Integrand size = 81, antiderivative size = 24 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {x^2}{\frac {2}{5}+\frac {3125 \left (-2+3 x^4\right )^5}{e}} \]
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\[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-5000000 e+20 e^2\right ) x-37500000 e x^5+337500000 e x^9-843750000 e x^{13}+885937500 e x^{17}-341718750 e x^{21}}{250000000000-2000000 e+4 e^2+(-3750000000000+15000000 e) x^4+(25312500000000-45000000 e) x^8+(-101250000000000+67500000 e) x^{12}+(265781250000000-50625000 e) x^{16}+(-478406250000000+15187500 e) x^{20}+598007812500000 x^{24}-512578125000000 x^{28}+288325195312500 x^{32}-96108398437500 x^{36}+14416259765625 x^{40}} \, dx \\ & = \int \left (\frac {200 e x \left (-250000 \left (1-\frac {e}{250000}\right )+1500000 x^4-3375000 x^8+3375000 x^{12}-1265625 x^{16}\right )}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^4+11250000 x^8-16875000 x^{12}+12656250 x^{16}-3796875 x^{20}\right )^2}+\frac {90 e x}{500000 \left (1-\frac {e}{250000}\right )-3750000 x^4+11250000 x^8-16875000 x^{12}+12656250 x^{16}-3796875 x^{20}}\right ) \, dx \\ & = (90 e) \int \frac {x}{500000 \left (1-\frac {e}{250000}\right )-3750000 x^4+11250000 x^8-16875000 x^{12}+12656250 x^{16}-3796875 x^{20}} \, dx+(200 e) \int \frac {x \left (-250000 \left (1-\frac {e}{250000}\right )+1500000 x^4-3375000 x^8+3375000 x^{12}-1265625 x^{16}\right )}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^4+11250000 x^8-16875000 x^{12}+12656250 x^{16}-3796875 x^{20}\right )^2} \, dx \\ & = (90 e) \int \frac {x}{-2 e-15625 \left (-2+3 x^4\right )^5} \, dx+(200 e) \int \frac {x \left (e-15625 \left (2-3 x^4\right )^4\right )}{\left (2 e+15625 \left (-2+3 x^4\right )^5\right )^2} \, dx \\ & = (45 e) \text {Subst}\left (\int \frac {1}{-2 e-15625 \left (-2+3 x^2\right )^5} \, dx,x,x^2\right )+(100 e) \text {Subst}\left (\int \frac {e-15625 \left (2-3 x^2\right )^4}{\left (2 e+15625 \left (-2+3 x^2\right )^5\right )^2} \, dx,x,x^2\right ) \\ & = -\left (\frac {9}{2} \text {Subst}\left (\int -\frac {2}{-2-10\ 2^{4/5} \sqrt [5]{-\frac {5}{e}}+15\ 2^{4/5} \sqrt [5]{-\frac {5}{e}} x^2} \, dx,x,x^2\right )\right )-\frac {9}{2} \text {Subst}\left (\int \frac {2}{2-10 (-2)^{4/5} \sqrt [5]{\frac {5}{e}}+15 (-2)^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )-\frac {9}{2} \text {Subst}\left (\int \frac {2}{2-10\ 2^{4/5} \sqrt [5]{\frac {5}{e}}+15\ 2^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )-\frac {9}{2} \text {Subst}\left (\int \frac {2}{2-10 (-1)^{2/5} 2^{4/5} \sqrt [5]{\frac {5}{e}}+15 (-1)^{2/5} 2^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )-\frac {9}{2} \text {Subst}\left (\int -\frac {2}{-2-10 (-1)^{3/5} 2^{4/5} \sqrt [5]{\frac {5}{e}}+15 (-1)^{3/5} 2^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )+(100 e) \text {Subst}\left (\int \left (-\frac {250000 \left (1-\frac {e}{250000}\right )}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2}+\frac {1500000 x^2}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2}-\frac {3375000 x^4}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2}+\frac {3375000 x^6}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2}-\frac {1265625 x^8}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2}\right ) \, dx,x,x^2\right ) \\ & = 9 \text {Subst}\left (\int \frac {1}{-2-10\ 2^{4/5} \sqrt [5]{-\frac {5}{e}}+15\ 2^{4/5} \sqrt [5]{-\frac {5}{e}} x^2} \, dx,x,x^2\right )-9 \text {Subst}\left (\int \frac {1}{2-10 (-2)^{4/5} \sqrt [5]{\frac {5}{e}}+15 (-2)^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )-9 \text {Subst}\left (\int \frac {1}{2-10\ 2^{4/5} \sqrt [5]{\frac {5}{e}}+15\ 2^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )-9 \text {Subst}\left (\int \frac {1}{2-10 (-1)^{2/5} 2^{4/5} \sqrt [5]{\frac {5}{e}}+15 (-1)^{2/5} 2^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )+9 \text {Subst}\left (\int \frac {1}{-2-10 (-1)^{3/5} 2^{4/5} \sqrt [5]{\frac {5}{e}}+15 (-1)^{3/5} 2^{4/5} \sqrt [5]{\frac {5}{e}} x^2} \, dx,x,x^2\right )-(126562500 e) \text {Subst}\left (\int \frac {x^8}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2} \, dx,x,x^2\right )+(150000000 e) \text {Subst}\left (\int \frac {x^2}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2} \, dx,x,x^2\right )-(337500000 e) \text {Subst}\left (\int \frac {x^4}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2} \, dx,x,x^2\right )+(337500000 e) \text {Subst}\left (\int \frac {x^6}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2} \, dx,x,x^2\right )-(100 (250000-e) e) \text {Subst}\left (\int \frac {1}{\left (500000 \left (1-\frac {e}{250000}\right )-3750000 x^2+11250000 x^4-16875000 x^6+12656250 x^8-3796875 x^{10}\right )^2} \, dx,x,x^2\right ) \\ & = -\frac {3 \sqrt {3 \left (50+5^{4/5} \sqrt [5]{-2 e}\right )} \text {arctanh}\left (\frac {5 x^2}{\sqrt {\frac {1}{3} \left (50+5^{4/5} \sqrt [5]{-2 e}\right )}}\right )}{10 \left (1-5 (-2)^{4/5} \sqrt [5]{\frac {5}{e}}\right )}+\frac {3 \sqrt {\frac {3}{-1+5\ 2^{4/5} \sqrt [5]{\frac {5}{e}}}} \sqrt [10]{e} \text {arctanh}\left (\frac {5^{3/5} \sqrt {\frac {3}{-1+5\ 2^{4/5} \sqrt [5]{\frac {5}{e}}}} x^2}{\sqrt [10]{2 e}}\right )}{2^{9/10} 5^{3/5}}-\frac {3 \sqrt {3 \left (50-(-5)^{4/5} \sqrt [5]{2 e}\right )} \text {arctanh}\left (\frac {5 x^2}{\sqrt {\frac {1}{3} \left (50-(-5)^{4/5} \sqrt [5]{2 e}\right )}}\right )}{10 \left (1+5\ 2^{4/5} \sqrt [5]{-\frac {5}{e}}\right )}-\frac {3 \sqrt {3 \left (50-(-1)^{2/5} 5^{4/5} \sqrt [5]{2 e}\right )} \text {arctanh}\left (\frac {5 x^2}{\sqrt {\frac {1}{3} \left (50-(-1)^{2/5} 5^{4/5} \sqrt [5]{2 e}\right )}}\right )}{10 \left (1+5 (-1)^{3/5} 2^{4/5} \sqrt [5]{\frac {5}{e}}\right )}-\frac {3 \sqrt {3 \left (50+(-1)^{3/5} 5^{4/5} \sqrt [5]{2 e}\right )} \text {arctanh}\left (\frac {5 x^2}{\sqrt {\frac {1}{3} \left (50+(-1)^{3/5} 5^{4/5} \sqrt [5]{2 e}\right )}}\right )}{10 \left (1-5 (-1)^{2/5} 2^{4/5} \sqrt [5]{\frac {5}{e}}\right )}-(126562500 e) \text {Subst}\left (\int \frac {x^8}{\left (2 e+15625 \left (-2+3 x^2\right )^5\right )^2} \, dx,x,x^2\right )+(150000000 e) \text {Subst}\left (\int \frac {x^2}{\left (2 e+15625 \left (-2+3 x^2\right )^5\right )^2} \, dx,x,x^2\right )-(337500000 e) \text {Subst}\left (\int \frac {x^4}{\left (2 e+15625 \left (-2+3 x^2\right )^5\right )^2} \, dx,x,x^2\right )+(337500000 e) \text {Subst}\left (\int \frac {x^6}{\left (2 e+15625 \left (-2+3 x^2\right )^5\right )^2} \, dx,x,x^2\right )-(100 (250000-e) e) \text {Subst}\left (\int \frac {1}{\left (2 e+15625 \left (-2+3 x^2\right )^5\right )^2} \, dx,x,x^2\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {10 e x^2}{4 e+31250 \left (-2+3 x^4\right )^5} \]
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Time = 4.56 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\frac {5 x^{2}}{2+5 \,{\mathrm e}^{5 \ln \left (15 x^{4}-10\right )-1}}\) | \(25\) |
risch | \(\frac {x^{2}}{759375 \,{\mathrm e}^{-1} x^{20}-2531250 \,{\mathrm e}^{-1} x^{16}+3375000 \,{\mathrm e}^{-1} x^{12}-2250000 \,{\mathrm e}^{-1} x^{8}+750000 \,{\mathrm e}^{-1} x^{4}-100000 \,{\mathrm e}^{-1}+\frac {2}{5}}\) | \(48\) |
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Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {5 \, x^{2} e}{3796875 \, x^{20} - 12656250 \, x^{16} + 16875000 \, x^{12} - 11250000 \, x^{8} + 3750000 \, x^{4} + 2 \, e - 500000} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 4.40 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {5 e x^{2}}{3796875 x^{20} - 12656250 x^{16} + 16875000 x^{12} - 11250000 x^{8} + 3750000 x^{4} - 500000 + 2 e} \]
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Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {5 \, x^{2} e}{3796875 \, x^{20} - 12656250 \, x^{16} + 16875000 \, x^{12} - 11250000 \, x^{8} + 3750000 \, x^{4} + 2 \, e - 500000} \]
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Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {5 \, x^{2} e}{3796875 \, x^{20} - 12656250 \, x^{16} + 16875000 \, x^{12} - 11250000 \, x^{8} + 3750000 \, x^{4} + 2 \, e - 500000} \]
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Time = 16.72 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {-40 x+60 x^5+\frac {\left (-10+15 x^4\right )^5 \left (-100 x-1350 x^5\right )}{e}}{-8+12 x^4+\frac {\left (-10+15 x^4\right )^5 \left (-40+60 x^4\right )}{e}+\frac {\left (-10+15 x^4\right )^{10} \left (-50+75 x^4\right )}{e^2}} \, dx=\frac {5\,x^2\,\mathrm {e}}{3796875\,x^{20}-12656250\,x^{16}+16875000\,x^{12}-11250000\,x^8+3750000\,x^4+2\,\mathrm {e}-500000} \]
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