Integrand size = 24, antiderivative size = 30 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=-\frac {1}{3} \sqrt {19} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|\frac {55}{76}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {732, 435} \[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=-\frac {1}{3} \sqrt {19} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|\frac {55}{76}\right ) \]
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Rule 435
Rule 732
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \sqrt {19} \text {Subst}\left (\int \frac {\sqrt {1-\frac {55 x^2}{76}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {16-24 x}}{\sqrt {22}}\right )\right ) \\ & = -\frac {1}{3} \sqrt {19} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|\frac {55}{76}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(86\) vs. \(2(30)=60\).
Time = 30.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.87 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=\frac {\sqrt {19} \sqrt {-1-4 x} \sqrt {2-3 x} \left (-E\left (\arcsin \left (\frac {2 \sqrt {3+5 x}}{\sqrt {7}}\right )|\frac {21}{76}\right )+\operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {3+5 x}}{\sqrt {7}}\right ),\frac {21}{76}\right )\right )}{3 \sqrt {2+5 x-12 x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs. \(2(23)=46\).
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.57
method | result | size |
default | \(-\frac {\left (F\left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )-E\left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )\right ) \sqrt {190-285 x}\, \sqrt {-35-140 x}\, \sqrt {57}\, \sqrt {-12 x^{2}+5 x +2}}{570 \left (12 x^{2}-5 x -2\right )}\) | \(77\) |
elliptic | \(\frac {\sqrt {-\left (12 x^{2}-5 x -2\right ) \left (3+5 x \right )}\, \left (\frac {2 \sqrt {171+285 x}\, \sqrt {-35-140 x}\, \sqrt {190-285 x}\, F\left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )}{665 \sqrt {-60 x^{3}-11 x^{2}+25 x +6}}+\frac {2 \sqrt {171+285 x}\, \sqrt {-35-140 x}\, \sqrt {190-285 x}\, \left (-\frac {7 E\left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )}{20}-\frac {F\left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )}{4}\right )}{399 \sqrt {-60 x^{3}-11 x^{2}+25 x +6}}\right )}{\sqrt {3+5 x}\, \sqrt {-12 x^{2}+5 x +2}}\) | \(171\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=-\frac {97}{540} \, \sqrt {-15} {\rm weierstrassPInverse}\left (\frac {4621}{2700}, \frac {216019}{729000}, x + \frac {11}{180}\right ) + \frac {1}{3} \, \sqrt {-15} {\rm weierstrassZeta}\left (\frac {4621}{2700}, \frac {216019}{729000}, {\rm weierstrassPInverse}\left (\frac {4621}{2700}, \frac {216019}{729000}, x + \frac {11}{180}\right )\right ) \]
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\[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=\int \frac {\sqrt {5 x + 3}}{\sqrt {- \left (3 x - 2\right ) \left (4 x + 1\right )}}\, dx \]
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\[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{\sqrt {-12 \, x^{2} + 5 \, x + 2}} \,d x } \]
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\[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{\sqrt {-12 \, x^{2} + 5 \, x + 2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=\int \frac {\sqrt {5\,x+3}}{\sqrt {-12\,x^2+5\,x+2}} \,d x \]
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