \right |}\right ) - 4 \, \log \left ({\left | x \right |}\right ) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-16+9 x+3 x^2}{4 x+x^2} \, dx=3\,x+\ln \left (x+4\right )-4\,\ln \left (x\right ) \]
[In]
[Out]