Integrand size = 25, antiderivative size = 159 \[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{3/2}} \, dx=\frac {22 \sqrt {x} (2+3 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {2 (6+5 x) \sqrt {2+5 x+3 x^2}}{3 \sqrt {x}}-\frac {22 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{9 \sqrt {2+5 x+3 x^2}}+\frac {10 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{3 \sqrt {2+5 x+3 x^2}} \]
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Time = 0.07 (sec) , antiderivative size = 159, 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 = {826, 853, 1203, 1114, 1150} \[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{3/2}} \, dx=\frac {10 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {22 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{9 \sqrt {3 x^2+5 x+2}}+\frac {22 \sqrt {x} (3 x+2)}{9 \sqrt {3 x^2+5 x+2}}-\frac {2 (5 x+6) \sqrt {3 x^2+5 x+2}}{3 \sqrt {x}} \]
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Rule 826
Rule 853
Rule 1114
Rule 1150
Rule 1203
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (6+5 x) \sqrt {2+5 x+3 x^2}}{3 \sqrt {x}}-\frac {2}{3} \int \frac {-5-\frac {11 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (6+5 x) \sqrt {2+5 x+3 x^2}}{3 \sqrt {x}}-\frac {4}{3} \text {Subst}\left (\int \frac {-5-\frac {11 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 (6+5 x) \sqrt {2+5 x+3 x^2}}{3 \sqrt {x}}+\frac {20}{3} \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {22}{3} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {22 \sqrt {x} (2+3 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {2 (6+5 x) \sqrt {2+5 x+3 x^2}}{3 \sqrt {x}}-\frac {22 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{9 \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 )}{3 \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 20.02 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.96 \[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{3/2}} \, dx=\frac {-2 \left (14+65 x+96 x^2+45 x^3\right )+22 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+8 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{9 \sqrt {x} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.71
method | result | size |
default | \(-\frac {3 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-11 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+270 x^{3}+774 x^{2}+720 x +216}{27 \sqrt {3 x^{2}+5 x +2}\, \sqrt {x}}\) | \(113\) |
risch | \(-\frac {2 \left (15 x^{3}+43 x^{2}+40 x +12\right )}{3 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}-\frac {\left (-\frac {10 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{9 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {11 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{9 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(193\) |
elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {4 \left (3 x^{2}+5 x +2\right )}{\sqrt {x \left (3 x^{2}+5 x +2\right )}}-\frac {10 \sqrt {3 x^{3}+5 x^{2}+2 x}}{3}+\frac {10 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{9 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {11 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{9 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(203\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.35 \[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{3/2}} \, dx=\frac {2 \, {\left (35 \, \sqrt {3} x {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 99 \, \sqrt {3} x {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 27 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x + 6\right )} \sqrt {x}\right )}}{81 \, x} \]
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\[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{3/2}} \, dx=- \int \left (- \frac {2 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {3}{2}}}\right )\, dx - \int \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {x}}\, dx \]
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\[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{3/2}} \, dx=\int { -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )}}{x^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{3/2}} \, dx=\int { -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )}}{x^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{3/2}} \, dx=\int -\frac {\left (5\,x-2\right )\,\sqrt {3\,x^2+5\,x+2}}{x^{3/2}} \,d x \]
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