Integrand size = 22, antiderivative size = 88 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=\frac {455}{24} x \sqrt {2+3 x^2}+\frac {455}{72} x \left (2+3 x^2\right )^{3/2}+\frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}} \]
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Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {794, 201, 221} \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=\frac {455 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}}+\frac {1}{12} (4-x) \left (3 x^2+2\right )^{7/2}+\frac {91}{36} x \left (3 x^2+2\right )^{5/2}+\frac {455}{72} x \left (3 x^2+2\right )^{3/2}+\frac {455}{24} x \sqrt {3 x^2+2} \]
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Rule 201
Rule 221
Rule 794
Rubi steps \begin{align*} \text {integral}& = \frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {91}{6} \int \left (2+3 x^2\right )^{5/2} \, dx \\ & = \frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455}{18} \int \left (2+3 x^2\right )^{3/2} \, dx \\ & = \frac {455}{72} x \left (2+3 x^2\right )^{3/2}+\frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455}{12} \int \sqrt {2+3 x^2} \, dx \\ & = \frac {455}{24} x \sqrt {2+3 x^2}+\frac {455}{72} x \left (2+3 x^2\right )^{3/2}+\frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455}{12} \int \frac {1}{\sqrt {2+3 x^2}} \, dx \\ & = \frac {455}{24} x \sqrt {2+3 x^2}+\frac {455}{72} x \left (2+3 x^2\right )^{3/2}+\frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.92 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{24} \sqrt {2+3 x^2} \left (-64-985 x-288 x^2-1111 x^3-432 x^4-438 x^5-216 x^6+54 x^7\right )-\frac {455 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{12 \sqrt {3}} \]
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Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {\left (54 x^{7}-216 x^{6}-438 x^{5}-432 x^{4}-1111 x^{3}-288 x^{2}-985 x -64\right ) \sqrt {3 x^{2}+2}}{24}+\frac {455 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{36}\) | \(60\) |
default | \(\frac {91 x \left (3 x^{2}+2\right )^{\frac {5}{2}}}{36}+\frac {455 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{72}+\frac {455 x \sqrt {3 x^{2}+2}}{24}+\frac {455 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{36}+\frac {\left (3 x^{2}+2\right )^{\frac {7}{2}}}{3}-\frac {x \left (3 x^{2}+2\right )^{\frac {7}{2}}}{12}\) | \(73\) |
trager | \(\left (-\frac {9}{4} x^{7}+9 x^{6}+\frac {73}{4} x^{5}+18 x^{4}+\frac {1111}{24} x^{3}+12 x^{2}+\frac {985}{24} x +\frac {8}{3}\right ) \sqrt {3 x^{2}+2}-\frac {455 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{36}\) | \(77\) |
meijerg | \(-\frac {75 \sqrt {3}\, \left (-\frac {8 \sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \left (\frac {3}{8} x^{4}+\frac {13}{16} x^{2}+\frac {11}{16}\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{15}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{3}\right )}{2 \sqrt {\pi }}-\frac {35 \sqrt {2}\, \left (\frac {16 \sqrt {\pi }}{105}-\frac {8 \sqrt {\pi }\, \left (\frac {27}{4} x^{6}+\frac {27}{2} x^{4}+9 x^{2}+2\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{105}\right )}{2 \sqrt {\pi }}+\frac {10 \sqrt {3}\, \left (-\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (162 x^{6}+306 x^{4}+177 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{720}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{24}\right )}{3 \sqrt {\pi }}\) | \(163\) |
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Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.85 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{24} \, {\left (54 \, x^{7} - 216 \, x^{6} - 438 \, x^{5} - 432 \, x^{4} - 1111 \, x^{3} - 288 \, x^{2} - 985 \, x - 64\right )} \sqrt {3 \, x^{2} + 2} + \frac {455}{72} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]
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Time = 2.87 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.62 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=- \frac {9 x^{7} \sqrt {3 x^{2} + 2}}{4} + 9 x^{6} \sqrt {3 x^{2} + 2} + \frac {73 x^{5} \sqrt {3 x^{2} + 2}}{4} + 18 x^{4} \sqrt {3 x^{2} + 2} + \frac {1111 x^{3} \sqrt {3 x^{2} + 2}}{24} + 12 x^{2} \sqrt {3 x^{2} + 2} + \frac {985 x \sqrt {3 x^{2} + 2}}{24} + \frac {8 \sqrt {3 x^{2} + 2}}{3} + \frac {455 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{36} \]
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Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.82 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x + \frac {1}{3} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} + \frac {91}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {455}{72} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {455}{24} \, \sqrt {3 \, x^{2} + 2} x + \frac {455}{36} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{24} \, {\left ({\left ({\left ({\left (6 \, {\left ({\left (9 \, {\left (x - 4\right )} x - 73\right )} x - 72\right )} x - 1111\right )} x - 288\right )} x - 985\right )} x - 64\right )} \sqrt {3 \, x^{2} + 2} - \frac {455}{36} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]
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Time = 10.58 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=\frac {455\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{36}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {27\,x^7}{4}+27\,x^6+\frac {219\,x^5}{4}+54\,x^4+\frac {1111\,x^3}{8}+36\,x^2+\frac {985\,x}{8}+8\right )}{3} \]
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