Optimal. Leaf size=20 \[ e^3 \left (3+4 e^x-\frac {3 e}{x}+x\right )^2 \]
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Rubi [B] time = 0.16, antiderivative size = 56, normalized size of antiderivative = 2.80, number of steps used = 12, number of rules used = 7, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.101, Rules used = {14, 2194, 1590, 2199, 2177, 2178, 2176} \begin {gather*} \frac {e^3 \left (-x^2-3 x+3 e\right )^2}{x^2}+24 e^{x+3}+16 e^{2 x+3}+8 e^{x+3} x-\frac {24 e^{x+4}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1590
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (32 e^{3+2 x}-\frac {2 e^3 \left (3 e-3 x-x^2\right ) \left (3 e+x^2\right )}{x^3}-\frac {8 e^{3+x} \left (-3 e+3 e x-4 x^2-x^3\right )}{x^2}\right ) \, dx\\ &=-\left (8 \int \frac {e^{3+x} \left (-3 e+3 e x-4 x^2-x^3\right )}{x^2} \, dx\right )+32 \int e^{3+2 x} \, dx-\left (2 e^3\right ) \int \frac {\left (3 e-3 x-x^2\right ) \left (3 e+x^2\right )}{x^3} \, dx\\ &=16 e^{3+2 x}+\frac {e^3 \left (3 e-3 x-x^2\right )^2}{x^2}-8 \int \left (-4 e^{3+x}-\frac {3 e^{4+x}}{x^2}+\frac {3 e^{4+x}}{x}-e^{3+x} x\right ) \, dx\\ &=16 e^{3+2 x}+\frac {e^3 \left (3 e-3 x-x^2\right )^2}{x^2}+8 \int e^{3+x} x \, dx+24 \int \frac {e^{4+x}}{x^2} \, dx-24 \int \frac {e^{4+x}}{x} \, dx+32 \int e^{3+x} \, dx\\ &=32 e^{3+x}+16 e^{3+2 x}-\frac {24 e^{4+x}}{x}+8 e^{3+x} x+\frac {e^3 \left (3 e-3 x-x^2\right )^2}{x^2}-24 e^4 \text {Ei}(x)-8 \int e^{3+x} \, dx+24 \int \frac {e^{4+x}}{x} \, dx\\ &=24 e^{3+x}+16 e^{3+2 x}-\frac {24 e^{4+x}}{x}+8 e^{3+x} x+\frac {e^3 \left (3 e-3 x-x^2\right )^2}{x^2}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.12, size = 54, normalized size = 2.70 \begin {gather*} -2 e^3 \left (-8 e^{2 x}+e^x \left (-12+\frac {12 e}{x}-4 x\right )-\frac {9 e^2}{2 x^2}+\frac {9 e}{x}-3 x-\frac {x^2}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 64, normalized size = 3.20 \begin {gather*} \frac {{\left (16 \, x^{2} e^{\left (2 \, x + 6\right )} - 18 \, x e^{7} + {\left (x^{4} + 6 \, x^{3}\right )} e^{6} - 8 \, {\left (3 \, x e^{4} - {\left (x^{3} + 3 \, x^{2}\right )} e^{3}\right )} e^{\left (x + 3\right )} + 9 \, e^{8}\right )} e^{\left (-3\right )}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 63, normalized size = 3.15 \begin {gather*} \frac {x^{4} e^{3} + 6 \, x^{3} e^{3} + 8 \, x^{3} e^{\left (x + 3\right )} + 16 \, x^{2} e^{\left (2 \, x + 3\right )} + 24 \, x^{2} e^{\left (x + 3\right )} - 18 \, x e^{4} - 24 \, x e^{\left (x + 4\right )} + 9 \, e^{5}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 57, normalized size = 2.85
method | result | size |
risch | \(x^{2} {\mathrm e}^{3}+6 x \,{\mathrm e}^{3}+\frac {9 \,{\mathrm e}^{5}-18 x \,{\mathrm e}^{4}}{x^{2}}+16 \,{\mathrm e}^{2 x +3}-\frac {8 \left (-x^{2}+3 \,{\mathrm e}-3 x \right ) {\mathrm e}^{3+x}}{x}\) | \(57\) |
norman | \(\frac {x^{4} {\mathrm e}^{3}+6 x^{3} {\mathrm e}^{3}+9 \,{\mathrm e}^{2} {\mathrm e}^{3}-18 x \,{\mathrm e} \,{\mathrm e}^{3}+24 x^{2} {\mathrm e}^{3} {\mathrm e}^{x}+16 x^{2} {\mathrm e}^{3} {\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} {\mathrm e}^{3} x^{3}-24 \,{\mathrm e}^{x} {\mathrm e}^{3} x \,{\mathrm e}}{x^{2}}\) | \(72\) |
default | \(x^{2} {\mathrm e}^{3}+16 \,{\mathrm e}^{3} {\mathrm e}^{2 x}+32 \,{\mathrm e}^{x} {\mathrm e}^{3}+\frac {9 \,{\mathrm e}^{2} {\mathrm e}^{3}}{x^{2}}-\frac {18 \,{\mathrm e} \,{\mathrm e}^{3}}{x}+8 \,{\mathrm e}^{3} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+24 \,{\mathrm e} \,{\mathrm e}^{3} \left (-\frac {{\mathrm e}^{x}}{x}-\expIntegralEi \left (1, -x \right )\right )+24 \,{\mathrm e} \,{\mathrm e}^{3} \expIntegralEi \left (1, -x \right )+6 x \,{\mathrm e}^{3}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, size = 68, normalized size = 3.40 \begin {gather*} x^{2} e^{3} - 24 \, {\rm Ei}\relax (x) e^{4} + 6 \, x e^{3} + 8 \, {\left (x e^{3} - e^{3}\right )} e^{x} + 24 \, e^{4} \Gamma \left (-1, -x\right ) - \frac {18 \, e^{4}}{x} + \frac {9 \, e^{5}}{x^{2}} + 16 \, e^{\left (2 \, x + 3\right )} + 32 \, e^{\left (x + 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 56, normalized size = 2.80 \begin {gather*} \frac {9\,{\mathrm {e}}^5-x\,{\mathrm {e}}^3\,\left (24\,{\mathrm {e}}^{x+1}+18\,\mathrm {e}\right )}{x^2}+x^2\,{\mathrm {e}}^3+{\mathrm {e}}^3\,\left (16\,{\mathrm {e}}^{2\,x}+24\,{\mathrm {e}}^x\right )+x\,{\mathrm {e}}^3\,\left (8\,{\mathrm {e}}^x+6\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.23, size = 65, normalized size = 3.25 \begin {gather*} x^{2} e^{3} + 6 x e^{3} + \frac {16 x e^{3} e^{2 x} + \left (8 x^{2} e^{3} + 24 x e^{3} - 24 e^{4}\right ) e^{x}}{x} + \frac {- 18 x e^{4} + 9 e^{5}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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