Integrand size = 55, antiderivative size = 29 \[ \int \frac {-108-72 x-48 x^2-24 x^3-4 x^4+e^5 \left (11 x^2+6 x^3+x^4\right )}{9 x^2+6 x^3+x^4} \, dx=-3+\frac {12}{x}-e^5 \left (-x+\frac {2}{3+x}\right )-4 (x+\log (3)) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {1608, 27, 1634} \[ \int \frac {-108-72 x-48 x^2-24 x^3-4 x^4+e^5 \left (11 x^2+6 x^3+x^4\right )}{9 x^2+6 x^3+x^4} \, dx=-\left (\left (4-e^5\right ) x\right )-\frac {2 e^5}{x+3}+\frac {12}{x} \]
[In]
[Out]
Rule 27
Rule 1608
Rule 1634
Rubi steps \begin{align*} \text {integral}& = \int \frac {-108-72 x-48 x^2-24 x^3-4 x^4+e^5 \left (11 x^2+6 x^3+x^4\right )}{x^2 \left (9+6 x+x^2\right )} \, dx \\ & = \int \frac {-108-72 x-48 x^2-24 x^3-4 x^4+e^5 \left (11 x^2+6 x^3+x^4\right )}{x^2 (3+x)^2} \, dx \\ & = \int \left (-4 \left (1-\frac {e^5}{4}\right )-\frac {12}{x^2}+\frac {2 e^5}{(3+x)^2}\right ) \, dx \\ & = \frac {12}{x}-\left (4-e^5\right ) x-\frac {2 e^5}{3+x} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {-108-72 x-48 x^2-24 x^3-4 x^4+e^5 \left (11 x^2+6 x^3+x^4\right )}{9 x^2+6 x^3+x^4} \, dx=\frac {12}{x}+\left (-4+e^5\right ) x-\frac {2 e^5}{3+x} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79
method | result | size |
default | \(x \,{\mathrm e}^{5}-4 x +\frac {12}{x}-\frac {2 \,{\mathrm e}^{5}}{3+x}\) | \(23\) |
norman | \(\frac {\left ({\mathrm e}^{5}-4\right ) x^{3}+36+\left (-11 \,{\mathrm e}^{5}+48\right ) x}{\left (3+x \right ) x}\) | \(28\) |
risch | \(x \,{\mathrm e}^{5}-4 x +\frac {\left (-2 \,{\mathrm e}^{5}+12\right ) x +36}{\left (3+x \right ) x}\) | \(28\) |
gosper | \(\frac {x^{3} {\mathrm e}^{5}-4 x^{3}-11 x \,{\mathrm e}^{5}+48 x +36}{x \left (3+x \right )}\) | \(31\) |
parallelrisch | \(\frac {x^{3} {\mathrm e}^{5}-4 x^{3}-11 x \,{\mathrm e}^{5}+48 x +36}{x \left (3+x \right )}\) | \(31\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {-108-72 x-48 x^2-24 x^3-4 x^4+e^5 \left (11 x^2+6 x^3+x^4\right )}{9 x^2+6 x^3+x^4} \, dx=-\frac {4 \, x^{3} + 12 \, x^{2} - {\left (x^{3} + 3 \, x^{2} - 2 \, x\right )} e^{5} - 12 \, x - 36}{x^{2} + 3 \, x} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-108-72 x-48 x^2-24 x^3-4 x^4+e^5 \left (11 x^2+6 x^3+x^4\right )}{9 x^2+6 x^3+x^4} \, dx=- x \left (4 - e^{5}\right ) - \frac {x \left (-12 + 2 e^{5}\right ) - 36}{x^{2} + 3 x} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-108-72 x-48 x^2-24 x^3-4 x^4+e^5 \left (11 x^2+6 x^3+x^4\right )}{9 x^2+6 x^3+x^4} \, dx=x {\left (e^{5} - 4\right )} - \frac {2 \, {\left (x {\left (e^{5} - 6\right )} - 18\right )}}{x^{2} + 3 \, x} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {-108-72 x-48 x^2-24 x^3-4 x^4+e^5 \left (11 x^2+6 x^3+x^4\right )}{9 x^2+6 x^3+x^4} \, dx=x e^{5} - 4 \, x - \frac {2 \, {\left (x e^{5} - 6 \, x - 18\right )}}{x^{2} + 3 \, x} \]
[In]
[Out]
Time = 14.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-108-72 x-48 x^2-24 x^3-4 x^4+e^5 \left (11 x^2+6 x^3+x^4\right )}{9 x^2+6 x^3+x^4} \, dx=x\,\left ({\mathrm {e}}^5-4\right )-\frac {x\,\left (2\,{\mathrm {e}}^5-12\right )-36}{x\,\left (x+3\right )} \]
[In]
[Out]