Integrand size = 29, antiderivative size = 234 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\frac {5083 \sqrt {2+5 x+3 x^2}}{247500 \sqrt {3+2 x}}+\frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {5083 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{165000 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {9421 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{231000 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {824, 848, 857, 732, 435, 430} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\frac {9421 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{231000 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {5083 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{165000 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{6930 (2 x+3)^{9/2}}+\frac {(17833 x+21492) \sqrt {3 x^2+5 x+2}}{346500 (2 x+3)^{5/2}}-\frac {5083 \sqrt {3 x^2+5 x+2}}{247500 \sqrt {2 x+3}} \]
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Rule 430
Rule 435
Rule 732
Rule 824
Rule 848
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}-\frac {1}{286} \int \frac {(104+39 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx \\ & = \frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {\int \frac {(-4979-6357 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{7/2}} \, dx}{60060} \\ & = \frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}-\frac {\int \frac {319852+367419 x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx}{9009000} \\ & = -\frac {5083 \sqrt {2+5 x+3 x^2}}{247500 \sqrt {3+2 x}}+\frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {\int \frac {\frac {1162941}{2}+\frac {1387659 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{22522500} \\ & = -\frac {5083 \sqrt {2+5 x+3 x^2}}{247500 \sqrt {3+2 x}}+\frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {5083 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{330000}-\frac {9421 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{462000} \\ & = -\frac {5083 \sqrt {2+5 x+3 x^2}}{247500 \sqrt {3+2 x}}+\frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {\left (5083 \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 )}{165000 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {\left (9421 \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 )}{231000 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ & = -\frac {5083 \sqrt {2+5 x+3 x^2}}{247500 \sqrt {3+2 x}}+\frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {5083 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{165000 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {9421 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{231000 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Time = 31.42 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.99 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\frac {8 \left (2+5 x+3 x^2\right ) \left (11865789+41339721 x+54318160 x^2+33648370 x^3+12953760 x^4+6409516 x^5+2277184 x^6\right )-4 (3+2 x)^6 \left (71162 \left (2+5 x+3 x^2\right )+35581 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-7318 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )}{13860000 (3+2 x)^{13/2} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.41 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.46
method | result | size |
elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {25 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{4096 \left (x +\frac {3}{2}\right )^{7}}+\frac {1105 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{22528 \left (x +\frac {3}{2}\right )^{6}}-\frac {3929 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{25344 \left (x +\frac {3}{2}\right )^{5}}+\frac {487 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2016 \left (x +\frac {3}{2}\right )^{4}}-\frac {278249 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1478400 \left (x +\frac {3}{2}\right )^{3}}+\frac {704257 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{11088000 \left (x +\frac {3}{2}\right )^{2}}-\frac {5083 \left (6 x^{2}+10 x +4\right )}{495000 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {29819 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{17325000 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {5083 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{2475000 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(342\) |
default | \(\frac {1106304 F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x^{6} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-2277184 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{6} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+9956736 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{5} \sqrt {3+2 x}\, \sqrt {-20-30 x}\, \sqrt {3+3 x}-20494656 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{5} \sqrt {3+2 x}\, \sqrt {-20-30 x}\, \sqrt {3+3 x}+37337760 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-76854960 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+74675520 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-153709920 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-204946560 x^{8}+84009960 \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}-172923660 \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}-918434040 x^{7}+50405976 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}-103754196 \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}-2263896840 x^{6}+12601494 \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 )-25938549 \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 )-5355988260 x^{5}-10713115500 x^{4}-13887201090 x^{3}-10527968760 x^{2}-4260251610 x -711947340}{51975000 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {13}{2}}}\) | \(668\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.79 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\frac {139297 \, \sqrt {6} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 640458 \, \sqrt {6} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 36 \, {\left (2277184 \, x^{6} + 6409516 \, x^{5} + 12953760 \, x^{4} + 33648370 \, x^{3} + 54318160 \, x^{2} + 41339721 \, x + 11865789\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{62370000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]
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Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {15}{2}}} \,d x } \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {15}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{15/2}} \,d x \]
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