Optimal. Leaf size=34 \[ \left (\frac {5}{x^2}-\frac {-2+x}{x}+x\right ) \left (e^5+3 x^2 \left (-e^{2 x}+x\right )\right ) \]
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Rubi [B] time = 0.14, antiderivative size = 74, normalized size of antiderivative = 2.18, number of steps used = 16, number of rules used = 4, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14, 2196, 2194, 2176} \begin {gather*} 3 x^4-3 e^{2 x} x^3-3 x^3+3 e^{2 x} x^2+6 x^2+\frac {5 e^5}{x^2}-6 e^{2 x} x+\left (15+e^5\right ) x-15 e^{2 x}+\frac {2 e^5}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-3 e^{2 x} \left (12+2 x+x^2+2 x^3\right )+\frac {-10 e^5-2 e^5 x+15 \left (1+\frac {e^5}{15}\right ) x^3+12 x^4-9 x^5+12 x^6}{x^3}\right ) \, dx\\ &=-\left (3 \int e^{2 x} \left (12+2 x+x^2+2 x^3\right ) \, dx\right )+\int \frac {-10 e^5-2 e^5 x+15 \left (1+\frac {e^5}{15}\right ) x^3+12 x^4-9 x^5+12 x^6}{x^3} \, dx\\ &=-\left (3 \int \left (12 e^{2 x}+2 e^{2 x} x+e^{2 x} x^2+2 e^{2 x} x^3\right ) \, dx\right )+\int \left (15 \left (1+\frac {e^5}{15}\right )-\frac {10 e^5}{x^3}-\frac {2 e^5}{x^2}+12 x-9 x^2+12 x^3\right ) \, dx\\ &=\frac {5 e^5}{x^2}+\frac {2 e^5}{x}+\left (15+e^5\right ) x+6 x^2-3 x^3+3 x^4-3 \int e^{2 x} x^2 \, dx-6 \int e^{2 x} x \, dx-6 \int e^{2 x} x^3 \, dx-36 \int e^{2 x} \, dx\\ &=-18 e^{2 x}+\frac {5 e^5}{x^2}+\frac {2 e^5}{x}-3 e^{2 x} x+\left (15+e^5\right ) x+6 x^2-\frac {3}{2} e^{2 x} x^2-3 x^3-3 e^{2 x} x^3+3 x^4+3 \int e^{2 x} \, dx+3 \int e^{2 x} x \, dx+9 \int e^{2 x} x^2 \, dx\\ &=-\frac {33 e^{2 x}}{2}+\frac {5 e^5}{x^2}+\frac {2 e^5}{x}-\frac {3}{2} e^{2 x} x+\left (15+e^5\right ) x+6 x^2+3 e^{2 x} x^2-3 x^3-3 e^{2 x} x^3+3 x^4-\frac {3}{2} \int e^{2 x} \, dx-9 \int e^{2 x} x \, dx\\ &=-\frac {69 e^{2 x}}{4}+\frac {5 e^5}{x^2}+\frac {2 e^5}{x}-6 e^{2 x} x+\left (15+e^5\right ) x+6 x^2+3 e^{2 x} x^2-3 x^3-3 e^{2 x} x^3+3 x^4+\frac {9}{2} \int e^{2 x} \, dx\\ &=-15 e^{2 x}+\frac {5 e^5}{x^2}+\frac {2 e^5}{x}-6 e^{2 x} x+\left (15+e^5\right ) x+6 x^2+3 e^{2 x} x^2-3 x^3-3 e^{2 x} x^3+3 x^4\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 60, normalized size = 1.76 \begin {gather*} \frac {5 e^5}{x^2}+\frac {2 e^5}{x}+15 x+e^5 x+6 x^2-3 x^3+3 x^4-3 e^{2 x} \left (5+2 x-x^2+x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 61, normalized size = 1.79 \begin {gather*} \frac {3 \, x^{6} - 3 \, x^{5} + 6 \, x^{4} + 15 \, x^{3} + {\left (x^{3} + 2 \, x + 5\right )} e^{5} - 3 \, {\left (x^{5} - x^{4} + 2 \, x^{3} + 5 \, x^{2}\right )} e^{\left (2 \, x\right )}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 76, normalized size = 2.24 \begin {gather*} \frac {3 \, x^{6} - 3 \, x^{5} e^{\left (2 \, x\right )} - 3 \, x^{5} + 3 \, x^{4} e^{\left (2 \, x\right )} + 6 \, x^{4} + x^{3} e^{5} - 6 \, x^{3} e^{\left (2 \, x\right )} + 15 \, x^{3} - 15 \, x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{5} + 5 \, e^{5}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 58, normalized size = 1.71
method | result | size |
risch | \(3 x^{4}-3 x^{3}+x \,{\mathrm e}^{5}+6 x^{2}+15 x +\frac {2 x \,{\mathrm e}^{5}+5 \,{\mathrm e}^{5}}{x^{2}}+\left (-3 x^{3}+3 x^{2}-6 x -15\right ) {\mathrm e}^{2 x}\) | \(58\) |
derivativedivides | \(-3 x^{3}+6 x^{2}+15 x +3 x^{4}+\frac {5 \,{\mathrm e}^{5}}{x^{2}}+\frac {2 \,{\mathrm e}^{5}}{x}-6 x \,{\mathrm e}^{2 x}-15 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{2 x} x^{2}-3 \,{\mathrm e}^{2 x} x^{3}+x \,{\mathrm e}^{5}\) | \(69\) |
default | \(-3 x^{3}+6 x^{2}+15 x +3 x^{4}+\frac {5 \,{\mathrm e}^{5}}{x^{2}}+\frac {2 \,{\mathrm e}^{5}}{x}-6 x \,{\mathrm e}^{2 x}-15 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{2 x} x^{2}-3 \,{\mathrm e}^{2 x} x^{3}+x \,{\mathrm e}^{5}\) | \(69\) |
norman | \(\frac {\left ({\mathrm e}^{5}+15\right ) x^{3}+6 x^{4}-3 x^{5}+3 x^{6}+2 x \,{\mathrm e}^{5}-3 x^{5} {\mathrm e}^{2 x}-15 \,{\mathrm e}^{2 x} x^{2}-6 \,{\mathrm e}^{2 x} x^{3}+3 \,{\mathrm e}^{2 x} x^{4}+5 \,{\mathrm e}^{5}}{x^{2}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 91, normalized size = 2.68 \begin {gather*} 3 \, x^{4} - 3 \, x^{3} + 6 \, x^{2} + x e^{5} - \frac {3}{4} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} - \frac {3}{4} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} - \frac {3}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + 15 \, x + \frac {2 \, e^{5}}{x} + \frac {5 \, e^{5}}{x^{2}} - 18 \, e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.05, size = 63, normalized size = 1.85 \begin {gather*} x^2\,\left (3\,{\mathrm {e}}^{2\,x}+6\right )-x^3\,\left (3\,{\mathrm {e}}^{2\,x}+3\right )-15\,{\mathrm {e}}^{2\,x}+x\,\left ({\mathrm {e}}^5-6\,{\mathrm {e}}^{2\,x}+15\right )+\frac {5\,{\mathrm {e}}^5+2\,x\,{\mathrm {e}}^5}{x^2}+3\,x^4 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.22, size = 56, normalized size = 1.65 \begin {gather*} 3 x^{4} - 3 x^{3} + 6 x^{2} + x \left (15 + e^{5}\right ) + \left (- 3 x^{3} + 3 x^{2} - 6 x - 15\right ) e^{2 x} + \frac {2 x e^{5} + 5 e^{5}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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