Optimal. Leaf size=33 \[ x^2 \left (-e+4 x+\left (\frac {2 x}{e^3-x}+\frac {1+x}{5}\right )^2\right ) \]
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Rubi [B] time = 0.14, antiderivative size = 84, normalized size of antiderivative = 2.55, number of steps used = 3, number of rules used = 2, integrand size = 124, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6, 2074} \begin {gather*} \frac {x^4}{25}+\frac {82 x^3}{25}+\frac {1}{25} \left (81-25 e-20 e^3\right ) x^2+\frac {4}{5} e^3 \left (9-e^3\right ) x-\frac {4 e^9 \left (19-e^3\right )}{5 \left (e^3-x\right )}+\frac {4 e^{12}}{\left (e^3-x\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(-162+50 e) x^4-246 x^5-4 x^6+e^9 \left (2 x-50 e x+306 x^2+4 x^3\right )+e^6 \left (54 x^2+150 e x^2-838 x^3-12 x^4\right )+e^3 \left (306 x^3-150 e x^3+778 x^4+12 x^5\right )}{25 e^9-75 e^6 x+75 e^3 x^2-25 x^3} \, dx\\ &=\int \left (-\frac {4}{5} e^3 \left (-9+e^3\right )+\frac {8 e^{12}}{\left (e^3-x\right )^3}+\frac {4 e^9 \left (-19+e^3\right )}{5 \left (e^3-x\right )^2}-\frac {2}{25} \left (-81+25 e+20 e^3\right ) x+\frac {246 x^2}{25}+\frac {4 x^3}{25}\right ) \, dx\\ &=\frac {4 e^{12}}{\left (e^3-x\right )^2}-\frac {4 e^9 \left (19-e^3\right )}{5 \left (e^3-x\right )}+\frac {4}{5} e^3 \left (9-e^3\right ) x+\frac {1}{25} \left (81-25 e-20 e^3\right ) x^2+\frac {82 x^3}{25}+\frac {x^4}{25}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.08, size = 94, normalized size = 2.85 \begin {gather*} \frac {1}{25} \left (25 e^7-e^{12} \left (1-\frac {100}{\left (e^3-x\right )^2}-\frac {20}{e^3-x}\right )-e^9 \left (42+\frac {380}{e^3-x}\right )-20 e^3 (-9+x) x-25 e x^2-e^6 (261+20 x)+x^2 \left (81+82 x+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 107, normalized size = 3.24 \begin {gather*} \frac {x^{6} + 82 \, x^{5} - 25 \, x^{4} e + 81 \, x^{4} + 50 \, x^{3} e^{4} - 25 \, x^{2} e^{7} - 40 \, {\left (x + 7\right )} e^{12} + 20 \, {\left (x^{2} + 28 \, x\right )} e^{9} + {\left (x^{4} + 102 \, x^{3} - 279 \, x^{2}\right )} e^{6} - 2 \, {\left (x^{5} + 92 \, x^{4} - 9 \, x^{3}\right )} e^{3} + 20 \, e^{15}}{25 \, {\left (x^{2} - 2 \, x e^{3} + e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (2 \, x^{6} + 123 \, x^{5} - 25 \, x^{4} e + 81 \, x^{4} - {\left (2 \, x^{3} + 153 \, x^{2} - 25 \, x e + x\right )} e^{9} + {\left (6 \, x^{4} + 419 \, x^{3} - 75 \, x^{2} e - 27 \, x^{2}\right )} e^{6} - {\left (6 \, x^{5} + 389 \, x^{4} - 75 \, x^{3} e + 153 \, x^{3}\right )} e^{3}\right )}}{25 \, {\left (x^{3} - 3 \, x^{2} e^{3} + 3 \, x e^{6} - e^{9}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 76, normalized size = 2.30
method | result | size |
risch | \(-\frac {4 x \,{\mathrm e}^{6}}{5}-\frac {4 x^{2} {\mathrm e}^{3}}{5}+\frac {36 x \,{\mathrm e}^{3}}{5}+\frac {x^{4}}{25}-x^{2} {\mathrm e}+\frac {82 x^{3}}{25}+\frac {81 x^{2}}{25}+\frac {\frac {\left (-20 \,{\mathrm e}^{12}+380 \,{\mathrm e}^{9}\right ) x}{25}+\frac {4 \,{\mathrm e}^{15}}{5}-\frac {56 \,{\mathrm e}^{12}}{5}}{{\mathrm e}^{6}-2 x \,{\mathrm e}^{3}+x^{2}}\) | \(76\) |
norman | \(\frac {\left (\frac {82}{25}-\frac {2 \,{\mathrm e}^{3}}{25}\right ) x^{5}+\left (2 \,{\mathrm e} \,{\mathrm e}^{3}+\frac {102 \,{\mathrm e}^{6}}{25}+\frac {18 \,{\mathrm e}^{3}}{25}\right ) x^{3}+\left (-2 \,{\mathrm e} \,{\mathrm e}^{9}+\frac {2 \,{\mathrm e}^{9}}{25}\right ) x +\left (\frac {{\mathrm e}^{6}}{25}-{\mathrm e}-\frac {184 \,{\mathrm e}^{3}}{25}+\frac {81}{25}\right ) x^{4}+\frac {x^{6}}{25}+\frac {{\mathrm e}^{12} \left (25 \,{\mathrm e}-1\right )}{25}}{\left (-x +{\mathrm e}^{3}\right )^{2}}\) | \(96\) |
gosper | \(\frac {x^{4} {\mathrm e}^{6}-2 x^{5} {\mathrm e}^{3}+x^{6}+25 \,{\mathrm e} \,{\mathrm e}^{12}-50 \,{\mathrm e} \,{\mathrm e}^{9} x +50 \,{\mathrm e} \,{\mathrm e}^{3} x^{3}-25 x^{4} {\mathrm e}+102 x^{3} {\mathrm e}^{6}-184 x^{4} {\mathrm e}^{3}+82 x^{5}-{\mathrm e}^{12}+2 x \,{\mathrm e}^{9}+18 x^{3} {\mathrm e}^{3}+81 x^{4}}{25 \,{\mathrm e}^{6}-50 x \,{\mathrm e}^{3}+25 x^{2}}\) | \(116\) |
default | \(\frac {x^{4}}{25}-6 x \,{\mathrm e} \,{\mathrm e}^{3}-x^{2} {\mathrm e}-\frac {4 x \,{\mathrm e}^{6}}{5}-\frac {4 x^{2} {\mathrm e}^{3}}{5}+\frac {82 x^{3}}{25}+6 x \,{\mathrm e}^{4}+\frac {36 x \,{\mathrm e}^{3}}{5}+\frac {81 x^{2}}{25}-\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (-3 \textit {\_Z}^{2} {\mathrm e}^{3}+\textit {\_Z}^{3}+3 \textit {\_Z} \,{\mathrm e}^{6}-{\mathrm e}^{9}\right )}{\sum }\frac {\left (-9 \,{\mathrm e}^{9} {\mathrm e}^{3}-\textit {\_R} \,{\mathrm e}^{12}+{\mathrm e}^{9} {\mathrm e}^{6}+19 \textit {\_R} \,{\mathrm e}^{9}\right ) \ln \left (x -\textit {\_R} \right )}{-2 \textit {\_R} \,{\mathrm e}^{3}+\textit {\_R}^{2}+{\mathrm e}^{6}}\right )}{15}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 69, normalized size = 2.09 \begin {gather*} \frac {1}{25} \, x^{4} + \frac {82}{25} \, x^{3} - \frac {1}{25} \, x^{2} {\left (20 \, e^{3} + 25 \, e - 81\right )} - \frac {4}{5} \, x {\left (e^{6} - 9 \, e^{3}\right )} - \frac {4 \, {\left (x {\left (e^{12} - 19 \, e^{9}\right )} - e^{15} + 14 \, e^{12}\right )}}{5 \, {\left (x^{2} - 2 \, x e^{3} + e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 106, normalized size = 3.21 \begin {gather*} \frac {4\,{\mathrm {e}}^{15}-56\,{\mathrm {e}}^{12}+x\,\left (76\,{\mathrm {e}}^9-4\,{\mathrm {e}}^{12}\right )}{5\,x^2-10\,{\mathrm {e}}^3\,x+5\,{\mathrm {e}}^6}-x^2\,\left (\mathrm {e}+\frac {4\,{\mathrm {e}}^3}{5}-\frac {81}{25}\right )+\frac {82\,x^3}{25}+\frac {x^4}{25}-x\,\left (\frac {738\,{\mathrm {e}}^6}{25}-\frac {4\,{\mathrm {e}}^9}{25}-\frac {2\,{\mathrm {e}}^3\,\left (75\,\mathrm {e}+419\,{\mathrm {e}}^3-2\,{\mathrm {e}}^6-153\right )}{25}+3\,{\mathrm {e}}^3\,\left (2\,\mathrm {e}+\frac {8\,{\mathrm {e}}^3}{5}-\frac {162}{25}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.48, size = 82, normalized size = 2.48 \begin {gather*} \frac {x^{4}}{25} + \frac {82 x^{3}}{25} + x^{2} \left (- \frac {4 e^{3}}{5} - e + \frac {81}{25}\right ) + x \left (- \frac {4 e^{6}}{5} + \frac {36 e^{3}}{5}\right ) + \frac {x \left (- 4 e^{12} + 76 e^{9}\right ) - 56 e^{12} + 4 e^{15}}{5 x^{2} - 10 x e^{3} + 5 e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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