Optimal. Leaf size=32 \[ \left (-\frac {(5+e)^2}{25 x^2}-x+\frac {5+\frac {x}{e^5}}{4 x}\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(32)=64\).
time = 0.05, antiderivative size = 88, normalized size of antiderivative = 2.75, number of steps
used = 5, number of rules used = 3, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6, 12, 14}
\begin {gather*} \frac {(5+e)^4}{625 x^4}-\frac {(5+e)^2}{10 x^3}+x^2-\frac {200+80 e+8 e^2-625 e^5}{400 e^5 x^2}-\frac {x}{2 e^5}+\frac {125+400 e^5+160 e^6+16 e^7}{200 e^5 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 14
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {200 e^2 x^2+(5000+2000 e) x^2-3125 x^3-2500 x^5+e^5 \left (-20000-640 e^3-32 e^4+37500 x-15625 x^2-10000 x^3+10000 x^6+e \left (-16000+15000 x-4000 x^3\right )+e^2 \left (-4800+1500 x-400 x^3\right )\right )}{5000 e^5 x^5} \, dx\\ &=\int \frac {\left (5000+2000 e+200 e^2\right ) x^2-3125 x^3-2500 x^5+e^5 \left (-20000-640 e^3-32 e^4+37500 x-15625 x^2-10000 x^3+10000 x^6+e \left (-16000+15000 x-4000 x^3\right )+e^2 \left (-4800+1500 x-400 x^3\right )\right )}{5000 e^5 x^5} \, dx\\ &=\frac {\int \frac {\left (5000+2000 e+200 e^2\right ) x^2-3125 x^3-2500 x^5+e^5 \left (-20000-640 e^3-32 e^4+37500 x-15625 x^2-10000 x^3+10000 x^6+e \left (-16000+15000 x-4000 x^3\right )+e^2 \left (-4800+1500 x-400 x^3\right )\right )}{x^5} \, dx}{5000 e^5}\\ &=\frac {\int \left (-2500-\frac {32 e^5 (5+e)^4}{x^5}+\frac {1500 e^5 (5+e)^2}{x^4}-\frac {25 \left (-200-80 e-8 e^2+625 e^5\right )}{x^3}-\frac {25 \left (125+400 e^5+160 e^6+16 e^7\right )}{x^2}+10000 e^5 x\right ) \, dx}{5000 e^5}\\ &=\frac {(5+e)^4}{625 x^4}-\frac {(5+e)^2}{10 x^3}-\frac {200+80 e+8 e^2-625 e^5}{400 e^5 x^2}+\frac {125+400 e^5+160 e^6+16 e^7}{200 e^5 x}-\frac {x}{2 e^5}+x^2\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(32)=64\).
time = 0.02, size = 111, normalized size = 3.47 \begin {gather*} -\frac {-\frac {8 \left (625 e^5+500 e^6+150 e^7+20 e^8+e^9\right )}{x^4}+\frac {500 \left (25 e^5+10 e^6+e^7\right )}{x^3}-\frac {25 \left (-200-80 e-8 e^2+625 e^5\right )}{2 x^2}-\frac {25 \left (125+400 e^5+160 e^6+16 e^7\right )}{x}+2500 x-5000 e^5 x^2}{5000 e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs.
\(2(34)=68\).
time = 0.18, size = 100, normalized size = 3.12
method | result | size |
risch | \(x^{2}-\frac {x \,{\mathrm e}^{-5}}{2}+\frac {{\mathrm e}^{-5} \left (\left (10000 \,{\mathrm e}^{5}+400 \,{\mathrm e}^{7}+4000 \,{\mathrm e}^{6}+3125\right ) x^{3}+\left (\frac {15625 \,{\mathrm e}^{5}}{2}-100 \,{\mathrm e}^{2}-1000 \,{\mathrm e}-2500\right ) x^{2}+\left (-500 \,{\mathrm e}^{7}-5000 \,{\mathrm e}^{6}-12500 \,{\mathrm e}^{5}\right ) x +8 \,{\mathrm e}^{9}+160 \,{\mathrm e}^{8}+1200 \,{\mathrm e}^{7}+4000 \,{\mathrm e}^{6}+5000 \,{\mathrm e}^{5}\right )}{5000 x^{4}}\) | \(89\) |
default | \(\frac {{\mathrm e}^{-5} \left (-2500 x +5000 x^{2} {\mathrm e}^{5}-\frac {-20000 \,{\mathrm e}^{5}-4800 \,{\mathrm e}^{7}-32 \,{\mathrm e}^{9}-16000 \,{\mathrm e}^{6}-640 \,{\mathrm e}^{8}}{4 x^{4}}-\frac {37500 \,{\mathrm e}^{5}+1500 \,{\mathrm e}^{7}+15000 \,{\mathrm e}^{6}}{3 x^{3}}-\frac {-15625 \,{\mathrm e}^{5}+200 \,{\mathrm e}^{2}+2000 \,{\mathrm e}+5000}{2 x^{2}}-\frac {-10000 \,{\mathrm e}^{5}-400 \,{\mathrm e}^{7}-4000 \,{\mathrm e}^{6}-3125}{x}\right )}{5000}\) | \(100\) |
norman | \(\frac {x^{6}+\left (-\frac {{\mathrm e}^{2}}{10}-{\mathrm e}-\frac {5}{2}\right ) x -\frac {{\mathrm e}^{-5} x^{5}}{2}+\frac {\left (-8 \,{\mathrm e}^{2}+625 \,{\mathrm e}^{5}-80 \,{\mathrm e}-200\right ) {\mathrm e}^{-5} x^{2}}{400}+\frac {\left (16 \,{\mathrm e}^{2} {\mathrm e}^{5}+160 \,{\mathrm e} \,{\mathrm e}^{5}+400 \,{\mathrm e}^{5}+125\right ) {\mathrm e}^{-5} x^{3}}{200}+\frac {{\mathrm e}^{4}}{625}+\frac {4 \,{\mathrm e}^{3}}{125}+\frac {6 \,{\mathrm e}^{2}}{25}+\frac {4 \,{\mathrm e}}{5}+1}{x^{4}}\) | \(109\) |
gosper | \(\frac {\left (10000 x^{6} {\mathrm e}^{5}+800 \,{\mathrm e}^{5} {\mathrm e}^{2} x^{3}+16 \,{\mathrm e}^{4} {\mathrm e}^{5}+8000 \,{\mathrm e}^{5} {\mathrm e} x^{3}-5000 x^{5}+320 \,{\mathrm e}^{3} {\mathrm e}^{5}-1000 \,{\mathrm e}^{5} {\mathrm e}^{2} x +20000 x^{3} {\mathrm e}^{5}-200 x^{2} {\mathrm e}^{2}+2400 \,{\mathrm e}^{2} {\mathrm e}^{5}-10000 x \,{\mathrm e} \,{\mathrm e}^{5}+15625 x^{2} {\mathrm e}^{5}-2000 x^{2} {\mathrm e}+6250 x^{3}+8000 \,{\mathrm e} \,{\mathrm e}^{5}-25000 x \,{\mathrm e}^{5}-5000 x^{2}+10000 \,{\mathrm e}^{5}\right ) {\mathrm e}^{-5}}{10000 x^{4}}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs.
\(2 (30) = 60\).
time = 0.26, size = 92, normalized size = 2.88 \begin {gather*} \frac {1}{10000} \, {\left (10000 \, x^{2} e^{5} - 5000 \, x + \frac {50 \, x^{3} {\left (16 \, e^{7} + 160 \, e^{6} + 400 \, e^{5} + 125\right )} + 25 \, x^{2} {\left (625 \, e^{5} - 8 \, e^{2} - 80 \, e - 200\right )} - 1000 \, x {\left (e^{7} + 10 \, e^{6} + 25 \, e^{5}\right )} + 16 \, e^{9} + 320 \, e^{8} + 2400 \, e^{7} + 8000 \, e^{6} + 10000 \, e^{5}}{x^{4}}\right )} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs.
\(2 (30) = 60\).
time = 0.34, size = 97, normalized size = 3.03 \begin {gather*} -\frac {{\left (5000 \, x^{5} - 6250 \, x^{3} + 200 \, x^{2} e^{2} + 2000 \, x^{2} e + 5000 \, x^{2} - 200 \, {\left (4 \, x^{3} - 5 \, x + 12\right )} e^{7} - 2000 \, {\left (4 \, x^{3} - 5 \, x + 4\right )} e^{6} - 625 \, {\left (16 \, x^{6} + 32 \, x^{3} + 25 \, x^{2} - 40 \, x + 16\right )} e^{5} - 16 \, e^{9} - 320 \, e^{8}\right )} e^{\left (-5\right )}}{10000 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 105 vs.
\(2 (26) = 52\).
time = 2.49, size = 105, normalized size = 3.28 \begin {gather*} \frac {5000 x^{2} e^{5} - 2500 x + \frac {x^{3} \cdot \left (6250 + 800 e^{7} + 20000 e^{5} + 8000 e^{6}\right ) + x^{2} \left (- 2000 e - 5000 - 200 e^{2} + 15625 e^{5}\right ) + x \left (- 10000 e^{6} - 25000 e^{5} - 1000 e^{7}\right ) + 16 e^{9} + 320 e^{8} + 10000 e^{5} + 2400 e^{7} + 8000 e^{6}}{2 x^{4}}}{5000 e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs.
\(2 (30) = 60\).
time = 0.39, size = 107, normalized size = 3.34 \begin {gather*} \frac {1}{10000} \, {\left (10000 \, x^{2} e^{5} - 5000 \, x + \frac {800 \, x^{3} e^{7} + 8000 \, x^{3} e^{6} + 20000 \, x^{3} e^{5} + 6250 \, x^{3} + 15625 \, x^{2} e^{5} - 200 \, x^{2} e^{2} - 2000 \, x^{2} e - 5000 \, x^{2} - 1000 \, x e^{7} - 10000 \, x e^{6} - 25000 \, x e^{5} + 16 \, e^{9} + 320 \, e^{8} + 2400 \, e^{7} + 8000 \, e^{6} + 10000 \, e^{5}}{x^{4}}\right )} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.69, size = 90, normalized size = 2.81 \begin {gather*} x^2-\frac {x\,{\mathrm {e}}^{-5}}{2}+\frac {{\mathrm {e}}^{-5}\,\left (\left (10000\,{\mathrm {e}}^5+4000\,{\mathrm {e}}^6+400\,{\mathrm {e}}^7+3125\right )\,x^3+\left (\frac {15625\,{\mathrm {e}}^5}{2}-100\,{\mathrm {e}}^2-1000\,\mathrm {e}-2500\right )\,x^2+\left (-12500\,{\mathrm {e}}^5-5000\,{\mathrm {e}}^6-500\,{\mathrm {e}}^7\right )\,x+5000\,{\mathrm {e}}^5+4000\,{\mathrm {e}}^6+1200\,{\mathrm {e}}^7+160\,{\mathrm {e}}^8+8\,{\mathrm {e}}^9\right )}{5000\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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