∫ −5124800+1310720e15x3−131072e20x4+e5(19200+8199680x)+e10(−7680x−4917504x2)_______________________________________ 23011209+15350400x+2560000x2+e5(−36840960x−24568320x2−4096000x3)+e10(22113792x2+14744064x3+2457600x4)+e15(−5898240x3−3932160x4−655360x5)+e20(589824x4+393216x5+65536x6) dx [7981]

Optimal. Leaf size=23 \[ \frac {5+x}{3+x-\frac {3}{64 \left (-5+2 e^5 x\right )^2}} \]

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Rubi [C] Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
time = 104.20, antiderivative size = 13828, normalized size of antiderivative = 601.22, number of steps used = 20, number of rules used = 10, integrand size = 131, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.076, Rules used = {2099, 2126, 2106, 2104, 836, 814, 648, 632, 212, 642} \begin {gather*} \text {Too large to display} \end {gather*}

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {64 \left (-80075-191580 e^5-115128 e^{10}+8 e^5 \left (8015+19182 e^5+11520 e^{10}\right ) x-4 e^{10} \left (3209+7680 e^5+4608 e^{10}\right ) x^2\right )}{\left (4797+320 \left (5-12 e^5\right ) x-256 e^5 \left (5-3 e^5\right ) x^2+256 e^{10} x^3\right )^2}+\frac {512 e^5 \left (5+3 e^5-e^5 x\right )}{4797+320 \left (5-12 e^5\right ) x-256 e^5 \left (5-3 e^5\right ) x^2+256 e^{10} x^3}\right ) \, dx\\ &=64 \int \frac {-80075-191580 e^5-115128 e^{10}+8 e^5 \left (8015+19182 e^5+11520 e^{10}\right ) x-4 e^{10} \left (3209+7680 e^5+4608 e^{10}\right ) x^2}{\left (4797+320 \left (5-12 e^5\right ) x-256 e^5 \left (5-3 e^5\right ) x^2+256 e^{10} x^3\right )^2} \, dx+\left (512 e^5\right ) \int \frac {5+3 e^5-e^5 x}{4797+320 \left (5-12 e^5\right ) x-256 e^5 \left (5-3 e^5\right ) x^2+256 e^{10} x^3} \, dx\\ &=\frac {3209+7680 e^5+4608 e^{10}}{3 \left (4797+320 \left (5-12 e^5\right ) x-256 e^5 \left (5-3 e^5\right ) x^2+256 e^{10} x^3\right )}+\frac {\int \frac {-2048 e^{10} \left (20000+71910 e^5+86373 e^{10}+34560 e^{15}\right )+2048 e^{15} \left (8000+28773 e^5+34560 e^{10}+13824 e^{15}\right ) x}{\left (4797+320 \left (5-12 e^5\right ) x-256 e^5 \left (5-3 e^5\right ) x^2+256 e^{10} x^3\right )^2} \, dx}{12 e^{10}}+\left (512 e^5\right ) \text {Subst}\left (\int \frac {\frac {-256 e^{10} \left (5-3 e^5\right )+768 e^{10} \left (5+3 e^5\right )}{768 e^{10}}-e^5 x}{\frac {1}{27} \left (28719+\frac {8000}{e^5}+34560 e^5+13824 e^{10}\right )-\frac {64}{3} \left (5+6 e^5\right )^2 x+256 e^{10} x^3} \, dx,x,-\frac {5-3 e^5}{3 e^5}+x\right )\\ &=\frac {3209+7680 e^5+4608 e^{10}}{3 \left (4797+320 \left (5-12 e^5\right ) x-256 e^5 \left (5-3 e^5\right ) x^2+256 e^{10} x^3\right )}+\frac {\text {Subst}\left (\int \frac {\frac {524288 e^{20} \left (5-3 e^5\right ) \left (8000+28773 e^5+34560 e^{10}+13824 e^{15}\right )-1572864 e^{20} \left (20000+71910 e^5+86373 e^{10}+34560 e^{15}\right )}{768 e^{10}}+2048 e^{15} \left (8000+28773 e^5+34560 e^{10}+13824 e^{15}\right ) x}{\left (\frac {1}{27} \left (28719+\frac {8000}{e^5}+34560 e^5+13824 e^{10}\right )-\frac {64}{3} \left (5+6 e^5\right )^2 x+256 e^{10} x^3\right )^2} \, dx,x,-\frac {5-3 e^5}{3 e^5}+x\right )}{12 e^{10}}+\left (33554432 e^{25}\right ) \text {Subst}\left (\int \frac {\frac {-256 e^{10} \left (5-3 e^5\right )+768 e^{10} \left (5+3 e^5\right )}{768 e^{10}}-e^5 x}{\left (\frac {32 e^5 \left (16 \left (5+6 e^5\right )^2+\left (8000+28719 e^5+34560 e^{10}+13824 e^{15}-9 i e^{5/2} \sqrt {16000+57519 e^5+69120 e^{10}+27648 e^{15}}\right )^{2/3}\right )}{3 \sqrt [3]{8000+28719 e^5+34560 e^{10}+13824 e^{15}-9 i e^{5/2} \sqrt {16000+57519 e^5+69120 e^{10}+27648 e^{15}}}}+256 e^{10} x\right ) \left (-\frac {1024}{9} e^{10} \left (16 \left (5+6 e^5\right )^2-\frac {256 \left (5+6 e^5\right )^4}{\left (8000+28719 e^5+34560 e^{10}+13824 e^{15}-9 i e^{5/2} \sqrt {16000+57519 e^5+69120 e^{10}+27648 e^{15}}\right )^{2/3}}-\left (8000+28719 e^5+34560 e^{10}+13824 e^{15}-9 i e^{5/2} \sqrt {16000+57519 e^5+69120 e^{10}+27648 e^{15}}\right )^{2/3}\right )-\frac {8192 e^{15} \left (400+960 e^5+576 e^{10}+\left (8000+28719 e^5+34560 e^{10}+13824 e^{15}-9 i e^{5/2} \sqrt {16000+57519 e^5+69120 e^{10}+27648 e^{15}}\right )^{2/3}\right ) x}{3 \sqrt [3]{8000+28719 e^5+34560 e^{10}+13824 e^{15}-9 i e^{5/2} \sqrt {16000+57519 e^5+69120 e^{10}+27648 e^{15}}}}+65536 e^{20} x^2\right )} \, dx,x,-\frac {5-3 e^5}{3 e^5}+x\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(23)=46\).
time = 0.05, size = 55, normalized size = 2.39 \begin {gather*} -\frac {-3203+2560 e^5 x-512 e^{10} x^2}{4797+1600 x-3840 e^5 x-1280 e^5 x^2+768 e^{10} x^2+256 e^{10} x^3} \end {gather*}

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.25, size = 220, normalized size = 9.57


method	result	size

risch	\(\frac {2 \,{\mathrm e}^{10} x^{2}-10 x \,{\mathrm e}^{5}+\frac {3203}{256}}{x^{3} {\mathrm e}^{10}+3 \,{\mathrm e}^{10} x^{2}-5 x^{2} {\mathrm e}^{5}-15 x \,{\mathrm e}^{5}+\frac {25 x}{4}+\frac {4797}{256}}\)	\(48\)
gosper	\(\frac {512 \,{\mathrm e}^{10} x^{2}-2560 x \,{\mathrm e}^{5}+3203}{256 x^{3} {\mathrm e}^{10}+768 \,{\mathrm e}^{10} x^{2}-1280 x^{2} {\mathrm e}^{5}-3840 x \,{\mathrm e}^{5}+1600 x +4797}\)	\(55\)
norman	\(\frac {-\frac {819968 x^{3} {\mathrm e}^{10}}{4797}+\left (-\frac {5124800}{4797}+\frac {6400 \,{\mathrm e}^{5}}{1599}\right ) x +\left (-\frac {1280 \,{\mathrm e}^{10}}{1599}+\frac {4099840 \,{\mathrm e}^{5}}{4797}\right ) x^{2}}{256 x^{3} {\mathrm e}^{10}+768 \,{\mathrm e}^{10} x^{2}-1280 x^{2} {\mathrm e}^{5}-3840 x \,{\mathrm e}^{5}+1600 x +4797}\)	\(72\)
default	\(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (23011209+65536 \,{\mathrm e}^{20} \textit {\_Z}^{6}-\left (-393216 \,{\mathrm e}^{20}+655360 \,{\mathrm e}^{15}\right ) \textit {\_Z}^{5}-\left (-2457600 \,{\mathrm e}^{10}-589824 \,{\mathrm e}^{20}+3932160 \,{\mathrm e}^{15}\right ) \textit {\_Z}^{4}-\left (4096000 \,{\mathrm e}^{5}-14744064 \,{\mathrm e}^{10}+5898240 \,{\mathrm e}^{15}\right ) \textit {\_Z}^{3}-\left (24568320 \,{\mathrm e}^{5}-22113792 \,{\mathrm e}^{10}-2560000\right ) \textit {\_Z}^{2}-\left (36840960 \,{\mathrm e}^{5}-15350400\right ) \textit {\_Z} \right )}{\sum }\frac {\left (-80075-2048 \,{\mathrm e}^{20} \textit {\_R}^{4}+20480 \textit {\_R}^{3} {\mathrm e}^{15}-76836 \,{\mathrm e}^{10} \textit {\_R}^{2}+40 \left (3203 \,{\mathrm e}^{5}-3 \,{\mathrm e}^{10}\right ) \textit {\_R} +300 \,{\mathrm e}^{5}\right ) \ln \left (x -\textit {\_R} \right )}{-119925-3072 \textit {\_R}^{5} {\mathrm e}^{20}-15360 \,{\mathrm e}^{20} \textit {\_R}^{4}-18432 \textit {\_R}^{3} {\mathrm e}^{20}+25600 \,{\mathrm e}^{15} \textit {\_R}^{4}+122880 \textit {\_R}^{3} {\mathrm e}^{15}+138240 \textit {\_R}^{2} {\mathrm e}^{15}-76800 \textit {\_R}^{3} {\mathrm e}^{10}-345564 \,{\mathrm e}^{10} \textit {\_R}^{2}-345528 \textit {\_R} \,{\mathrm e}^{10}+96000 \textit {\_R}^{2} {\mathrm e}^{5}+383880 \textit {\_R} \,{\mathrm e}^{5}+287820 \,{\mathrm e}^{5}-40000 \textit {\_R}}\right )}{2}\)	\(220\)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).
time = 0.30, size = 49, normalized size = 2.13 \begin {gather*} \frac {512 \, x^{2} e^{10} - 2560 \, x e^{5} + 3203}{256 \, x^{3} e^{10} + 256 \, x^{2} {\left (3 \, e^{10} - 5 \, e^{5}\right )} - 320 \, x {\left (12 \, e^{5} - 5\right )} + 4797} \end {gather*}

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Fricas [A]
time = 0.37, size = 46, normalized size = 2.00 \begin {gather*} \frac {512 \, x^{2} e^{10} - 2560 \, x e^{5} + 3203}{256 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{10} - 1280 \, {\left (x^{2} + 3 \, x\right )} e^{5} + 1600 \, x + 4797} \end {gather*}

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Sympy [A]
time = 1.74, size = 49, normalized size = 2.13 \begin {gather*} - \frac {- 512 x^{2} e^{10} + 2560 x e^{5} - 3203}{256 x^{3} e^{10} + x^{2} \left (- 1280 e^{5} + 768 e^{10}\right ) + x \left (1600 - 3840 e^{5}\right ) + 4797} \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (23) = 46\).
time = 0.41, size = 48, normalized size = 2.09 \begin {gather*} \frac {512 \, x^{2} e^{10} - 2560 \, x e^{5} + 3203}{256 \, x^{3} e^{10} + 768 \, x^{2} e^{10} - 1280 \, x^{2} e^{5} - 3840 \, x e^{5} + 1600 \, x + 4797} \end {gather*}

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Mupad [B]
time = 0.43, size = 48, normalized size = 2.09 \begin {gather*} -\frac {512\,{\mathrm {e}}^{10}\,x^2-2560\,{\mathrm {e}}^5\,x+3203}{-256\,{\mathrm {e}}^{10}\,x^3+\left (1280\,{\mathrm {e}}^5-768\,{\mathrm {e}}^{10}\right )\,x^2+\left (3840\,{\mathrm {e}}^5-1600\right )\,x-4797} \end {gather*}

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