Integrand size = 30, antiderivative size = 21 \[ \int \frac {-1+2 x+2 x^2}{(-1+x) x \sqrt {-x+x^4}} \, dx=-\frac {2 \sqrt {-x+x^4}}{(-1+x) x} \]
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\[ \int \frac {-1+2 x+2 x^2}{(-1+x) x \sqrt {-x+x^4}} \, dx=\int \frac {-1+2 x+2 x^2}{(-1+x) x \sqrt {-x+x^4}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {-1+2 x+2 x^2}{(-1+x) x^{3/2} \sqrt {-1+x^3}} \, dx}{\sqrt {-x+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {-1+2 x^2+2 x^4}{x^2 \left (-1+x^2\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \left (\frac {2}{\sqrt {-1+x^6}}+\frac {1}{x^2 \sqrt {-1+x^6}}+\frac {3}{\left (-1+x^2\right ) \sqrt {-1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}+\frac {\left (4 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}+\frac {\left (6 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}} \\ & = -\frac {2 \left (1-x^3\right )}{\sqrt {-x+x^4}}+\frac {2 (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}-\frac {\left (4 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}+\frac {\left (6 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \left (\frac {1}{2 (-1+x) \sqrt {-1+x^6}}-\frac {1}{2 (1+x) \sqrt {-1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}} \\ & = -\frac {2 \left (1-x^3\right )}{\sqrt {-x+x^4}}+\frac {2 (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {-1+\sqrt {3}-2 x^4}{\sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}+\frac {\left (3 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}-\frac {\left (3 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}-\frac {\left (2 \left (-1+\sqrt {3}\right ) \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}} \\ & = -\frac {2 \left (1-x^3\right )}{\sqrt {-x+x^4}}-\frac {2 \left (1+\sqrt {3}\right ) x \left (1-x^3\right )}{\left (1-\left (1+\sqrt {3}\right ) x\right ) \sqrt {-x+x^4}}+\frac {2 \sqrt [4]{3} (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} E\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {2 (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {\left (1-\sqrt {3}\right ) (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}+\frac {\left (3 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}}-\frac {\left (3 \sqrt {x} \sqrt {-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^4}} \\ \end{align*}
Time = 8.42 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-1+2 x+2 x^2}{(-1+x) x \sqrt {-x+x^4}} \, dx=-\frac {2 \left (1+x+x^2\right )}{\sqrt {x \left (-1+x^3\right )}} \]
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Time = 2.66 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86
method | result | size |
gosper | \(-\frac {2 \left (x^{2}+x +1\right )}{\sqrt {x^{4}-x}}\) | \(18\) |
trager | \(-\frac {2 \sqrt {x^{4}-x}}{\left (x -1\right ) x}\) | \(20\) |
default | \(\frac {2 x^{3}-2}{\sqrt {x \left (x^{3}-1\right )}}-\frac {2 \left (x^{3}+x^{2}+x \right )}{\sqrt {\left (x -1\right ) \left (x^{3}+x^{2}+x \right )}}\) | \(42\) |
elliptic | \(\frac {2 x^{3}-2}{\sqrt {x \left (x^{3}-1\right )}}-\frac {2 \left (x^{3}+x^{2}+x \right )}{\sqrt {\left (x -1\right ) \left (x^{3}+x^{2}+x \right )}}\) | \(42\) |
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Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-1+2 x+2 x^2}{(-1+x) x \sqrt {-x+x^4}} \, dx=-\frac {2 \, \sqrt {x^{4} - x}}{x^{2} - x} \]
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\[ \int \frac {-1+2 x+2 x^2}{(-1+x) x \sqrt {-x+x^4}} \, dx=\int \frac {2 x^{2} + 2 x - 1}{x \sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1\right )}\, dx \]
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\[ \int \frac {-1+2 x+2 x^2}{(-1+x) x \sqrt {-x+x^4}} \, dx=\int { \frac {2 \, x^{2} + 2 \, x - 1}{\sqrt {x^{4} - x} {\left (x - 1\right )} x} \,d x } \]
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\[ \int \frac {-1+2 x+2 x^2}{(-1+x) x \sqrt {-x+x^4}} \, dx=\int { \frac {2 \, x^{2} + 2 \, x - 1}{\sqrt {x^{4} - x} {\left (x - 1\right )} x} \,d x } \]
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Time = 5.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-1+2 x+2 x^2}{(-1+x) x \sqrt {-x+x^4}} \, dx=-\frac {2\,\sqrt {x^4-x}}{x\,\left (x-1\right )} \]
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