Integrand size = 47, antiderivative size = 28 \[ \int \frac {12-12 x-13 x^2+20 x^3+12 x^4+e^x \left (18+24 x+8 x^2\right )}{18+24 x+8 x^2} \, dx=4+e^x+x-\frac {1}{4} (4-2 x) x^2-\frac {x}{3+2 x} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {27, 12, 6874, 2225, 45} \[ \int \frac {12-12 x-13 x^2+20 x^3+12 x^4+e^x \left (18+24 x+8 x^2\right )}{18+24 x+8 x^2} \, dx=\frac {x^3}{2}-x^2+x+e^x+\frac {3}{2 (2 x+3)} \]
[In]
[Out]
Rule 12
Rule 27
Rule 45
Rule 2225
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {12-12 x-13 x^2+20 x^3+12 x^4+e^x \left (18+24 x+8 x^2\right )}{2 (3+2 x)^2} \, dx \\ & = \frac {1}{2} \int \frac {12-12 x-13 x^2+20 x^3+12 x^4+e^x \left (18+24 x+8 x^2\right )}{(3+2 x)^2} \, dx \\ & = \frac {1}{2} \int \left (2 e^x+\frac {12}{(3+2 x)^2}-\frac {12 x}{(3+2 x)^2}-\frac {13 x^2}{(3+2 x)^2}+\frac {20 x^3}{(3+2 x)^2}+\frac {12 x^4}{(3+2 x)^2}\right ) \, dx \\ & = -\frac {3}{3+2 x}-6 \int \frac {x}{(3+2 x)^2} \, dx+6 \int \frac {x^4}{(3+2 x)^2} \, dx-\frac {13}{2} \int \frac {x^2}{(3+2 x)^2} \, dx+10 \int \frac {x^3}{(3+2 x)^2} \, dx+\int e^x \, dx \\ & = e^x-\frac {3}{3+2 x}+6 \int \left (\frac {27}{16}-\frac {3 x}{4}+\frac {x^2}{4}+\frac {81}{16 (3+2 x)^2}-\frac {27}{4 (3+2 x)}\right ) \, dx-6 \int \left (-\frac {3}{2 (3+2 x)^2}+\frac {1}{2 (3+2 x)}\right ) \, dx-\frac {13}{2} \int \left (\frac {1}{4}+\frac {9}{4 (3+2 x)^2}-\frac {3}{2 (3+2 x)}\right ) \, dx+10 \int \left (-\frac {3}{4}+\frac {x}{4}-\frac {27}{8 (3+2 x)^2}+\frac {27}{8 (3+2 x)}\right ) \, dx \\ & = e^x+x-x^2+\frac {x^3}{2}+\frac {3}{2 (3+2 x)} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {12-12 x-13 x^2+20 x^3+12 x^4+e^x \left (18+24 x+8 x^2\right )}{18+24 x+8 x^2} \, dx=\frac {1}{2} \left (2 e^x+\frac {285+222 x-16 x^2-8 x^3+16 x^4}{24+16 x}\right ) \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {x^{3}}{2}-x^{2}+x +\frac {3}{4 \left (x +\frac {3}{2}\right )}+{\mathrm e}^{x}\) | \(22\) |
default | \(\frac {3}{2 \left (3+2 x \right )}+x -x^{2}+\frac {x^{3}}{2}+{\mathrm e}^{x}\) | \(24\) |
parts | \(\frac {3}{2 \left (3+2 x \right )}+x -x^{2}+\frac {x^{3}}{2}+{\mathrm e}^{x}\) | \(24\) |
norman | \(\frac {x^{4}-x^{2}-\frac {x^{3}}{2}+2 \,{\mathrm e}^{x} x +3 \,{\mathrm e}^{x}-3}{3+2 x}\) | \(33\) |
parallelrisch | \(\frac {4 x^{4}-2 x^{3}-4 x^{2}+8 \,{\mathrm e}^{x} x -12+12 \,{\mathrm e}^{x}}{8 x +12}\) | \(36\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {12-12 x-13 x^2+20 x^3+12 x^4+e^x \left (18+24 x+8 x^2\right )}{18+24 x+8 x^2} \, dx=\frac {2 \, x^{4} - x^{3} - 2 \, x^{2} + 2 \, {\left (2 \, x + 3\right )} e^{x} + 6 \, x + 3}{2 \, {\left (2 \, x + 3\right )}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {12-12 x-13 x^2+20 x^3+12 x^4+e^x \left (18+24 x+8 x^2\right )}{18+24 x+8 x^2} \, dx=\frac {x^{3}}{2} - x^{2} + x + e^{x} + \frac {3}{4 x + 6} \]
[In]
[Out]
\[ \int \frac {12-12 x-13 x^2+20 x^3+12 x^4+e^x \left (18+24 x+8 x^2\right )}{18+24 x+8 x^2} \, dx=\int { \frac {12 \, x^{4} + 20 \, x^{3} - 13 \, x^{2} + 2 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} e^{x} - 12 \, x + 12}{2 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \,d x } \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {12-12 x-13 x^2+20 x^3+12 x^4+e^x \left (18+24 x+8 x^2\right )}{18+24 x+8 x^2} \, dx=\frac {2 \, x^{4} - x^{3} - 2 \, x^{2} + 4 \, x e^{x} + 6 \, x + 6 \, e^{x} + 3}{2 \, {\left (2 \, x + 3\right )}} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {12-12 x-13 x^2+20 x^3+12 x^4+e^x \left (18+24 x+8 x^2\right )}{18+24 x+8 x^2} \, dx=x+{\mathrm {e}}^x+\frac {3}{4\,x+6}-x^2+\frac {x^3}{2} \]
[In]
[Out]