Integrand size = 25, antiderivative size = 233 \[ \int \frac {(2-5 x) x^{11/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {33608 \sqrt {x} (2+3 x)}{729 \sqrt {2+5 x+3 x^2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {16040}{243} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {2348}{27} x^{3/2} \sqrt {2+5 x+3 x^2}-\frac {33608 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{729 \sqrt {2+5 x+3 x^2}}+\frac {16040 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{243 \sqrt {2+5 x+3 x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {832, 846, 853, 1203, 1114, 1150} \[ \int \frac {(2-5 x) x^{11/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {16040 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{243 \sqrt {3 x^2+5 x+2}}-\frac {33608 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{729 \sqrt {3 x^2+5 x+2}}-\frac {16040}{243} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {33608 (3 x+2) \sqrt {x}}{729 \sqrt {3 x^2+5 x+2}}+\frac {2 (95 x+74) x^{9/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {8 (905 x+773) x^{5/2}}{27 \sqrt {3 x^2+5 x+2}}+\frac {2348}{27} \sqrt {3 x^2+5 x+2} x^{3/2} \]
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Rule 832
Rule 846
Rule 853
Rule 1114
Rule 1150
Rule 1203
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {(-333-245 x) x^{7/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx \\ & = \frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {4}{27} \int \frac {x^{3/2} \left (3865+\frac {8805 x}{2}\right )}{\sqrt {2+5 x+3 x^2}} \, dx \\ & = \frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {2348}{27} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {8}{405} \int \frac {\left (-\frac {26415}{2}-\frac {30075 x}{2}\right ) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx \\ & = \frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {16040}{243} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {2348}{27} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {16 \int \frac {\frac {30075}{2}+\frac {63015 x}{4}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{3645} \\ & = \frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {16040}{243} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {2348}{27} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {32 \text {Subst}\left (\int \frac {\frac {30075}{2}+\frac {63015 x^2}{4}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{3645} \\ & = \frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {16040}{243} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {2348}{27} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {32080}{243} \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {33608}{243} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 x^{9/2} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {33608 \sqrt {x} (2+3 x)}{729 \sqrt {2+5 x+3 x^2}}-\frac {8 x^{5/2} (773+905 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {16040}{243} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {2348}{27} x^{3/2} \sqrt {2+5 x+3 x^2}-\frac {33608 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{729 \sqrt {2+5 x+3 x^2}}+\frac {16040 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{243 \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 21.23 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.77 \[ \int \frac {(2-5 x) x^{11/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {134432+479680 x+534680 x^2+161784 x^3-21276 x^4+2484 x^5-486 x^6+33608 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+14512 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{729 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}} \]
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Time = 0.21 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.08
method | result | size |
elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (\frac {\left (\frac {7828}{6561}+\frac {11230 x}{6561}\right ) \sqrt {3 x^{3}+5 x^{2}+2 x}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 x \left (\frac {50533}{2187}+\frac {18635 x}{729}\right ) \sqrt {3}}{\sqrt {x \left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )}}-\frac {2 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{27}+\frac {152 \sqrt {3 x^{3}+5 x^{2}+2 x}}{243}+\frac {16040 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{729 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {16804 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{729 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(251\) |
default | \(-\frac {2 \left (3438 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}-25206 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}+5730 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x -42010 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x +729 x^{6}+2292 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-16804 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-3726 x^{5}+485622 x^{4}+1269684 x^{3}+1063224 x^{2}+288720 x \right ) \sqrt {3 x^{2}+5 x +2}}{2187 \sqrt {x}\, \left (2+3 x \right )^{2} \left (1+x \right )^{2}}\) | \(307\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.57 \[ \int \frac {(2-5 x) x^{11/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (60340 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 151236 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 27 \, {\left (81 \, x^{5} - 414 \, x^{4} + 53958 \, x^{3} + 141076 \, x^{2} + 118136 \, x + 32080\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{6561 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
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Timed out. \[ \int \frac {(2-5 x) x^{11/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(2-5 x) x^{11/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (5 \, x - 2\right )} x^{\frac {11}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(2-5 x) x^{11/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (5 \, x - 2\right )} x^{\frac {11}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(2-5 x) x^{11/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {x^{11/2}\,\left (5\,x-2\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]
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