Optimal. Leaf size=28 \[ x-(5+x)^2+\frac {x}{3 x-\left (-2+\frac {x}{e^5}\right )^2} \]
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Rubi [B] time = 0.41, antiderivative size = 67, normalized size of antiderivative = 2.39, number of steps used = 4, number of rules used = 3, integrand size = 140, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {1680, 1814, 1586} \begin {gather*} \frac {e^{10} x}{-x^2+e^5 \left (4+3 e^5\right ) x-4 e^{10}}-\left (x+\frac {1}{4} \left (-8 e^5-6 e^{10}\right )\right )^2-\left (9+4 e^5+3 e^{10}\right ) x \end {gather*}
Antiderivative was successfully verified.
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Rule 1586
Rule 1680
Rule 1814
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {-1620 e^{45} \left (1+\frac {3 e^5}{20}\right )-64 e^5 x^4-16 x^4 (9+2 x)-144 e^{35} (43+6 x)-27 e^{40} (155+6 x)+36 e^{30} \left (-143-32 x+6 x^2\right )+64 e^{15} x \left (1+27 x+6 x^2\right )+24 e^{20} x \left (2+59 x+6 x^2\right )+96 e^{25} \left (1+9 x^2\right )+e^{10} \left (16 x^2-48 x^4\right )}{\left (24 e^{15}+9 e^{20}-4 x^2\right )^2} \, dx,x,\frac {1}{4} \left (-8 e^5-6 e^{10}\right )+x\right )\\ &=\frac {e^{10} x}{-4 e^{10}+e^5 \left (4+3 e^5\right ) x-x^2}-\frac {\operatorname {Subst}\left (\int \frac {18 e^{30} \left (8+3 e^5\right )^2 \left (9+4 e^5+3 e^{10}\right )+36 e^{30} \left (8+3 e^5\right )^2 x-24 e^{15} \left (72+59 e^5+36 e^{10}+9 e^{15}\right ) x^2-48 e^{15} \left (8+3 e^5\right ) x^3}{24 e^{15}+9 e^{20}-4 x^2} \, dx,x,\frac {1}{4} \left (-8 e^5-6 e^{10}\right )+x\right )}{6 e^{15} \left (8+3 e^5\right )}\\ &=\frac {e^{10} x}{-4 e^{10}+e^5 \left (4+3 e^5\right ) x-x^2}-\frac {\operatorname {Subst}\left (\int \left (432 e^{15}+354 e^{20}+216 e^{25}+54 e^{30}+\left (96 e^{15}+36 e^{20}\right ) x\right ) \, dx,x,\frac {1}{4} \left (-8 e^5-6 e^{10}\right )+x\right )}{6 e^{15} \left (8+3 e^5\right )}\\ &=-\left (\left (9+4 e^5+3 e^{10}\right ) x\right )-\left (-\frac {1}{2} e^5 \left (4+3 e^5\right )+x\right )^2+\frac {e^{10} x}{-4 e^{10}+e^5 \left (4+3 e^5\right ) x-x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 34, normalized size = 1.21 \begin {gather*} x \left (-9-x+\frac {e^{10}}{4 e^5 x-x^2+e^{10} (-4+3 x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 62, normalized size = 2.21 \begin {gather*} -\frac {x^{4} + 9 \, x^{3} - {\left (3 \, x^{3} + 23 \, x^{2} - 37 \, x\right )} e^{10} - 4 \, {\left (x^{3} + 9 \, x^{2}\right )} e^{5}}{x^{2} - {\left (3 \, x - 4\right )} e^{10} - 4 \, x e^{5}} \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 \, x^{5} + 9 \, x^{4} + {\left (18 \, x^{3} + 33 \, x^{2} - 184 \, x + 148\right )} e^{20} + 8 \, {\left (6 \, x^{3} + 19 \, x^{2} - 36 \, x\right )} e^{15} - {\left (12 \, x^{4} + 6 \, x^{3} - 215 \, x^{2}\right )} e^{10} - 8 \, {\left (2 \, x^{4} + 9 \, x^{3}\right )} e^{5}}{x^{4} - 8 \, x^{3} e^{5} + {\left (9 \, x^{2} - 24 \, x + 16\right )} e^{20} + 8 \, {\left (3 \, x^{2} - 4 \, x\right )} e^{15} - 6 \, {\left (x^{3} - 4 \, x^{2}\right )} e^{10}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 36, normalized size = 1.29
method | result | size |
risch | \(-x^{2}-9 x +\frac {x \,{\mathrm e}^{10}}{3 x \,{\mathrm e}^{10}-4 \,{\mathrm e}^{10}+4 x \,{\mathrm e}^{5}-x^{2}}\) | \(36\) |
norman | \(\frac {x^{4}+\left (9-3 \,{\mathrm e}^{10}-4 \,{\mathrm e}^{5}\right ) x^{3}+\left (-69 \,{\mathrm e}^{20}-200 \,{\mathrm e}^{15}-107 \,{\mathrm e}^{10}\right ) x +4 \,{\mathrm e}^{15} \left (23 \,{\mathrm e}^{5}+36\right )}{3 x \,{\mathrm e}^{10}-4 \,{\mathrm e}^{10}+4 x \,{\mathrm e}^{5}-x^{2}}\) | \(81\) |
gosper | \(-\frac {69 x \,{\mathrm e}^{20}+3 x^{3} {\mathrm e}^{10}-92 \,{\mathrm e}^{20}+200 x \,{\mathrm e}^{15}+4 x^{3} {\mathrm e}^{5}-x^{4}-144 \,{\mathrm e}^{15}+107 x \,{\mathrm e}^{10}-9 x^{3}}{3 x \,{\mathrm e}^{10}-4 \,{\mathrm e}^{10}+4 x \,{\mathrm e}^{5}-x^{2}}\) | \(89\) |
default | \(-x^{2}-9 x -\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+\left (-8 \,{\mathrm e}^{5}-6 \,{\mathrm e}^{10}\right ) \textit {\_Z}^{3}+\left (24 \,{\mathrm e}^{10}+9 \,{\mathrm e}^{20}+24 \,{\mathrm e}^{15}\right ) \textit {\_Z}^{2}+\left (-24 \,{\mathrm e}^{20}-32 \,{\mathrm e}^{15}\right ) \textit {\_Z} +16 \,{\mathrm e}^{20}\right )}{\sum }\frac {\left ({\mathrm e}^{10} \textit {\_R}^{2}-4 \,{\mathrm e}^{20}\right ) \ln \left (x -\textit {\_R} \right )}{12 \textit {\_R}^{2} {\mathrm e}^{5}+9 \,{\mathrm e}^{10} \textit {\_R}^{2}-2 \textit {\_R}^{3}-9 \textit {\_R} \,{\mathrm e}^{20}-24 \,{\mathrm e}^{15} \textit {\_R} -24 \,{\mathrm e}^{10} \textit {\_R} +12 \,{\mathrm e}^{20}+16 \,{\mathrm e}^{15}}\right )}{2}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 36, normalized size = 1.29 \begin {gather*} -x^{2} - 9 \, x - \frac {x e^{10}}{x^{2} - x {\left (3 \, e^{10} + 4 \, e^{5}\right )} + 4 \, e^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.98, size = 55, normalized size = 1.96 \begin {gather*} x\,\left (16\,{\mathrm {e}}^5+12\,{\mathrm {e}}^{10}-4\,{\mathrm {e}}^5\,\left (3\,{\mathrm {e}}^5+4\right )-9\right )-x^2-\frac {x\,{\mathrm {e}}^{10}}{x^2+\left (-4\,{\mathrm {e}}^5-3\,{\mathrm {e}}^{10}\right )\,x+4\,{\mathrm {e}}^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 34, normalized size = 1.21 \begin {gather*} - x^{2} - 9 x - \frac {x e^{10}}{x^{2} + x \left (- 3 e^{10} - 4 e^{5}\right ) + 4 e^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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