Integrand size = 29, antiderivative size = 207 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=-\frac {(10763+3117 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}+\frac {(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}+\frac {2333 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{40 \sqrt {2+5 x+3 x^2}}-\frac {12857 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{56 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {826, 857, 732, 435, 430} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=-\frac {12857 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{56 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {2333 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{40 \sqrt {3 x^2+5 x+2}}-\frac {(5 x+53) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{5/2}}+\frac {(879 x+2291) \left (3 x^2+5 x+2\right )^{3/2}}{210 (2 x+3)^{3/2}}-\frac {(3117 x+10763) \sqrt {3 x^2+5 x+2}}{140 \sqrt {2 x+3}} \]
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Rule 430
Rule 435
Rule 732
Rule 826
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}-\frac {1}{14} \int \frac {(-245-293 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{5/2}} \, dx \\ & = \frac {(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}+\frac {1}{140} \int \frac {(-7939-9351 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{3/2}} \, dx \\ & = -\frac {(10763+3117 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}+\frac {(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}-\frac {1}{840} \int \frac {-124041-146979 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {(10763+3117 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}+\frac {(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}+\frac {6999}{80} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx-\frac {12857}{112} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {(10763+3117 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}+\frac {(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}-\frac {\left (12857 \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 )}{56 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (2333 \sqrt {3} \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 )}{40 \sqrt {2+5 x+3 x^2}} \\ & = -\frac {(10763+3117 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}+\frac {(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}+\frac {2333 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{40 \sqrt {2+5 x+3 x^2}}-\frac {12857 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{56 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Time = 31.60 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.98 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=\frac {701136+2556580 x+3262382 x^2+1717690 x^3+335988 x^4+41220 x^5+12744 x^6-3240 x^7+48993 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{7/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-10422 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{7/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{840 (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.34 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.48
method | result | size |
elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {65 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{256 \left (x +\frac {3}{2}\right )^{3}}+\frac {1169 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{384 \left (x +\frac {3}{2}\right )^{2}}-\frac {6889 \left (6 x^{2}+10 x +4\right )}{240 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {9 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{56}+\frac {909 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{560}-\frac {1559 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{280}-\frac {41347 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{4200 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {2333 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{200 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(307\) |
default | \(\frac {30584 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-65324 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-16200 x^{7}+91752 F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-195972 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+63720 x^{6}+68814 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-146979 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+206100 x^{5}-4199220 x^{4}-18847630 x^{3}-30231440 x^{2}-21022270 x -5313060}{4200 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {5}{2}}}\) | \(311\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.58 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=-\frac {62207 \, \sqrt {6} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 293958 \, \sqrt {6} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 12 \, {\left (540 \, x^{5} - 3024 \, x^{4} - 2190 \, x^{3} + 145640 \, x^{2} + 386981 \, x + 265653\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{5040 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\, dx \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{7/2}} \,d x \]
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