Integrand size = 29, antiderivative size = 143 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=\frac {1}{45} (88-9 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {761 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{90 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {191 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{18 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {828, 857, 732, 435, 430} \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=\frac {191 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{18 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {761 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{90 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (88-9 x) \]
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Rule 430
Rule 435
Rule 732
Rule 828
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {1}{45} (88-9 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {1}{90} \int \frac {664+761 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = \frac {1}{45} (88-9 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {761}{180} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {191}{36} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = \frac {1}{45} (88-9 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {\left (761 \sqrt {-2-5 x-3 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 x^2}{3}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{90 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (191 \sqrt {-2-5 x-3 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 x^2}{3}}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{18 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ & = \frac {1}{45} (88-9 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {761 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{90 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {191 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{18 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Time = 25.77 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.35 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=-\frac {2 \sqrt {3+2 x} \left (-62-1049 x-2220 x^2-1071 x^3+162 x^4\right )+761 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-188 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{270 (3+2 x) \sqrt {2+5 x+3 x^2}} \]
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Time = 0.42 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {3+2 x}\, \left (291 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-761 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+4860 x^{4}-32130 x^{3}-135090 x^{2}-145620 x -47520\right )}{4050 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\) | \(141\) |
risch | \(-\frac {\left (-88+9 x \right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{45}-\frac {\left (\frac {332 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{675 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {761 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {E\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{1350 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(198\) |
elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5}+\frac {88 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{45}+\frac {332 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{675 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {761 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{1350 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(209\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.36 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=-\frac {1}{45} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (9 \, x - 88\right )} \sqrt {2 \, x + 3} + \frac {2507}{4860} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {761}{270} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
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\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\, dx \]
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\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=\int { -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (x - 5\right )}}{\sqrt {2 \, x + 3}} \,d x } \]
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\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=\int { -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (x - 5\right )}}{\sqrt {2 \, x + 3}} \,d x } \]
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Timed out. \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=-\int \frac {\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2}}{\sqrt {2\,x+3}} \,d x \]
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