Integrand size = 29, antiderivative size = 202 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=-\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {23 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{7500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {7 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{1500 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.09 (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 = {824, 848, 857, 732, 435, 430} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\frac {7 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{1500 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {23 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{7500 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}-\frac {(211 x+189) \sqrt {3 x^2+5 x+2}}{2250 (2 x+3)^{5/2}}-\frac {23 \sqrt {3 x^2+5 x+2}}{11250 \sqrt {2 x+3}} \]
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Rule 430
Rule 435
Rule 732
Rule 824
Rule 848
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}-\frac {1}{210} \int \frac {(28-21 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{7/2}} \, dx \\ & = -\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {\int \frac {301+147 x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx}{31500} \\ & = -\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}-\frac {\int \frac {-546-\frac {483 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{78750} \\ & = -\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {23 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{15000}+\frac {7 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{3000} \\ & = -\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {\left (23 \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 )}{7500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (7 \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 )}{1500 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ & = -\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {23 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{7500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {7 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{1500 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Time = 31.39 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.95 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\frac {53980+373610 x+998860 x^2+1297210 x^3+822160 x^4+204180 x^5+23 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{11/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-44 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{11/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{22500 (3+2 x)^{9/2} \sqrt {2+5 x+3 x^2}} \]
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Time = 0.38 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.46
method | result | size |
elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {65 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2304 \left (x +\frac {3}{2}\right )^{5}}+\frac {155 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1152 \left (x +\frac {3}{2}\right )^{4}}-\frac {971 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{4800 \left (x +\frac {3}{2}\right )^{3}}+\frac {3403 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{36000 \left (x +\frac {3}{2}\right )^{2}}-\frac {23 \left (6 x^{2}+10 x +4\right )}{22500 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {13 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{28125 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {23 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{112500 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(294\) |
default | \(-\frac {1392 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+368 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+8352 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+2208 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+18792 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+4968 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+18792 F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+4968 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+33120 x^{6}+7047 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+1863 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-2808780 x^{5}-11532000 x^{4}-18133350 x^{3}-13771950 x^{2}-5026620 x -697920}{337500 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {9}{2}}}\) | \(482\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.72 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\frac {499 \, \sqrt {6} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) - 414 \, \sqrt {6} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 36 \, {\left (368 \, x^{4} - 31822 \, x^{3} - 75342 \, x^{2} - 54697 \, x - 11632\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{405000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\, dx \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {11}{2}}} \,d x } \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {11}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^{11/2}} \,d x \]
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