Integrand size = 29, antiderivative size = 202 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {110516 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {148780 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {832, 846, 857, 732, 435, 430} \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {148780 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {110516 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {2 (139 x+121) (2 x+3)^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 (2571 x+2164) (2 x+3)^{3/2}}{9 \sqrt {3 x^2+5 x+2}}-\frac {59512}{81} \sqrt {3 x^2+5 x+2} \sqrt {2 x+3} \]
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Rule 430
Rule 435
Rule 732
Rule 832
Rule 846
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {(3+2 x)^{5/2} (4+411 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx \\ & = -\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}+\frac {4}{27} \int \frac {(-18243-22317 x) \sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}+\frac {8}{243} \int \frac {-34269-\frac {82887 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {55258}{81} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {74390}{81} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {\left (110516 \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 )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (148780 \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 )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ & = -\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {110516 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {148780 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Time = 31.42 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.09 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 \left (3 (3+2 x) \left (-85285-330053 x-411640 x^2-166566 x^3+144 x^4\right )+2 \left (2+5 x+3 x^2\right ) \left (55258 \left (2+5 x+3 x^2\right )+27629 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-5312 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )\right )}{243 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \]
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Time = 0.39 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.24
method | result | size |
elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (\frac {\left (-\frac {23194}{2187}-\frac {24166 x}{2187}\right ) \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 \left (9+6 x \right ) \left (-\frac {133822}{729}-\frac {55682 x}{243}\right )}{\sqrt {\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right ) \left (9+6 x \right )}}-\frac {32 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{81}+\frac {91384 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{1215 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {110516 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{1215 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(250\) |
default | \(-\frac {2 \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}\, \left (86094 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-165774 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+143490 F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-276290 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+57396 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-110516 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+12960 x^{5}-14971500 x^{4}-59534010 x^{3}-85276170 x^{2}-52232805 x -11513475\right )}{3645 \left (1+x \right ) \left (2 x^{2}+5 x +3\right ) \left (2+3 x \right )^{2}}\) | \(325\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.65 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (113723 \, \sqrt {6} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 497322 \, \sqrt {6} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 27 \, {\left (144 \, x^{4} - 166566 \, x^{3} - 411640 \, x^{2} - 330053 \, x - 85285\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}\right )}}{2187 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
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Timed out. \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (2 \, x + 3\right )}^{\frac {9}{2}} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (2 \, x + 3\right )}^{\frac {9}{2}} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {{\left (2\,x+3\right )}^{9/2}\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]
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