Integrand size = 35, antiderivative size = 18 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{c d (a e+c d x)} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 32} \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{c d (a e+c d x)} \]
[In]
[Out]
Rule 32
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a e+c d x)^2} \, dx \\ & = -\frac {1}{c d (a e+c d x)} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{c d (a e+c d x)} \]
[In]
[Out]
Time = 2.76 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
method | result | size |
gosper | \(-\frac {1}{c d \left (c d x +a e \right )}\) | \(19\) |
default | \(-\frac {1}{c d \left (c d x +a e \right )}\) | \(19\) |
risch | \(-\frac {1}{c d \left (c d x +a e \right )}\) | \(19\) |
parallelrisch | \(-\frac {1}{c d \left (c d x +a e \right )}\) | \(19\) |
norman | \(\frac {-\frac {1}{c}-\frac {e x}{c d}}{\left (c d x +a e \right ) \left (e x +d \right )}\) | \(35\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{c^{2} d^{2} x + a c d e} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=- \frac {1}{a c d e + c^{2} d^{2} x} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{c^{2} d^{2} x + a c d e} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{{\left (c d x + a e\right )} c d} \]
[In]
[Out]
Time = 9.70 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {1}{c\,d\,\left (a\,e+c\,d\,x\right )} \]
[In]
[Out]