Optimal. Leaf size=30 \[ -\frac {x^2}{2}+x^2 \left (e^{2 x}+\frac {9 x}{-4+3 x}\right )^2 \]
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Rubi [B] time = 0.95, antiderivative size = 97, normalized size of antiderivative = 3.23, number of steps used = 28, number of rules used = 10, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {6688, 6742, 2196, 2176, 2194, 37, 43, 2199, 2177, 2178} \begin {gather*} 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 43
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2196
Rule 2199
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \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} \text {Ei}\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 {aligned} \end {gather*}
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Mathematica [A] time = 0.25, size = 46, normalized size = 1.53 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.36, size = 73, normalized size = 2.43 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.52, size = 79, normalized size = 2.63 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 50, normalized size = 1.67
method | result | size |
risch | \(\frac {17 x^{2}}{2}+24 x +\frac {\frac {256 x}{3}-\frac {256}{3}}{x^{2}-\frac {8}{3} x +\frac {16}{9}}+x^{2} {\mathrm e}^{4 x}+\frac {18 x^{3} {\mathrm e}^{2 x}}{3 x -4}\) | \(50\) |
derivativedivides | \(\frac {1024}{\left (6 x -8\right )^{2}}+\frac {512}{6 x -8}+24 x +\frac {17 x^{2}}{2}+\frac {256 \,{\mathrm e}^{2 x}}{9 \left (2 x -\frac {8}{3}\right )}+\frac {32 \,{\mathrm e}^{2 x}}{3}+8 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x} x^{2}+x^{2} {\mathrm e}^{4 x}\) | \(73\) |
default | \(\frac {1024}{\left (6 x -8\right )^{2}}+\frac {512}{6 x -8}+24 x +\frac {17 x^{2}}{2}+\frac {256 \,{\mathrm e}^{2 x}}{9 \left (2 x -\frac {8}{3}\right )}+\frac {32 \,{\mathrm e}^{2 x}}{3}+8 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x} x^{2}+x^{2} {\mathrm e}^{4 x}\) | \(73\) |
norman | \(\frac {-\frac {64 x}{3}+12 x^{3}+\frac {153 x^{4}}{2}+16 x^{2} {\mathrm e}^{4 x}-24 x^{3} {\mathrm e}^{4 x}+9 x^{4} {\mathrm e}^{4 x}-72 \,{\mathrm e}^{2 x} x^{3}+54 \,{\mathrm e}^{2 x} x^{4}+\frac {128}{9}}{\left (3 x -4\right )^{2}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 117, normalized size = 3.90 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 59, normalized size = 1.97 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.20, size = 54, normalized size = 1.80 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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