Integrand size = 47, antiderivative size = 27 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=\log \left (\frac {x \left (-2-\frac {16}{3 \left (5+\frac {4}{x}+3 x\right )}\right )}{12 e^5}\right ) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2099, 642} \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=-\log \left (3 x^2+5 x+4\right )+\log \left (9 x^2+23 x+12\right )+\log (x) \]
[In]
[Out]
Rule 642
Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x}+\frac {-5-6 x}{4+5 x+3 x^2}+\frac {23+18 x}{12+23 x+9 x^2}\right ) \, dx \\ & = \log (x)+\int \frac {-5-6 x}{4+5 x+3 x^2} \, dx+\int \frac {23+18 x}{12+23 x+9 x^2} \, dx \\ & = \log (x)-\log \left (4+5 x+3 x^2\right )+\log \left (12+23 x+9 x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=\log (x)-\log \left (4+5 x+3 x^2\right )+\log \left (12+23 x+9 x^2\right ) \]
[In]
[Out]
Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
parallelrisch | \(\ln \left (x \right )-\ln \left (x^{2}+\frac {5}{3} x +\frac {4}{3}\right )+\ln \left (x^{2}+\frac {23}{9} x +\frac {4}{3}\right )\) | \(24\) |
default | \(-\ln \left (3 x^{2}+5 x +4\right )+\ln \left (9 x^{2}+23 x +12\right )+\ln \left (x \right )\) | \(28\) |
norman | \(-\ln \left (3 x^{2}+5 x +4\right )+\ln \left (9 x^{2}+23 x +12\right )+\ln \left (x \right )\) | \(28\) |
risch | \(-\ln \left (3 x^{2}+5 x +4\right )+\ln \left (9 x^{3}+23 x^{2}+12 x \right )\) | \(30\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=\log \left (9 \, x^{3} + 23 \, x^{2} + 12 \, x\right ) - \log \left (3 \, x^{2} + 5 \, x + 4\right ) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=- \log {\left (3 x^{2} + 5 x + 4 \right )} + \log {\left (9 x^{3} + 23 x^{2} + 12 x \right )} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=\log \left (9 \, x^{2} + 23 \, x + 12\right ) - \log \left (3 \, x^{2} + 5 \, x + 4\right ) + \log \left (x\right ) \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=-\log \left (3 \, x^{2} + 5 \, x + 4\right ) + \log \left ({\left | 9 \, x^{2} + 23 \, x + 12 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx=\ln \left (x\,\left (9\,x^2+23\,x+12\right )\right )-\ln \left (x^2+\frac {5\,x}{3}+\frac {4}{3}\right ) \]
[In]
[Out]