Integrand size = 73, antiderivative size = 22 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15}{4} e^{-\frac {2 x^4}{\left (x+(5+2 x)^2\right )^2}} \]
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Time = 0.45 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {1607, 6820, 12, 6838} \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15}{4} e^{-\frac {2 x^4}{\left (4 x^2+21 x+25\right )^2}} \]
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Rule 12
Rule 1607
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} (-750-315 x) x^3}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx \\ & = \int \frac {15 e^{-\frac {2 x^4}{\left (25+21 x+4 x^2\right )^2}} (-50-21 x) x^3}{\left (25+21 x+4 x^2\right )^3} \, dx \\ & = 15 \int \frac {e^{-\frac {2 x^4}{\left (25+21 x+4 x^2\right )^2}} (-50-21 x) x^3}{\left (25+21 x+4 x^2\right )^3} \, dx \\ & = \frac {15}{4} e^{-\frac {2 x^4}{\left (25+21 x+4 x^2\right )^2}} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15}{4} e^{-\frac {2 x^4}{\left (25+21 x+4 x^2\right )^2}} \]
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {15 \,{\mathrm e}^{-\frac {2 x^{4}}{\left (4 x^{2}+21 x +25\right )^{2}}}}{4}\) | \(21\) |
gosper | \(\frac {15 \,{\mathrm e}^{-\frac {2 x^{4}}{16 x^{4}+168 x^{3}+641 x^{2}+1050 x +625}}}{4}\) | \(32\) |
norman | \(\frac {\left (\frac {9375}{4}+\frac {7875}{2} x +\frac {9615}{4} x^{2}+630 x^{3}+60 x^{4}\right ) {\mathrm e}^{-\frac {2 x^{4}}{16 x^{4}+168 x^{3}+641 x^{2}+1050 x +625}}}{\left (4 x^{2}+21 x +25\right )^{2}}\) | \(63\) |
parallelrisch | \(\frac {\left (2965028618774880 x^{4}+31132800497136240 x^{3}+118786459039668630 x^{2}+194580003107101500 x +115821430420893750\right ) {\mathrm e}^{-\frac {2 x^{4}}{16 x^{4}+168 x^{3}+641 x^{2}+1050 x +625}}}{49417143646248 \left (4 x^{2}+21 x +25\right )^{2}}\) | \(64\) |
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Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15}{4} \, e^{\left (-\frac {2 \, x^{4}}{16 \, x^{4} + 168 \, x^{3} + 641 \, x^{2} + 1050 \, x + 625}\right )} \]
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Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15 e^{- \frac {2 x^{4}}{16 x^{4} + 168 x^{3} + 641 x^{2} + 1050 x + 625}}}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (19) = 38\).
Time = 4.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.77 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15}{4} \, e^{\left (\frac {5061 \, x}{32 \, {\left (16 \, x^{4} + 168 \, x^{3} + 641 \, x^{2} + 1050 \, x + 625\right )}} + \frac {21 \, x}{4 \, {\left (4 \, x^{2} + 21 \, x + 25\right )}} + \frac {8525}{32 \, {\left (16 \, x^{4} + 168 \, x^{3} + 641 \, x^{2} + 1050 \, x + 625\right )}} - \frac {241}{32 \, {\left (4 \, x^{2} + 21 \, x + 25\right )}} - \frac {1}{8}\right )} \]
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15}{4} \, e^{\left (-\frac {2 \, x^{4}}{16 \, x^{4} + 168 \, x^{3} + 641 \, x^{2} + 1050 \, x + 625}\right )} \]
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Time = 12.56 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-\frac {2 x^4}{625+1050 x+641 x^2+168 x^3+16 x^4}} \left (-750 x^3-315 x^4\right )}{15625+39375 x+40575 x^2+21861 x^3+6492 x^4+1008 x^5+64 x^6} \, dx=\frac {15\,{\mathrm {e}}^{-\frac {2\,x^4}{16\,x^4+168\,x^3+641\,x^2+1050\,x+625}}}{4} \]
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