Integrand size = 68, antiderivative size = 24 \[ \int \frac {-2 \sqrt {1+x^3}+5 x^4 \sqrt {1+x^3}-3 x^2 \sqrt {1-2 x+x^5}}{2 \sqrt {1+x^3} \sqrt {1-2 x+x^5}} \, dx=-\sqrt {1+x^3}+\sqrt {1-2 x+x^5} \]
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {12, 6820, 267, 2124} \[ \int \frac {-2 \sqrt {1+x^3}+5 x^4 \sqrt {1+x^3}-3 x^2 \sqrt {1-2 x+x^5}}{2 \sqrt {1+x^3} \sqrt {1-2 x+x^5}} \, dx=\sqrt {x^5-2 x+1}-\sqrt {x^3+1} \]
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Rule 12
Rule 267
Rule 2124
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {-2 \sqrt {1+x^3}+5 x^4 \sqrt {1+x^3}-3 x^2 \sqrt {1-2 x+x^5}}{\sqrt {1+x^3} \sqrt {1-2 x+x^5}} \, dx \\ & = \frac {1}{2} \int \left (-\frac {3 x^2}{\sqrt {1+x^3}}-\frac {2}{\sqrt {1-2 x+x^5}}+\frac {5 x^4}{\sqrt {1-2 x+x^5}}\right ) \, dx \\ & = -\left (\frac {3}{2} \int \frac {x^2}{\sqrt {1+x^3}} \, dx\right )+\frac {5}{2} \int \frac {x^4}{\sqrt {1-2 x+x^5}} \, dx-\int \frac {1}{\sqrt {1-2 x+x^5}} \, dx \\ & = -\sqrt {1+x^3}+\sqrt {1-2 x+x^5} \\ \end{align*}
Time = 14.67 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-2 \sqrt {1+x^3}+5 x^4 \sqrt {1+x^3}-3 x^2 \sqrt {1-2 x+x^5}}{2 \sqrt {1+x^3} \sqrt {1-2 x+x^5}} \, dx=-\sqrt {1+x^3}+\sqrt {1-2 x+x^5} \]
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Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\sqrt {x^{3}+1}+\sqrt {x^{5}-2 x +1}\) | \(21\) |
elliptic | \(-\sqrt {x^{3}+1}+\frac {\left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x -1\right )}{\sqrt {x^{5}-2 x +1}}\) | \(37\) |
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Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-2 \sqrt {1+x^3}+5 x^4 \sqrt {1+x^3}-3 x^2 \sqrt {1-2 x+x^5}}{2 \sqrt {1+x^3} \sqrt {1-2 x+x^5}} \, dx=\sqrt {x^{5} - 2 \, x + 1} - \sqrt {x^{3} + 1} \]
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Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-2 \sqrt {1+x^3}+5 x^4 \sqrt {1+x^3}-3 x^2 \sqrt {1-2 x+x^5}}{2 \sqrt {1+x^3} \sqrt {1-2 x+x^5}} \, dx=- \sqrt {x^{3} + 1} + \sqrt {x^{5} - 2 x + 1} \]
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Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {-2 \sqrt {1+x^3}+5 x^4 \sqrt {1+x^3}-3 x^2 \sqrt {1-2 x+x^5}}{2 \sqrt {1+x^3} \sqrt {1-2 x+x^5}} \, dx=\sqrt {x^{4} + x^{3} + x^{2} + x - 1} \sqrt {x - 1} - \sqrt {x^{3} + 1} \]
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\[ \int \frac {-2 \sqrt {1+x^3}+5 x^4 \sqrt {1+x^3}-3 x^2 \sqrt {1-2 x+x^5}}{2 \sqrt {1+x^3} \sqrt {1-2 x+x^5}} \, dx=\int { \frac {5 \, \sqrt {x^{3} + 1} x^{4} - 3 \, \sqrt {x^{5} - 2 \, x + 1} x^{2} - 2 \, \sqrt {x^{3} + 1}}{2 \, \sqrt {x^{5} - 2 \, x + 1} \sqrt {x^{3} + 1}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-2 \sqrt {1+x^3}+5 x^4 \sqrt {1+x^3}-3 x^2 \sqrt {1-2 x+x^5}}{2 \sqrt {1+x^3} \sqrt {1-2 x+x^5}} \, dx=\sqrt {x^5-2\,x+1}-\sqrt {x^3+1} \]
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