Optimal. Leaf size=34 \[ \frac {\left (2 x+\frac {\left (e^{16}+\frac {x}{5}\right ) \left (4+\frac {x^2}{25}\right )^2}{x}\right )^2}{x^2} \]
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Rubi [B] time = 0.11, antiderivative size = 126, normalized size of antiderivative = 3.71, number of steps used = 3, number of rules used = 2, integrand size = 102, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {12, 14} \begin {gather*} \frac {x^6}{9765625}+\frac {2 e^{16} x^5}{1953125}+\frac {\left (16+e^{32}\right ) x^4}{390625}+\frac {256 e^{32}}{x^4}+\frac {4 \left (25+8 e^{16}\right ) x^3}{78125}+\frac {512 e^{16}}{5 x^3}+\frac {4 \left (24+25 e^{16}+4 e^{32}\right ) x^2}{15625}+\frac {64 \left (4+25 e^{16}+4 e^{32}\right )}{25 x^2}+\frac {32 \left (25+6 e^{16}\right ) x}{3125}+\frac {64 \left (25+8 e^{16}\right )}{125 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-200000000 x^2-125000000 x^3+2500000 x^5+120000 x^6+37500 x^7+1600 x^8+6 x^{10}+e^{32} \left (-10000000000-200000000 x^2+20000 x^6+100 x^8\right )+e^{16} \left (-3000000000 x-1250000000 x^2-40000000 x^3+600000 x^5+125000 x^6+12000 x^7+50 x^9\right )}{x^5} \, dx}{9765625}\\ &=\frac {\int \left (100000 \left (25+6 e^{16}\right )-\frac {10000000000 e^{32}}{x^5}-\frac {3000000000 e^{16}}{x^4}-\frac {50000000 \left (4+25 e^{16}+4 e^{32}\right )}{x^3}-\frac {5000000 \left (25+8 e^{16}\right )}{x^2}+5000 \left (24+25 e^{16}+4 e^{32}\right ) x+1500 \left (25+8 e^{16}\right ) x^2+100 \left (16+e^{32}\right ) x^3+50 e^{16} x^4+6 x^5\right ) \, dx}{9765625}\\ &=\frac {256 e^{32}}{x^4}+\frac {512 e^{16}}{5 x^3}+\frac {64 \left (4+25 e^{16}+4 e^{32}\right )}{25 x^2}+\frac {64 \left (25+8 e^{16}\right )}{125 x}+\frac {32 \left (25+6 e^{16}\right ) x}{3125}+\frac {4 \left (24+25 e^{16}+4 e^{32}\right ) x^2}{15625}+\frac {4 \left (25+8 e^{16}\right ) x^3}{78125}+\frac {\left (16+e^{32}\right ) x^4}{390625}+\frac {2 e^{16} x^5}{1953125}+\frac {x^6}{9765625}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.02, size = 152, normalized size = 4.47 \begin {gather*} \frac {256 e^{32}}{x^4}+\frac {512 e^{16}}{5 x^3}+\frac {256}{25 x^2}+\frac {64 e^{16}}{x^2}+\frac {256 e^{32}}{25 x^2}+\frac {64}{5 x}+\frac {512 e^{16}}{125 x}+\frac {32 x}{125}+\frac {192 e^{16} x}{3125}+\frac {96 x^2}{15625}+\frac {4 e^{16} x^2}{625}+\frac {16 e^{32} x^2}{15625}+\frac {4 x^3}{3125}+\frac {32 e^{16} x^3}{78125}+\frac {16 x^4}{390625}+\frac {e^{32} x^4}{390625}+\frac {2 e^{16} x^5}{1953125}+\frac {x^6}{9765625} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 94, normalized size = 2.76 \begin {gather*} \frac {x^{10} + 400 \, x^{8} + 12500 \, x^{7} + 60000 \, x^{6} + 2500000 \, x^{5} + 125000000 \, x^{3} + 100000000 \, x^{2} + 25 \, {\left (x^{8} + 400 \, x^{6} + 4000000 \, x^{2} + 100000000\right )} e^{32} + 10 \, {\left (x^{9} + 400 \, x^{7} + 6250 \, x^{6} + 60000 \, x^{5} + 4000000 \, x^{3} + 62500000 \, x^{2} + 100000000 \, x\right )} e^{16}}{9765625 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 110, normalized size = 3.24 \begin {gather*} \frac {1}{9765625} \, x^{6} + \frac {2}{1953125} \, x^{5} e^{16} + \frac {1}{390625} \, x^{4} e^{32} + \frac {16}{390625} \, x^{4} + \frac {32}{78125} \, x^{3} e^{16} + \frac {4}{3125} \, x^{3} + \frac {16}{15625} \, x^{2} e^{32} + \frac {4}{625} \, x^{2} e^{16} + \frac {96}{15625} \, x^{2} + \frac {192}{3125} \, x e^{16} + \frac {32}{125} \, x + \frac {64 \, {\left (8 \, x^{3} e^{16} + 25 \, x^{3} + 20 \, x^{2} e^{32} + 125 \, x^{2} e^{16} + 20 \, x^{2} + 200 \, x e^{16} + 500 \, e^{32}\right )}}{125 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 104, normalized size = 3.06
method | result | size |
risch | \(\frac {x^{4} {\mathrm e}^{32}}{390625}+\frac {16 \,{\mathrm e}^{32} x^{2}}{15625}+\frac {2 x^{5} {\mathrm e}^{16}}{1953125}+\frac {32 x^{3} {\mathrm e}^{16}}{78125}+\frac {4 x^{2} {\mathrm e}^{16}}{625}+\frac {x^{6}}{9765625}+\frac {192 x \,{\mathrm e}^{16}}{3125}+\frac {16 x^{4}}{390625}+\frac {4 x^{3}}{3125}+\frac {96 x^{2}}{15625}+\frac {32 x}{125}+\frac {\left (40000000 \,{\mathrm e}^{16}+125000000\right ) x^{3}+\left (100000000 \,{\mathrm e}^{32}+625000000 \,{\mathrm e}^{16}+100000000\right ) x^{2}+1000000000 x \,{\mathrm e}^{16}+2500000000 \,{\mathrm e}^{32}}{9765625 x^{4}}\) | \(104\) |
default | \(\frac {2 x^{5} {\mathrm e}^{16}}{1953125}+\frac {x^{6}}{9765625}+\frac {x^{4} {\mathrm e}^{32}}{390625}+\frac {32 x^{3} {\mathrm e}^{16}}{78125}+\frac {16 x^{4}}{390625}+\frac {16 \,{\mathrm e}^{32} x^{2}}{15625}+\frac {4 x^{2} {\mathrm e}^{16}}{625}+\frac {4 x^{3}}{3125}+\frac {192 x \,{\mathrm e}^{16}}{3125}+\frac {96 x^{2}}{15625}+\frac {32 x}{125}+\frac {512 \,{\mathrm e}^{16}}{5 x^{3}}-\frac {-100000000 \,{\mathrm e}^{32}-625000000 \,{\mathrm e}^{16}-100000000}{9765625 x^{2}}-\frac {2 \left (-20000000 \,{\mathrm e}^{16}-62500000\right )}{9765625 x}+\frac {256 \,{\mathrm e}^{32}}{x^{4}}\) | \(105\) |
norman | \(\frac {\left (\frac {32 \,{\mathrm e}^{16}}{78125}+\frac {4}{3125}\right ) x^{7}+\left (\frac {192 \,{\mathrm e}^{16}}{3125}+\frac {32}{125}\right ) x^{5}+\left (\frac {512 \,{\mathrm e}^{16}}{125}+\frac {64}{5}\right ) x^{3}+\left (\frac {{\mathrm e}^{32}}{390625}+\frac {16}{390625}\right ) x^{8}+\left (\frac {16 \,{\mathrm e}^{32}}{15625}+\frac {4 \,{\mathrm e}^{16}}{625}+\frac {96}{15625}\right ) x^{6}+\left (\frac {256 \,{\mathrm e}^{32}}{25}+64 \,{\mathrm e}^{16}+\frac {256}{25}\right ) x^{2}+\frac {x^{10}}{9765625}+256 \,{\mathrm e}^{32}+\frac {512 x \,{\mathrm e}^{16}}{5}+\frac {2 x^{9} {\mathrm e}^{16}}{1953125}}{x^{4}}\) | \(117\) |
gosper | \(\frac {25 x^{8} {\mathrm e}^{32}+10000 \,{\mathrm e}^{32} x^{6}+10 x^{9} {\mathrm e}^{16}+4000 \,{\mathrm e}^{16} x^{7}+100000000 \,{\mathrm e}^{32} x^{2}+62500 \,{\mathrm e}^{16} x^{6}+x^{10}+600000 x^{5} {\mathrm e}^{16}+2500000000 \,{\mathrm e}^{32}+400 x^{8}+40000000 x^{3} {\mathrm e}^{16}+12500 x^{7}+625000000 x^{2} {\mathrm e}^{16}+60000 x^{6}+1000000000 x \,{\mathrm e}^{16}+2500000 x^{5}+125000000 x^{3}+100000000 x^{2}}{9765625 x^{4}}\) | \(134\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 97, normalized size = 2.85 \begin {gather*} \frac {1}{9765625} \, x^{6} + \frac {2}{1953125} \, x^{5} e^{16} + \frac {1}{390625} \, x^{4} {\left (e^{32} + 16\right )} + \frac {4}{78125} \, x^{3} {\left (8 \, e^{16} + 25\right )} + \frac {4}{15625} \, x^{2} {\left (4 \, e^{32} + 25 \, e^{16} + 24\right )} + \frac {32}{3125} \, x {\left (6 \, e^{16} + 25\right )} + \frac {64 \, {\left (x^{3} {\left (8 \, e^{16} + 25\right )} + 5 \, x^{2} {\left (4 \, e^{32} + 25 \, e^{16} + 4\right )} + 200 \, x e^{16} + 500 \, e^{32}\right )}}{125 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.32, size = 94, normalized size = 2.76 \begin {gather*} \frac {\left (512\,{\mathrm {e}}^{16}+1600\right )\,x^3+\left (8000\,{\mathrm {e}}^{16}+1280\,{\mathrm {e}}^{32}+1280\right )\,x^2+12800\,{\mathrm {e}}^{16}\,x+32000\,{\mathrm {e}}^{32}}{125\,x^4}+x^2\,\left (\frac {4\,{\mathrm {e}}^{16}}{625}+\frac {16\,{\mathrm {e}}^{32}}{15625}+\frac {96}{15625}\right )+x^3\,\left (\frac {32\,{\mathrm {e}}^{16}}{78125}+\frac {4}{3125}\right )+x^4\,\left (\frac {{\mathrm {e}}^{32}}{390625}+\frac {16}{390625}\right )+\frac {2\,x^5\,{\mathrm {e}}^{16}}{1953125}+\frac {x^6}{9765625}+x\,\left (\frac {192\,{\mathrm {e}}^{16}}{3125}+\frac {32}{125}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.78, size = 107, normalized size = 3.15 \begin {gather*} \frac {x^{6}}{9765625} + \frac {2 x^{5} e^{16}}{1953125} + \frac {x^{4} \left (400 + 25 e^{32}\right )}{9765625} + \frac {x^{3} \left (12500 + 4000 e^{16}\right )}{9765625} + \frac {x^{2} \left (60000 + 62500 e^{16} + 10000 e^{32}\right )}{9765625} + \frac {x \left (2500000 + 600000 e^{16}\right )}{9765625} + \frac {x^{3} \left (125000000 + 40000000 e^{16}\right ) + x^{2} \left (100000000 + 625000000 e^{16} + 100000000 e^{32}\right ) + 1000000000 x e^{16} + 2500000000 e^{32}}{9765625 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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