Integrand size = 28, antiderivative size = 56 \[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=-\frac {1}{2} \log \left (-90-76 x+89 x^2-18 x^3+x^4+\left (-38+13 x-x^2\right ) \sqrt {-8+4 x-3 x^2-10 x^3+x^4}\right ) \]
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\[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=\int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}}+\frac {2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}}\right ) \, dx \\ & = 2 \int \frac {x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx-\int \frac {1}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx \\ \end{align*}
Time = 3.95 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=-\frac {1}{2} \log \left (-90-76 x+89 x^2-18 x^3+x^4+\left (-38+13 x-x^2\right ) \sqrt {-8+4 x-3 x^2-10 x^3+x^4}\right ) \]
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Time = 1.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.62
method | result | size |
trager | \(\frac {\ln \left (x^{4}+\sqrt {x^{4}-10 x^{3}-3 x^{2}+4 x -8}\, x^{2}-18 x^{3}-13 x \sqrt {x^{4}-10 x^{3}-3 x^{2}+4 x -8}+89 x^{2}+38 \sqrt {x^{4}-10 x^{3}-3 x^{2}+4 x -8}-76 x -90\right )}{2}\) | \(91\) |
default | \(\text {Expression too large to display}\) | \(2700\) |
elliptic | \(\text {Expression too large to display}\) | \(2700\) |
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Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89 \[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=\frac {1}{2} \, \log \left (x^{4} - 18 \, x^{3} + 89 \, x^{2} + \sqrt {x^{4} - 10 \, x^{3} - 3 \, x^{2} + 4 \, x - 8} {\left (x^{2} - 13 \, x + 38\right )} - 76 \, x - 90\right ) \]
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\[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=\int \frac {2 x - 1}{\sqrt {x^{4} - 10 x^{3} - 3 x^{2} + 4 x - 8}}\, dx \]
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\[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=\int { \frac {2 \, x - 1}{\sqrt {x^{4} - 10 \, x^{3} - 3 \, x^{2} + 4 \, x - 8}} \,d x } \]
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\[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=\int { \frac {2 \, x - 1}{\sqrt {x^{4} - 10 \, x^{3} - 3 \, x^{2} + 4 \, x - 8}} \,d x } \]
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Timed out. \[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=\int \frac {2\,x-1}{\sqrt {x^4-10\,x^3-3\,x^2+4\,x-8}} \,d x \]
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