Integrand size = 23, antiderivative size = 146 \[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x^3} \, dx=-\frac {\sqrt {1+x} \sqrt {1-x+x^2}}{2 x^2}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )} \]
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Time = 0.03 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {929, 283, 224} \[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x^3} \, dx=\frac {3^{3/4} \sqrt {2+\sqrt {3}} (x+1)^{3/2} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )}-\frac {\sqrt {x+1} \sqrt {x^2-x+1}}{2 x^2} \]
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Rule 224
Rule 283
Rule 929
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {\sqrt {1+x^3}}{x^3} \, dx}{\sqrt {1+x^3}} \\ & = -\frac {\sqrt {1+x} \sqrt {1-x+x^2}}{2 x^2}+\frac {\left (3 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx}{4 \sqrt {1+x^3}} \\ & = -\frac {\sqrt {1+x} \sqrt {1-x+x^2}}{2 x^2}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 20.31 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x^3} \, dx=\frac {\sqrt {1+x} \left (-\frac {2 \left (1-x+x^2\right )}{x^2}-\frac {3 i \sqrt {2} \sqrt {\frac {i+\sqrt {3}-2 i x}{3 i+\sqrt {3}}} \sqrt {\frac {-i+\sqrt {3}+2 i x}{-3 i+\sqrt {3}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {i (1+x)}{3 i+\sqrt {3}}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i (1+x)}{3 i+\sqrt {3}}}}\right )}{4 \sqrt {1-x+x^2}} \]
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Time = 0.65 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.09
method | result | size |
elliptic | \(\frac {\sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (-\frac {\sqrt {x^{3}+1}}{2 x^{2}}+\frac {3 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}\right )}{\sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(159\) |
risch | \(-\frac {\sqrt {1+x}\, \sqrt {x^{2}-x +1}}{2 x^{2}}+\frac {3 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}}{2 \sqrt {x^{3}+1}\, \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(166\) |
default | \(-\frac {\sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \left (3 i \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) \sqrt {3}\, x^{2}-9 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x^{2}+2 x^{3}+2\right )}{4 \left (x^{3}+1\right ) x^{2}}\) | \(259\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x^3} \, dx=\frac {3 \, x^{2} {\rm weierstrassPInverse}\left (0, -4, x\right ) - \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{2 \, x^{2}} \]
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\[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x^3} \, dx=\int \frac {\sqrt {x + 1} \sqrt {x^{2} - x + 1}}{x^{3}}\, dx \]
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\[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x^3} \, dx=\int { \frac {\sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x^{3}} \,d x } \]
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\[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x^3} \, dx=\int { \frac {\sqrt {x^{2} - x + 1} \sqrt {x + 1}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1+x} \sqrt {1-x+x^2}}{x^3} \, dx=\int \frac {\sqrt {x+1}\,\sqrt {x^2-x+1}}{x^3} \,d x \]
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