Integrand size = 18, antiderivative size = 16 \[ \int \frac {-1+x^4}{x^2 \sqrt {x+x^3}} \, dx=\frac {2 \left (x+x^3\right )^{3/2}}{3 x^3} \]
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Result contains higher order function than in optimal. Order 4 vs. order 2 in optimal.
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 4.88, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2073, 2050, 2036, 335, 226} \[ \int \frac {-1+x^4}{x^2 \sqrt {x+x^3}} \, dx=\frac {\sqrt {x} (x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{3 \sqrt {x^3+x}}+\frac {2 \sqrt {x^3+x}}{3}+\frac {2 \sqrt {x^3+x}}{3 x^2} \]
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Rule 226
Rule 335
Rule 2036
Rule 2050
Rule 2073
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {x+x^3}}{3}-\int \frac {1}{x^2 \sqrt {x+x^3}} \, dx \\ & = \frac {2 \sqrt {x+x^3}}{3}+\frac {2 \sqrt {x+x^3}}{3 x^2}+\frac {1}{3} \int \frac {1}{\sqrt {x+x^3}} \, dx \\ & = \frac {2 \sqrt {x+x^3}}{3}+\frac {2 \sqrt {x+x^3}}{3 x^2}+\frac {\left (\sqrt {x} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x^2}} \, dx}{3 \sqrt {x+x^3}} \\ & = \frac {2 \sqrt {x+x^3}}{3}+\frac {2 \sqrt {x+x^3}}{3 x^2}+\frac {\left (2 \sqrt {x} \sqrt {1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^3}} \\ & = \frac {2 \sqrt {x+x^3}}{3}+\frac {2 \sqrt {x+x^3}}{3 x^2}+\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{3 \sqrt {x+x^3}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {-1+x^4}{x^2 \sqrt {x+x^3}} \, dx=\frac {2 \left (1+x^2\right )^2}{3 x \sqrt {x+x^3}} \]
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Time = 0.89 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12
method | result | size |
trager | \(\frac {2 \left (x^{2}+1\right ) \sqrt {x^{3}+x}}{3 x^{2}}\) | \(18\) |
gosper | \(\frac {2 \left (x^{2}+1\right )^{2}}{3 \sqrt {x^{3}+x}\, x}\) | \(20\) |
pseudoelliptic | \(\frac {2 \left (x^{2}+1\right ) \sqrt {\left (x^{2}+1\right ) x}}{3 x^{2}}\) | \(20\) |
default | \(\frac {2 \sqrt {x^{3}+x}}{3}+\frac {2 \sqrt {x^{3}+x}}{3 x^{2}}\) | \(23\) |
elliptic | \(\frac {2 \sqrt {x^{3}+x}}{3}+\frac {2 \sqrt {x^{3}+x}}{3 x^{2}}\) | \(23\) |
risch | \(\frac {\frac {2}{3} x^{4}+\frac {4}{3} x^{2}+\frac {2}{3}}{x \sqrt {\left (x^{2}+1\right ) x}}\) | \(25\) |
meijerg | \(\frac {2 x^{\frac {5}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {9}{4}\right ], -x^{2}\right )}{5}+\frac {2 \operatorname {hypergeom}\left (\left [-\frac {3}{4}, \frac {1}{2}\right ], \left [\frac {1}{4}\right ], -x^{2}\right )}{3 x^{\frac {3}{2}}}\) | \(34\) |
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Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {-1+x^4}{x^2 \sqrt {x+x^3}} \, dx=\frac {2 \, \sqrt {x^{3} + x} {\left (x^{2} + 1\right )}}{3 \, x^{2}} \]
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\[ \int \frac {-1+x^4}{x^2 \sqrt {x+x^3}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{2} \sqrt {x \left (x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {-1+x^4}{x^2 \sqrt {x+x^3}} \, dx=\int { \frac {x^{4} - 1}{\sqrt {x^{3} + x} x^{2}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \frac {-1+x^4}{x^2 \sqrt {x+x^3}} \, dx=\frac {2}{3} \, \sqrt {x^{3} + x} + \frac {2}{3} \, \sqrt {\frac {1}{x} + \frac {1}{x^{3}}} \]
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Time = 4.90 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \frac {-1+x^4}{x^2 \sqrt {x+x^3}} \, dx=\frac {4\,\sqrt {x^3+x}}{3} \]
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