Integrand size = 94, antiderivative size = 30 \[ \int \frac {64 x-144 x^2-1188 x^3+459 x^4+e^{4 x} \left (-128 x+32 x^2+360 x^3-378 x^4+108 x^5\right )+e^{2 x} \left (864 x^2-504 x^3-540 x^4+324 x^5\right )}{-64+144 x-108 x^2+27 x^3} \, dx=-\frac {x^2}{2}+x^2 \left (e^{2 x}+\frac {9 x}{-4+3 x}\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(97\) vs. \(2(30)=60\).
Time = 0.88 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.23, number of steps used = 28, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {6820, 6874, 2227, 2207, 2225, 37, 45, 2230, 2208, 2209} \[ \int \frac {64 x-144 x^2-1188 x^3+459 x^4+e^{4 x} \left (-128 x+32 x^2+360 x^3-378 x^4+108 x^5\right )+e^{2 x} \left (864 x^2-504 x^3-540 x^4+324 x^5\right )}{-64+144 x-108 x^2+27 x^3} \, dx=6 e^{2 x} x^2+e^{4 x} x^2-\frac {8 x^2}{(4-3 x)^2}+\frac {17 x^2}{2}+8 e^{2 x} x+24 x+\frac {32 e^{2 x}}{3}-\frac {128 e^{2 x}}{3 (4-3 x)}-\frac {2368}{9 (4-3 x)}+\frac {2432}{9 (4-3 x)^2} \]
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Rule 37
Rule 45
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2227
Rule 2230
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (-64+144 x+1188 x^2-459 x^3-2 e^{4 x} (1+2 x) (-4+3 x)^3-36 e^{2 x} x \left (24-14 x-15 x^2+9 x^3\right )\right )}{(4-3 x)^3} \, dx \\ & = \int \left (2 e^{4 x} x (1+2 x)+\frac {64 x}{(-4+3 x)^3}-\frac {144 x^2}{(-4+3 x)^3}-\frac {1188 x^3}{(-4+3 x)^3}+\frac {459 x^4}{(-4+3 x)^3}+\frac {36 e^{2 x} x^2 \left (-6-x+3 x^2\right )}{(-4+3 x)^2}\right ) \, dx \\ & = 2 \int e^{4 x} x (1+2 x) \, dx+36 \int \frac {e^{2 x} x^2 \left (-6-x+3 x^2\right )}{(-4+3 x)^2} \, dx+64 \int \frac {x}{(-4+3 x)^3} \, dx-144 \int \frac {x^2}{(-4+3 x)^3} \, dx+459 \int \frac {x^4}{(-4+3 x)^3} \, dx-1188 \int \frac {x^3}{(-4+3 x)^3} \, dx \\ & = -\frac {8 x^2}{(4-3 x)^2}+2 \int \left (e^{4 x} x+2 e^{4 x} x^2\right ) \, dx+36 \int \left (\frac {22 e^{2 x}}{27}+\frac {7}{9} e^{2 x} x+\frac {1}{3} e^{2 x} x^2-\frac {32 e^{2 x}}{9 (-4+3 x)^2}+\frac {64 e^{2 x}}{27 (-4+3 x)}\right ) \, dx-144 \int \left (\frac {16}{9 (-4+3 x)^3}+\frac {8}{9 (-4+3 x)^2}+\frac {1}{9 (-4+3 x)}\right ) \, dx+459 \int \left (\frac {4}{27}+\frac {x}{27}+\frac {256}{81 (-4+3 x)^3}+\frac {256}{81 (-4+3 x)^2}+\frac {32}{27 (-4+3 x)}\right ) \, dx-1188 \int \left (\frac {1}{27}+\frac {64}{27 (-4+3 x)^3}+\frac {16}{9 (-4+3 x)^2}+\frac {4}{9 (-4+3 x)}\right ) \, dx \\ & = \frac {2432}{9 (4-3 x)^2}-\frac {2368}{9 (4-3 x)}+24 x+\frac {17 x^2}{2}-\frac {8 x^2}{(4-3 x)^2}+2 \int e^{4 x} x \, dx+4 \int e^{4 x} x^2 \, dx+12 \int e^{2 x} x^2 \, dx+28 \int e^{2 x} x \, dx+\frac {88}{3} \int e^{2 x} \, dx+\frac {256}{3} \int \frac {e^{2 x}}{-4+3 x} \, dx-128 \int \frac {e^{2 x}}{(-4+3 x)^2} \, dx \\ & = \frac {44 e^{2 x}}{3}+\frac {2432}{9 (4-3 x)^2}-\frac {2368}{9 (4-3 x)}-\frac {128 e^{2 x}}{3 (4-3 x)}+24 x+14 e^{2 x} x+\frac {1}{2} e^{4 x} x+\frac {17 x^2}{2}+6 e^{2 x} x^2+e^{4 x} x^2-\frac {8 x^2}{(4-3 x)^2}+\frac {256}{9} e^{8/3} \operatorname {ExpIntegralEi}\left (-\frac {2}{3} (4-3 x)\right )-\frac {1}{2} \int e^{4 x} \, dx-2 \int e^{4 x} x \, dx-12 \int e^{2 x} x \, dx-14 \int e^{2 x} \, dx-\frac {256}{3} \int \frac {e^{2 x}}{-4+3 x} \, dx \\ & = \frac {23 e^{2 x}}{3}-\frac {e^{4 x}}{8}+\frac {2432}{9 (4-3 x)^2}-\frac {2368}{9 (4-3 x)}-\frac {128 e^{2 x}}{3 (4-3 x)}+24 x+8 e^{2 x} x+\frac {17 x^2}{2}+6 e^{2 x} x^2+e^{4 x} x^2-\frac {8 x^2}{(4-3 x)^2}+\frac {1}{2} \int e^{4 x} \, dx+6 \int e^{2 x} \, dx \\ & = \frac {32 e^{2 x}}{3}+\frac {2432}{9 (4-3 x)^2}-\frac {2368}{9 (4-3 x)}-\frac {128 e^{2 x}}{3 (4-3 x)}+24 x+8 e^{2 x} x+\frac {17 x^2}{2}+6 e^{2 x} x^2+e^{4 x} x^2-\frac {8 x^2}{(4-3 x)^2} \\ \end{align*}
Time = 2.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \frac {64 x-144 x^2-1188 x^3+459 x^4+e^{4 x} \left (-128 x+32 x^2+360 x^3-378 x^4+108 x^5\right )+e^{2 x} \left (864 x^2-504 x^3-540 x^4+324 x^5\right )}{-64+144 x-108 x^2+27 x^3} \, dx=\frac {768 (-1+x)}{(4-3 x)^2}+24 x+\left (\frac {17}{2}+e^{4 x}\right ) x^2+\frac {18 e^{2 x} x^3}{-4+3 x} \]
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Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67
method | result | size |
risch | \(\frac {17 x^{2}}{2}+24 x +\frac {\frac {256 x}{3}-\frac {256}{3}}{x^{2}-\frac {8}{3} x +\frac {16}{9}}+x^{2} {\mathrm e}^{4 x}+\frac {18 x^{3} {\mathrm e}^{2 x}}{-4+3 x}\) | \(50\) |
derivativedivides | \(\frac {512}{6 x -8}+\frac {1024}{\left (6 x -8\right )^{2}}+24 x +\frac {17 x^{2}}{2}+\frac {256 \,{\mathrm e}^{2 x}}{9 \left (2 x -\frac {8}{3}\right )}+\frac {32 \,{\mathrm e}^{2 x}}{3}+8 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x} x^{2}+x^{2} {\mathrm e}^{4 x}\) | \(73\) |
default | \(\frac {512}{6 x -8}+\frac {1024}{\left (6 x -8\right )^{2}}+24 x +\frac {17 x^{2}}{2}+\frac {256 \,{\mathrm e}^{2 x}}{9 \left (2 x -\frac {8}{3}\right )}+\frac {32 \,{\mathrm e}^{2 x}}{3}+8 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x} x^{2}+x^{2} {\mathrm e}^{4 x}\) | \(73\) |
parts | \(\frac {256}{-4+3 x}+\frac {256}{\left (-4+3 x \right )^{2}}+\frac {17 x^{2}}{2}+24 x +x^{2} {\mathrm e}^{4 x}+\frac {32 \,{\mathrm e}^{2 x}}{3}+\frac {256 \,{\mathrm e}^{2 x}}{9 \left (2 x -\frac {8}{3}\right )}+8 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x} x^{2}\) | \(73\) |
norman | \(\frac {-\frac {64 x}{3}+12 x^{3}+\frac {153 x^{4}}{2}+16 x^{2} {\mathrm e}^{4 x}-24 x^{3} {\mathrm e}^{4 x}+9 \,{\mathrm e}^{4 x} x^{4}-72 \,{\mathrm e}^{2 x} x^{3}+54 \,{\mathrm e}^{2 x} x^{4}+\frac {128}{9}}{\left (-4+3 x \right )^{2}}\) | \(75\) |
parallelrisch | \(\frac {162 \,{\mathrm e}^{4 x} x^{4}+972 \,{\mathrm e}^{2 x} x^{4}-432 x^{3} {\mathrm e}^{4 x}+1377 x^{4}-1296 \,{\mathrm e}^{2 x} x^{3}+288 x^{2} {\mathrm e}^{4 x}+216 x^{3}+256-384 x}{162 x^{2}-432 x +288}\) | \(81\) |
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.43 \[ \int \frac {64 x-144 x^2-1188 x^3+459 x^4+e^{4 x} \left (-128 x+32 x^2+360 x^3-378 x^4+108 x^5\right )+e^{2 x} \left (864 x^2-504 x^3-540 x^4+324 x^5\right )}{-64+144 x-108 x^2+27 x^3} \, dx=\frac {153 \, x^{4} + 24 \, x^{3} - 880 \, x^{2} + 2 \, {\left (9 \, x^{4} - 24 \, x^{3} + 16 \, x^{2}\right )} e^{\left (4 \, x\right )} + 36 \, {\left (3 \, x^{4} - 4 \, x^{3}\right )} e^{\left (2 \, x\right )} + 2304 \, x - 1536}{2 \, {\left (9 \, x^{2} - 24 \, x + 16\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {64 x-144 x^2-1188 x^3+459 x^4+e^{4 x} \left (-128 x+32 x^2+360 x^3-378 x^4+108 x^5\right )+e^{2 x} \left (864 x^2-504 x^3-540 x^4+324 x^5\right )}{-64+144 x-108 x^2+27 x^3} \, dx=\frac {17 x^{2}}{2} + 24 x + \frac {768 x - 768}{9 x^{2} - 24 x + 16} + \frac {18 x^{3} e^{2 x} + \left (3 x^{3} - 4 x^{2}\right ) e^{4 x}}{3 x - 4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (27) = 54\).
Time = 0.22 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.90 \[ \int \frac {64 x-144 x^2-1188 x^3+459 x^4+e^{4 x} \left (-128 x+32 x^2+360 x^3-378 x^4+108 x^5\right )+e^{2 x} \left (864 x^2-504 x^3-540 x^4+324 x^5\right )}{-64+144 x-108 x^2+27 x^3} \, dx=\frac {17}{2} \, x^{2} + 24 \, x + \frac {704 \, {\left (9 \, x - 10\right )}}{3 \, {\left (9 \, x^{2} - 24 \, x + 16\right )}} - \frac {2176 \, {\left (6 \, x - 7\right )}}{9 \, {\left (9 \, x^{2} - 24 \, x + 16\right )}} - \frac {64 \, {\left (3 \, x - 2\right )}}{9 \, {\left (9 \, x^{2} - 24 \, x + 16\right )}} + \frac {18 \, x^{3} e^{\left (2 \, x\right )} + {\left (3 \, x^{3} - 4 \, x^{2}\right )} e^{\left (4 \, x\right )}}{3 \, x - 4} + \frac {128 \, {\left (x - 1\right )}}{9 \, x^{2} - 24 \, x + 16} \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.63 \[ \int \frac {64 x-144 x^2-1188 x^3+459 x^4+e^{4 x} \left (-128 x+32 x^2+360 x^3-378 x^4+108 x^5\right )+e^{2 x} \left (864 x^2-504 x^3-540 x^4+324 x^5\right )}{-64+144 x-108 x^2+27 x^3} \, dx=\frac {18 \, x^{4} e^{\left (4 \, x\right )} + 108 \, x^{4} e^{\left (2 \, x\right )} + 153 \, x^{4} - 48 \, x^{3} e^{\left (4 \, x\right )} - 144 \, x^{3} e^{\left (2 \, x\right )} + 24 \, x^{3} + 32 \, x^{2} e^{\left (4 \, x\right )} - 880 \, x^{2} + 2304 \, x - 1536}{2 \, {\left (9 \, x^{2} - 24 \, x + 16\right )}} \]
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Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.97 \[ \int \frac {64 x-144 x^2-1188 x^3+459 x^4+e^{4 x} \left (-128 x+32 x^2+360 x^3-378 x^4+108 x^5\right )+e^{2 x} \left (864 x^2-504 x^3-540 x^4+324 x^5\right )}{-64+144 x-108 x^2+27 x^3} \, dx=\frac {x^2\,\left (12\,x+16\,{\mathrm {e}}^{4\,x}-72\,x\,{\mathrm {e}}^{2\,x}-24\,x\,{\mathrm {e}}^{4\,x}+54\,x^2\,{\mathrm {e}}^{2\,x}+9\,x^2\,{\mathrm {e}}^{4\,x}+\frac {153\,x^2}{2}-8\right )}{{\left (3\,x-4\right )}^2} \]
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