Integrand size = 23, antiderivative size = 21 \[ \int \frac {1+4 x+3 x^2}{\left (4+7 x+2 x^2\right )^2} \, dx=-\frac {2+3 x}{2 \left (4+7 x+2 x^2\right )} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1674, 8} \[ \int \frac {1+4 x+3 x^2}{\left (4+7 x+2 x^2\right )^2} \, dx=-\frac {3 x+2}{2 \left (2 x^2+7 x+4\right )} \]
[In]
[Out]
Rule 8
Rule 1674
Rubi steps \begin{align*} \text {integral}& = -\frac {2+3 x}{2 \left (4+7 x+2 x^2\right )}-\frac {\int 0 \, dx}{17} \\ & = -\frac {2+3 x}{2 \left (4+7 x+2 x^2\right )} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1+4 x+3 x^2}{\left (4+7 x+2 x^2\right )^2} \, dx=\frac {-2-3 x}{2 \left (4+7 x+2 x^2\right )} \]
[In]
[Out]
Time = 0.54 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {-\frac {3 x}{4}-\frac {1}{2}}{x^{2}+\frac {7}{2} x +2}\) | \(17\) |
risch | \(\frac {-\frac {3 x}{4}-\frac {1}{2}}{x^{2}+\frac {7}{2} x +2}\) | \(17\) |
norman | \(\frac {-\frac {3 x}{2}-1}{2 x^{2}+7 x +4}\) | \(19\) |
gosper | \(-\frac {2+3 x}{2 \left (2 x^{2}+7 x +4\right )}\) | \(20\) |
parallelrisch | \(\frac {-2-3 x}{4 x^{2}+14 x +8}\) | \(20\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1+4 x+3 x^2}{\left (4+7 x+2 x^2\right )^2} \, dx=-\frac {3 \, x + 2}{2 \, {\left (2 \, x^{2} + 7 \, x + 4\right )}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {1+4 x+3 x^2}{\left (4+7 x+2 x^2\right )^2} \, dx=\frac {- 3 x - 2}{4 x^{2} + 14 x + 8} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1+4 x+3 x^2}{\left (4+7 x+2 x^2\right )^2} \, dx=-\frac {3 \, x + 2}{2 \, {\left (2 \, x^{2} + 7 \, x + 4\right )}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1+4 x+3 x^2}{\left (4+7 x+2 x^2\right )^2} \, dx=-\frac {3 \, x + 2}{2 \, {\left (2 \, x^{2} + 7 \, x + 4\right )}} \]
[In]
[Out]
Time = 13.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1+4 x+3 x^2}{\left (4+7 x+2 x^2\right )^2} \, dx=-\frac {\frac {3\,x}{4}+\frac {1}{2}}{x^2+\frac {7\,x}{2}+2} \]
[In]
[Out]