Integrand size = 24, antiderivative size = 106 \[ \int \frac {(5-x) (3+2 x)^4}{\sqrt {2+3 x^2}} \, dx=\frac {1477}{270} (3+2 x)^2 \sqrt {2+3 x^2}+\frac {19}{30} (3+2 x)^3 \sqrt {2+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+3 x^2}+\frac {49}{81} (383+99 x) \sqrt {2+3 x^2}+\frac {343 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]
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Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {847, 794, 221} \[ \int \frac {(5-x) (3+2 x)^4}{\sqrt {2+3 x^2}} \, dx=\frac {343 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}-\frac {1}{15} \sqrt {3 x^2+2} (2 x+3)^4+\frac {19}{30} \sqrt {3 x^2+2} (2 x+3)^3+\frac {1477}{270} \sqrt {3 x^2+2} (2 x+3)^2+\frac {49}{81} (99 x+383) \sqrt {3 x^2+2} \]
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Rule 221
Rule 794
Rule 847
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{15} (3+2 x)^4 \sqrt {2+3 x^2}+\frac {1}{15} \int \frac {(3+2 x)^3 (241+114 x)}{\sqrt {2+3 x^2}} \, dx \\ & = \frac {19}{30} (3+2 x)^3 \sqrt {2+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+3 x^2}+\frac {1}{180} \int \frac {(3+2 x)^2 (7308+8862 x)}{\sqrt {2+3 x^2}} \, dx \\ & = \frac {1477}{270} (3+2 x)^2 \sqrt {2+3 x^2}+\frac {19}{30} (3+2 x)^3 \sqrt {2+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+3 x^2}+\frac {\int \frac {(3+2 x) (126420+291060 x)}{\sqrt {2+3 x^2}} \, dx}{1620} \\ & = \frac {1477}{270} (3+2 x)^2 \sqrt {2+3 x^2}+\frac {19}{30} (3+2 x)^3 \sqrt {2+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+3 x^2}+\frac {49}{81} (383+99 x) \sqrt {2+3 x^2}+\frac {343}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx \\ & = \frac {1477}{270} (3+2 x)^2 \sqrt {2+3 x^2}+\frac {19}{30} (3+2 x)^3 \sqrt {2+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+3 x^2}+\frac {49}{81} (383+99 x) \sqrt {2+3 x^2}+\frac {343 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.62 \[ \int \frac {(5-x) (3+2 x)^4}{\sqrt {2+3 x^2}} \, dx=-\frac {1}{405} \sqrt {2+3 x^2} \left (-118513-58860 x-12264 x^2+540 x^3+432 x^4\right )-\frac {343 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{3 \sqrt {3}} \]
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Time = 0.32 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.42
method | result | size |
risch | \(-\frac {\left (432 x^{4}+540 x^{3}-12264 x^{2}-58860 x -118513\right ) \sqrt {3 x^{2}+2}}{405}+\frac {343 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{9}\) | \(45\) |
trager | \(\left (-\frac {16}{15} x^{4}-\frac {4}{3} x^{3}+\frac {4088}{135} x^{2}+\frac {436}{3} x +\frac {118513}{405}\right ) \sqrt {3 x^{2}+2}+\frac {343 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{9}\) | \(61\) |
default | \(\frac {343 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{9}+\frac {118513 \sqrt {3 x^{2}+2}}{405}-\frac {16 x^{4} \sqrt {3 x^{2}+2}}{15}+\frac {4088 x^{2} \sqrt {3 x^{2}+2}}{135}-\frac {4 x^{3} \sqrt {3 x^{2}+2}}{3}+\frac {436 x \sqrt {3 x^{2}+2}}{3}\) | \(79\) |
meijerg | \(135 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )+\frac {333 \sqrt {2}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {\frac {3 x^{2}}{2}+1}\right )}{2 \sqrt {\pi }}+\frac {96 \sqrt {3}\, \left (\frac {\sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \sqrt {\frac {3 x^{2}}{2}+1}}{2}-\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )\right )}{\sqrt {\pi }}+\frac {88 \sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-6 x^{2}+8\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{6}\right )}{3 \sqrt {\pi }}-\frac {32 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \left (-15 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{40}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{4}\right )}{27 \sqrt {\pi }}-\frac {32 \sqrt {2}\, \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (\frac {27}{2} x^{4}-12 x^{2}+16\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{15}\right )}{27 \sqrt {\pi }}\) | \(217\) |
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Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.57 \[ \int \frac {(5-x) (3+2 x)^4}{\sqrt {2+3 x^2}} \, dx=-\frac {1}{405} \, {\left (432 \, x^{4} + 540 \, x^{3} - 12264 \, x^{2} - 58860 \, x - 118513\right )} \sqrt {3 \, x^{2} + 2} + \frac {343}{18} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]
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Time = 0.33 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92 \[ \int \frac {(5-x) (3+2 x)^4}{\sqrt {2+3 x^2}} \, dx=- \frac {16 x^{4} \sqrt {3 x^{2} + 2}}{15} - \frac {4 x^{3} \sqrt {3 x^{2} + 2}}{3} + \frac {4088 x^{2} \sqrt {3 x^{2} + 2}}{135} + \frac {436 x \sqrt {3 x^{2} + 2}}{3} + \frac {118513 \sqrt {3 x^{2} + 2}}{405} + \frac {343 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{9} \]
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Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.74 \[ \int \frac {(5-x) (3+2 x)^4}{\sqrt {2+3 x^2}} \, dx=-\frac {16}{15} \, \sqrt {3 \, x^{2} + 2} x^{4} - \frac {4}{3} \, \sqrt {3 \, x^{2} + 2} x^{3} + \frac {4088}{135} \, \sqrt {3 \, x^{2} + 2} x^{2} + \frac {436}{3} \, \sqrt {3 \, x^{2} + 2} x + \frac {343}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {118513}{405} \, \sqrt {3 \, x^{2} + 2} \]
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Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.50 \[ \int \frac {(5-x) (3+2 x)^4}{\sqrt {2+3 x^2}} \, dx=-\frac {1}{405} \, {\left (12 \, {\left ({\left (9 \, {\left (4 \, x + 5\right )} x - 1022\right )} x - 4905\right )} x - 118513\right )} \sqrt {3 \, x^{2} + 2} - \frac {343}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.42 \[ \int \frac {(5-x) (3+2 x)^4}{\sqrt {2+3 x^2}} \, dx=\frac {343\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{9}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {16\,x^4}{5}-4\,x^3+\frac {4088\,x^2}{45}+436\,x+\frac {118513}{135}\right )}{3} \]
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