Integrand size = 29, antiderivative size = 205 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{3/2}} \, dx=-\frac {(21871-471213 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{24948}+\frac {5 \sqrt {3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}+\frac {451331 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{7128 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {4145485 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{49896 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 205, 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 = {826, 828, 857, 732, 435, 430} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{3/2}} \, dx=-\frac {4145485 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{49896 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {451331 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{7128 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {(x+73) \left (3 x^2+5 x+2\right )^{5/2}}{11 \sqrt {2 x+3}}+\frac {5 \sqrt {2 x+3} (3031 x+218) \left (3 x^2+5 x+2\right )^{3/2}}{1386}-\frac {(21871-471213 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}{24948} \]
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Rule 430
Rule 435
Rule 732
Rule 826
Rule 828
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}-\frac {5}{22} \int \frac {(-361-433 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx \\ & = \frac {5 \sqrt {3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}+\frac {5 \int \frac {(43918+52357 x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx}{2772} \\ & = -\frac {(21871-471213 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{24948}+\frac {5 \sqrt {3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}-\frac {\int \frac {-2666233-3159317 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{49896} \\ & = -\frac {(21871-471213 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{24948}+\frac {5 \sqrt {3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}+\frac {451331 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{14256}-\frac {4145485 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{99792} \\ & = -\frac {(21871-471213 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{24948}+\frac {5 \sqrt {3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}+\frac {\left (451331 \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 )}{7128 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {\left (4145485 \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 )}{49896 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ & = -\frac {(21871-471213 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{24948}+\frac {5 \sqrt {3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac {(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt {3+2 x}}+\frac {451331 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{7128 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {4145485 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{49896 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Time = 30.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{3/2}} \, dx=\frac {3159317 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+2 \left (2657740+6998740 x+6834513 x^2+6935769 x^3+7163046 x^4+3305934 x^5+401436 x^6-183708 x^7-336013 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )}{149688 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.35 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {3+2 x}\, \left (-5511240 x^{7}+12043080 x^{6}+1479252 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-3159317 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+99178020 x^{5}+214891380 x^{4}+208073070 x^{3}-79303140 x^{2}-263935350 x -109826820\right )}{13471920 x^{3}+42661080 x^{2}+42661080 x +13471920}\) | \(156\) |
elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {325 \left (6 x^{2}+10 x +4\right )}{64 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {9 x^{4} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{22}+\frac {289 x^{3} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{132}+\frac {869 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{504}+\frac {43987 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{11088}-\frac {951389 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{199584}-\frac {2666233 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{748440 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {451331 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{106920 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(303\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.44 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{3/2}} \, dx=-\frac {12034829 \, \sqrt {6} {\left (2 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 56867706 \, \sqrt {6} {\left (2 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 108 \, {\left (20412 \, x^{5} - 78624 \, x^{4} - 249894 \, x^{3} - 326988 \, x^{2} - 59065 \, x + 610149\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{2694384 \, {\left (2 \, x + 3\right )}} \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{3/2}} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{2 x \sqrt {2 x + 3} + 3 \sqrt {2 x + 3}}\, dx \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{3/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{3/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{3/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{3/2}} \,d x \]
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