Integrand size = 74, antiderivative size = 25 \[ \int \frac {-2560000 x^2-96000 x^3-20100 x^4-360 x^5-36 x^6+e^x \left (-240000+231000 x+1800 x^2+900 x^3\right )}{640000 x^2+24000 x^3+5025 x^4+90 x^5+9 x^6} \, dx=-4 x+\frac {e^x}{x \left (\frac {8}{3}+\frac {1}{100} x (5+x)\right )} \]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.08 (sec) , antiderivative size = 554, normalized size of antiderivative = 22.16, number of steps used = 29, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {6820, 6874, 2208, 2209, 2302} \[ \int \frac {-2560000 x^2-96000 x^3-20100 x^4-360 x^5-36 x^6+e^x \left (-240000+231000 x+1800 x^2+900 x^3\right )}{640000 x^2+24000 x^3+5025 x^4+90 x^5+9 x^6} \, dx=\frac {3 \left (3-5 i \sqrt {15}\right ) e^{\frac {5}{6} i \left (5 \sqrt {15}+3 i\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (6 x+5 \left (3-5 i \sqrt {15}\right )\right )\right )}{2000}-\frac {3 \left (125-3 i \sqrt {15}\right ) e^{\frac {5}{6} i \left (5 \sqrt {15}+3 i\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (6 x+5 \left (3-5 i \sqrt {15}\right )\right )\right )}{2000}+\frac {3}{200} i \sqrt {\frac {3}{5}} e^{\frac {5}{6} i \left (5 \sqrt {15}+3 i\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (6 x+5 \left (3-5 i \sqrt {15}\right )\right )\right )+\frac {183 e^{\frac {5}{6} i \left (5 \sqrt {15}+3 i\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (6 x+5 \left (3-5 i \sqrt {15}\right )\right )\right )}{1000}+\frac {3 \left (3+5 i \sqrt {15}\right ) e^{-\frac {5}{6} \left (3+5 i \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (6 x+5 \left (3+5 i \sqrt {15}\right )\right )\right )}{2000}-\frac {3 \left (125+3 i \sqrt {15}\right ) e^{-\frac {5}{6} \left (3+5 i \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (6 x+5 \left (3+5 i \sqrt {15}\right )\right )\right )}{2000}-\frac {3}{200} i \sqrt {\frac {3}{5}} e^{-\frac {5}{6} \left (3+5 i \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (6 x+5 \left (3+5 i \sqrt {15}\right )\right )\right )+\frac {183 e^{-\frac {5}{6} \left (3+5 i \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (6 x+5 \left (3+5 i \sqrt {15}\right )\right )\right )}{1000}-4 x-\frac {9 \left (3-5 i \sqrt {15}\right ) e^x}{1000 \left (6 x+5 \left (3-5 i \sqrt {15}\right )\right )}-\frac {549 e^x}{500 \left (6 x+5 \left (3-5 i \sqrt {15}\right )\right )}-\frac {9 \left (3+5 i \sqrt {15}\right ) e^x}{1000 \left (6 x+5 \left (3+5 i \sqrt {15}\right )\right )}-\frac {549 e^x}{500 \left (6 x+5 \left (3+5 i \sqrt {15}\right )\right )}+\frac {3 e^x}{8 x} \]
[In]
[Out]
Rule 2208
Rule 2209
Rule 2302
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-4+\frac {300 e^x \left (-800+770 x+6 x^2+3 x^3\right )}{x^2 \left (800+15 x+3 x^2\right )^2}\right ) \, dx \\ & = -4 x+300 \int \frac {e^x \left (-800+770 x+6 x^2+3 x^3\right )}{x^2 \left (800+15 x+3 x^2\right )^2} \, dx \\ & = -4 x+300 \int \left (-\frac {e^x}{800 x^2}+\frac {e^x}{800 x}+\frac {3 e^x (-305+3 x)}{160 \left (800+15 x+3 x^2\right )^2}-\frac {3 e^x (4+x)}{800 \left (800+15 x+3 x^2\right )}\right ) \, dx \\ & = -4 x-\frac {3}{8} \int \frac {e^x}{x^2} \, dx+\frac {3}{8} \int \frac {e^x}{x} \, dx-\frac {9}{8} \int \frac {e^x (4+x)}{800+15 x+3 x^2} \, dx+\frac {45}{8} \int \frac {e^x (-305+3 x)}{\left (800+15 x+3 x^2\right )^2} \, dx \\ & = \frac {3 e^x}{8 x}-4 x+\frac {3 \operatorname {ExpIntegralEi}(x)}{8}-\frac {3}{8} \int \frac {e^x}{x} \, dx-\frac {9}{8} \int \left (\frac {\left (1-\frac {3}{25} i \sqrt {\frac {3}{5}}\right ) e^x}{15-25 i \sqrt {15}+6 x}+\frac {\left (1+\frac {3}{25} i \sqrt {\frac {3}{5}}\right ) e^x}{15+25 i \sqrt {15}+6 x}\right ) \, dx+\frac {45}{8} \int \left (-\frac {305 e^x}{\left (800+15 x+3 x^2\right )^2}+\frac {3 e^x x}{\left (800+15 x+3 x^2\right )^2}\right ) \, dx \\ & = \frac {3 e^x}{8 x}-4 x+\frac {135}{8} \int \frac {e^x x}{\left (800+15 x+3 x^2\right )^2} \, dx-\frac {13725}{8} \int \frac {e^x}{\left (800+15 x+3 x^2\right )^2} \, dx-\frac {\left (9 \left (125-3 i \sqrt {15}\right )\right ) \int \frac {e^x}{15-25 i \sqrt {15}+6 x} \, dx}{1000}-\frac {\left (9 \left (125+3 i \sqrt {15}\right )\right ) \int \frac {e^x}{15+25 i \sqrt {15}+6 x} \, dx}{1000} \\ & = \frac {3 e^x}{8 x}-4 x-\frac {3 \left (125-3 i \sqrt {15}\right ) e^{\frac {5}{6} i \left (3 i+5 \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3-5 i \sqrt {15}\right )+6 x\right )\right )}{2000}-\frac {3 \left (125+3 i \sqrt {15}\right ) e^{-\frac {5}{6} \left (3+5 i \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3+5 i \sqrt {15}\right )+6 x\right )\right )}{2000}+\frac {135}{8} \int \left (-\frac {2 \left (-15+25 i \sqrt {15}\right ) e^x}{3125 \left (-15+25 i \sqrt {15}-6 x\right )^2}-\frac {2 i \sqrt {\frac {3}{5}} e^x}{15625 \left (-15+25 i \sqrt {15}-6 x\right )}-\frac {2 \left (-15-25 i \sqrt {15}\right ) e^x}{3125 \left (15+25 i \sqrt {15}+6 x\right )^2}-\frac {2 i \sqrt {\frac {3}{5}} e^x}{15625 \left (15+25 i \sqrt {15}+6 x\right )}\right ) \, dx-\frac {13725}{8} \int \left (-\frac {12 e^x}{3125 \left (-15+25 i \sqrt {15}-6 x\right )^2}+\frac {4 i \sqrt {\frac {3}{5}} e^x}{78125 \left (-15+25 i \sqrt {15}-6 x\right )}-\frac {12 e^x}{3125 \left (15+25 i \sqrt {15}+6 x\right )^2}+\frac {4 i \sqrt {\frac {3}{5}} e^x}{78125 \left (15+25 i \sqrt {15}+6 x\right )}\right ) \, dx \\ & = \frac {3 e^x}{8 x}-4 x-\frac {3 \left (125-3 i \sqrt {15}\right ) e^{\frac {5}{6} i \left (3 i+5 \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3-5 i \sqrt {15}\right )+6 x\right )\right )}{2000}-\frac {3 \left (125+3 i \sqrt {15}\right ) e^{-\frac {5}{6} \left (3+5 i \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3+5 i \sqrt {15}\right )+6 x\right )\right )}{2000}+\frac {1647}{250} \int \frac {e^x}{\left (-15+25 i \sqrt {15}-6 x\right )^2} \, dx+\frac {1647}{250} \int \frac {e^x}{\left (15+25 i \sqrt {15}+6 x\right )^2} \, dx-\frac {\left (27 i \sqrt {\frac {3}{5}}\right ) \int \frac {e^x}{-15+25 i \sqrt {15}-6 x} \, dx}{12500}-\frac {\left (27 i \sqrt {\frac {3}{5}}\right ) \int \frac {e^x}{15+25 i \sqrt {15}+6 x} \, dx}{12500}-\frac {\left (549 i \sqrt {\frac {3}{5}}\right ) \int \frac {e^x}{-15+25 i \sqrt {15}-6 x} \, dx}{6250}-\frac {\left (549 i \sqrt {\frac {3}{5}}\right ) \int \frac {e^x}{15+25 i \sqrt {15}+6 x} \, dx}{6250}+\frac {1}{500} \left (27 \left (3-5 i \sqrt {15}\right )\right ) \int \frac {e^x}{\left (-15+25 i \sqrt {15}-6 x\right )^2} \, dx+\frac {1}{500} \left (27 \left (3+5 i \sqrt {15}\right )\right ) \int \frac {e^x}{\left (15+25 i \sqrt {15}+6 x\right )^2} \, dx \\ & = \frac {3 e^x}{8 x}-4 x-\frac {549 e^x}{500 \left (5 \left (3-5 i \sqrt {15}\right )+6 x\right )}-\frac {9 \left (3-5 i \sqrt {15}\right ) e^x}{1000 \left (5 \left (3-5 i \sqrt {15}\right )+6 x\right )}-\frac {549 e^x}{500 \left (5 \left (3+5 i \sqrt {15}\right )+6 x\right )}-\frac {9 \left (3+5 i \sqrt {15}\right ) e^x}{1000 \left (5 \left (3+5 i \sqrt {15}\right )+6 x\right )}+\frac {3}{200} i \sqrt {\frac {3}{5}} e^{\frac {5}{6} i \left (3 i+5 \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3-5 i \sqrt {15}\right )+6 x\right )\right )-\frac {3 \left (125-3 i \sqrt {15}\right ) e^{\frac {5}{6} i \left (3 i+5 \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3-5 i \sqrt {15}\right )+6 x\right )\right )}{2000}-\frac {3}{200} i \sqrt {\frac {3}{5}} e^{-\frac {5}{6} \left (3+5 i \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3+5 i \sqrt {15}\right )+6 x\right )\right )-\frac {3 \left (125+3 i \sqrt {15}\right ) e^{-\frac {5}{6} \left (3+5 i \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3+5 i \sqrt {15}\right )+6 x\right )\right )}{2000}-\frac {549}{500} \int \frac {e^x}{-15+25 i \sqrt {15}-6 x} \, dx+\frac {549}{500} \int \frac {e^x}{15+25 i \sqrt {15}+6 x} \, dx-\frac {\left (9 \left (3-5 i \sqrt {15}\right )\right ) \int \frac {e^x}{-15+25 i \sqrt {15}-6 x} \, dx}{1000}+\frac {\left (9 \left (3+5 i \sqrt {15}\right )\right ) \int \frac {e^x}{15+25 i \sqrt {15}+6 x} \, dx}{1000} \\ & = \frac {3 e^x}{8 x}-4 x-\frac {549 e^x}{500 \left (5 \left (3-5 i \sqrt {15}\right )+6 x\right )}-\frac {9 \left (3-5 i \sqrt {15}\right ) e^x}{1000 \left (5 \left (3-5 i \sqrt {15}\right )+6 x\right )}-\frac {549 e^x}{500 \left (5 \left (3+5 i \sqrt {15}\right )+6 x\right )}-\frac {9 \left (3+5 i \sqrt {15}\right ) e^x}{1000 \left (5 \left (3+5 i \sqrt {15}\right )+6 x\right )}+\frac {183 e^{\frac {5}{6} i \left (3 i+5 \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3-5 i \sqrt {15}\right )+6 x\right )\right )}{1000}+\frac {3}{200} i \sqrt {\frac {3}{5}} e^{\frac {5}{6} i \left (3 i+5 \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3-5 i \sqrt {15}\right )+6 x\right )\right )-\frac {3 \left (125-3 i \sqrt {15}\right ) e^{\frac {5}{6} i \left (3 i+5 \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3-5 i \sqrt {15}\right )+6 x\right )\right )}{2000}+\frac {3 \left (3-5 i \sqrt {15}\right ) e^{\frac {5}{6} i \left (3 i+5 \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3-5 i \sqrt {15}\right )+6 x\right )\right )}{2000}+\frac {183 e^{-\frac {5}{6} \left (3+5 i \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3+5 i \sqrt {15}\right )+6 x\right )\right )}{1000}-\frac {3}{200} i \sqrt {\frac {3}{5}} e^{-\frac {5}{6} \left (3+5 i \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3+5 i \sqrt {15}\right )+6 x\right )\right )-\frac {3 \left (125+3 i \sqrt {15}\right ) e^{-\frac {5}{6} \left (3+5 i \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3+5 i \sqrt {15}\right )+6 x\right )\right )}{2000}+\frac {3 \left (3+5 i \sqrt {15}\right ) e^{-\frac {5}{6} \left (3+5 i \sqrt {15}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{6} \left (5 \left (3+5 i \sqrt {15}\right )+6 x\right )\right )}{2000} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-2560000 x^2-96000 x^3-20100 x^4-360 x^5-36 x^6+e^x \left (-240000+231000 x+1800 x^2+900 x^3\right )}{640000 x^2+24000 x^3+5025 x^4+90 x^5+9 x^6} \, dx=-4 x+300 e^x \left (\frac {1}{800 x}-\frac {3 (5+x)}{800 \left (800+15 x+3 x^2\right )}\right ) \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
risch | \(-4 x +\frac {300 \,{\mathrm e}^{x}}{x \left (3 x^{2}+15 x +800\right )}\) | \(24\) |
norman | \(\frac {-2900 x^{2}+16000 x -12 x^{4}+300 \,{\mathrm e}^{x}}{x \left (3 x^{2}+15 x +800\right )}\) | \(35\) |
parallelrisch | \(-\frac {36 x^{4}+8700 x^{2}-48000 x -900 \,{\mathrm e}^{x}}{3 x \left (3 x^{2}+15 x +800\right )}\) | \(36\) |
parts | \(-4 x +\frac {72 \,{\mathrm e}^{x} \left (5+2 x \right )}{125 \left (3 x^{2}+15 x +800\right )}+\frac {3 \,{\mathrm e}^{x} \left (279 x^{2}+1635 x +50000\right )}{500 \left (3 x^{2}+15 x +800\right ) x}-\frac {231 \,{\mathrm e}^{x} \left (3 x -305\right )}{500 \left (3 x^{2}+15 x +800\right )}-\frac {12 \,{\mathrm e}^{x} \left (3 x +320\right )}{25 \left (3 x^{2}+15 x +800\right )}\) | \(97\) |
default | \(-\frac {4096 \left (6 x +15\right )}{15 \left (3 x^{2}+15 x +800\right )}-\frac {256 \left (-15 x -1600\right )}{25 \left (3 x^{2}+15 x +800\right )}-\frac {20100 \left (-\frac {61 x}{1125}+\frac {32}{225}\right )}{x^{2}+5 x +\frac {800}{3}}-\frac {360 \left (\frac {31 x}{75}+\frac {1952}{135}\right )}{x^{2}+5 x +\frac {800}{3}}-4 x +\frac {-\frac {6692 x}{15}+3968}{x^{2}+5 x +\frac {800}{3}}+\frac {3 \,{\mathrm e}^{x} \left (279 x^{2}+1635 x +50000\right )}{500 \left (3 x^{2}+15 x +800\right ) x}-\frac {231 \,{\mathrm e}^{x} \left (3 x -305\right )}{500 \left (3 x^{2}+15 x +800\right )}+\frac {72 \,{\mathrm e}^{x} \left (5+2 x \right )}{125 \left (3 x^{2}+15 x +800\right )}-\frac {12 \,{\mathrm e}^{x} \left (3 x +320\right )}{25 \left (3 x^{2}+15 x +800\right )}\) | \(186\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {-2560000 x^2-96000 x^3-20100 x^4-360 x^5-36 x^6+e^x \left (-240000+231000 x+1800 x^2+900 x^3\right )}{640000 x^2+24000 x^3+5025 x^4+90 x^5+9 x^6} \, dx=-\frac {4 \, {\left (3 \, x^{4} + 15 \, x^{3} + 800 \, x^{2} - 75 \, e^{x}\right )}}{3 \, x^{3} + 15 \, x^{2} + 800 \, x} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-2560000 x^2-96000 x^3-20100 x^4-360 x^5-36 x^6+e^x \left (-240000+231000 x+1800 x^2+900 x^3\right )}{640000 x^2+24000 x^3+5025 x^4+90 x^5+9 x^6} \, dx=- 4 x + \frac {300 e^{x}}{3 x^{3} + 15 x^{2} + 800 x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (21) = 42\).
Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.76 \[ \int \frac {-2560000 x^2-96000 x^3-20100 x^4-360 x^5-36 x^6+e^x \left (-240000+231000 x+1800 x^2+900 x^3\right )}{640000 x^2+24000 x^3+5025 x^4+90 x^5+9 x^6} \, dx=-4 \, x - \frac {4 \, {\left (1673 \, x - 14880\right )}}{5 \, {\left (3 \, x^{2} + 15 \, x + 800\right )}} - \frac {8 \, {\left (279 \, x + 9760\right )}}{5 \, {\left (3 \, x^{2} + 15 \, x + 800\right )}} + \frac {268 \, {\left (61 \, x - 160\right )}}{5 \, {\left (3 \, x^{2} + 15 \, x + 800\right )}} + \frac {256 \, {\left (3 \, x + 320\right )}}{5 \, {\left (3 \, x^{2} + 15 \, x + 800\right )}} - \frac {4096 \, {\left (2 \, x + 5\right )}}{5 \, {\left (3 \, x^{2} + 15 \, x + 800\right )}} + \frac {300 \, e^{x}}{3 \, x^{3} + 15 \, x^{2} + 800 \, x} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {-2560000 x^2-96000 x^3-20100 x^4-360 x^5-36 x^6+e^x \left (-240000+231000 x+1800 x^2+900 x^3\right )}{640000 x^2+24000 x^3+5025 x^4+90 x^5+9 x^6} \, dx=-\frac {4 \, {\left (3 \, x^{4} + 15 \, x^{3} + 800 \, x^{2} - 75 \, e^{x}\right )}}{3 \, x^{3} + 15 \, x^{2} + 800 \, x} \]
[In]
[Out]
Time = 15.58 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {-2560000 x^2-96000 x^3-20100 x^4-360 x^5-36 x^6+e^x \left (-240000+231000 x+1800 x^2+900 x^3\right )}{640000 x^2+24000 x^3+5025 x^4+90 x^5+9 x^6} \, dx=\frac {300\,{\mathrm {e}}^x}{x\,\left (3\,x^2+15\,x+800\right )}-4\,x \]
[In]
[Out]