Integrand size = 29, antiderivative size = 256 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {207851 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{6254820 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {1015187 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{8756748 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.13 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {846, 828, 857, 732, 435, 430} \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=\frac {1015187 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{8756748 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {207851 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{6254820 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {2}{45} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac {202}{351} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac {13318 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{11583}+\frac {\sqrt {2 x+3} (629153 x+534271) \left (3 x^2+5 x+2\right )^{3/2}}{243243}-\frac {\sqrt {2 x+3} (7817373 x+6006884) \sqrt {3 x^2+5 x+2}}{21891870} \]
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Rule 430
Rule 435
Rule 732
Rule 828
Rule 846
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {2}{45} \int (3+2 x)^{3/2} \left (385+\frac {505 x}{2}\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx \\ & = \frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {4 \int \sqrt {3+2 x} \left (\frac {46155}{4}+\frac {33295 x}{4}\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx}{1755} \\ & = \frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {8 \int \frac {\left (242380+\frac {1348185 x}{8}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx}{57915} \\ & = \frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {4 \int \frac {\left (\frac {17298645}{8}+\frac {13028955 x}{8}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx}{3648645} \\ & = -\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {2 \int \frac {\frac {1333245}{2}-\frac {21824355 x}{8}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{164189025} \\ & = -\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {207851 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{12509640}+\frac {1015187 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{17513496} \\ & = -\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {\left (207851 \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 )}{6254820 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (1015187 \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 )}{8756748 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ & = -\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {207851 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{6254820 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {1015187 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{8756748 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Time = 31.29 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.85 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {2 \sqrt {3+2 x} \left (-5523159638-44206631441 x-150475882830 x^2-283276026729 x^3-319887585072 x^4-217661096106 x^5-82311172272 x^6-11776907520 x^7+1907623872 x^8+630485856 x^9\right )+1454957 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+1590604 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{131351220 (3+2 x) \sqrt {2+5 x+3 x^2}} \]
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Time = 0.39 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.65
method | result | size |
default | \(-\frac {\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}\, \left (18914575680 x^{9}+57228716160 x^{8}-353307225600 x^{7}-2469335168160 x^{6}-6529832883180 x^{5}+5431467 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-1454957 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-9596627552160 x^{4}-8498280801870 x^{3}-4514407431030 x^{2}-1326417186780 x -165782086560\right )}{1970268300 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\) | \(166\) |
risch | \(-\frac {\left (35026992 x^{6}-4939704 x^{5}-749549052 x^{4}-2218655502 x^{3}-2688028992 x^{2}-1484149221 x -307003864\right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{21891870}-\frac {\left (-\frac {88883 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{164189025 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {207851 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {E\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{93822300 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(223\) |
elliptic | \(\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {3+2 x}\, \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {8 x^{6} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5}+\frac {44 \sqrt {6 x^{3}+19 x^{2}+19 x +6}\, x^{5}}{195}+\frac {73442 x^{4} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2145}+\frac {5869459 x^{3} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{57915}+\frac {13575904 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{110565}+\frac {164905469 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2432430}+\frac {153501932 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{10945935}-\frac {88883 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{164189025 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {207851 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{93822300 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{6 x^{3}+19 x^{2}+19 x +6}\) | \(336\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.30 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{21891870} \, {\left (35026992 \, x^{6} - 4939704 \, x^{5} - 749549052 \, x^{4} - 2218655502 \, x^{3} - 2688028992 \, x^{2} - 1484149221 \, x - 307003864\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {34043759}{2364321960} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {207851}{18764460} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]
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\[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=- \int \left (- 90 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 327 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 406 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 185 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 4 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 12 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
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\[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=\int { -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (2 \, x + 3\right )}^{\frac {5}{2}} {\left (x - 5\right )} \,d x } \]
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\[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=\int { -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (2 \, x + 3\right )}^{\frac {5}{2}} {\left (x - 5\right )} \,d x } \]
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Timed out. \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\int {\left (2\,x+3\right )}^{5/2}\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \]
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