Integrand size = 25, antiderivative size = 154 \[ \int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx=\frac {136 \sqrt {x} (2+3 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {10}{9} \sqrt {x} \sqrt {2+5 x+3 x^2}-\frac {136 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{27 \sqrt {2+5 x+3 x^2}}+\frac {10 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{9 \sqrt {2+5 x+3 x^2}} \]
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Time = 0.07 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {846, 853, 1203, 1114, 1150} \[ \int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx=\frac {10 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{9 \sqrt {3 x^2+5 x+2}}-\frac {136 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{27 \sqrt {3 x^2+5 x+2}}+\frac {136 \sqrt {x} (3 x+2)}{27 \sqrt {3 x^2+5 x+2}}-\frac {10}{9} \sqrt {x} \sqrt {3 x^2+5 x+2} \]
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Rule 846
Rule 853
Rule 1114
Rule 1150
Rule 1203
Rubi steps \begin{align*} \text {integral}& = -\frac {10}{9} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {2}{9} \int \frac {5+34 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {10}{9} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {4}{9} \text {Subst}\left (\int \frac {5+34 x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {10}{9} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {20}{9} \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {136}{9} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {136 \sqrt {x} (2+3 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {10}{9} \sqrt {x} \sqrt {2+5 x+3 x^2}-\frac {136 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{27 \sqrt {2+5 x+3 x^2}}+\frac {10 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{9 \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 21.15 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.97 \[ \int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx=\frac {272+620 x+258 x^2-90 x^3+136 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-106 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{27 \sqrt {x} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.73
method | result | size |
default | \(-\frac {2 \left (87 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-34 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+135 x^{3}+225 x^{2}+90 x \right )}{81 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(112\) |
elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {10 \sqrt {3 x^{3}+5 x^{2}+2 x}}{9}+\frac {10 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{27 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {68 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{27 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(177\) |
risch | \(-\frac {10 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{9}-\frac {\left (-\frac {10 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{27 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {68 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{27 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(178\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.28 \[ \int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx=-\frac {500}{243} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - \frac {136}{27} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - \frac {10}{9} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} \]
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\[ \int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx=- \int \left (- \frac {2 \sqrt {x}}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {5 x^{\frac {3}{2}}}{\sqrt {3 x^{2} + 5 x + 2}}\, dx \]
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\[ \int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx=\int { -\frac {{\left (5 \, x - 2\right )} \sqrt {x}}{\sqrt {3 \, x^{2} + 5 \, x + 2}} \,d x } \]
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\[ \int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx=\int { -\frac {{\left (5 \, x - 2\right )} \sqrt {x}}{\sqrt {3 \, x^{2} + 5 \, x + 2}} \,d x } \]
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Timed out. \[ \int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx=-\int \frac {\sqrt {x}\,\left (5\,x-2\right )}{\sqrt {3\,x^2+5\,x+2}} \,d x \]
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