Optimal. Leaf size=110 \[ \frac {665}{12} x \sqrt {2+3 x^2}+\frac {665}{36} x \left (2+3 x^2\right )^{3/2}+\frac {133}{18} x \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac {665 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {847, 794, 201,
221} \begin {gather*} -\frac {1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}+\frac {1}{81} (63 x+226) \left (3 x^2+2\right )^{7/2}+\frac {133}{18} x \left (3 x^2+2\right )^{5/2}+\frac {665}{36} x \left (3 x^2+2\right )^{3/2}+\frac {665}{12} x \sqrt {3 x^2+2}+\frac {665 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rule 794
Rule 847
Rubi steps
\begin {align*} \int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{5/2} \, dx &=-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{27} \int (3+2 x) (413+252 x) \left (2+3 x^2\right )^{5/2} \, dx\\ &=-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac {133}{3} \int \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac {133}{18} x \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac {665}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {665}{36} x \left (2+3 x^2\right )^{3/2}+\frac {133}{18} x \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac {665}{6} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {665}{12} x \sqrt {2+3 x^2}+\frac {665}{36} x \left (2+3 x^2\right )^{3/2}+\frac {133}{18} x \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac {665}{6} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {665}{12} x \sqrt {2+3 x^2}+\frac {665}{36} x \left (2+3 x^2\right )^{3/2}+\frac {133}{18} x \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac {665 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 86, normalized size = 0.78 \begin {gather*} -\frac {1}{324} \sqrt {2+3 x^2} \left (-6368-40365 x-28272 x^2-50571 x^3-41256 x^4-27378 x^5-18900 x^6-2916 x^7+1296 x^8\right )-\frac {665 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{6 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.61, size = 87, normalized size = 0.79
method | result | size |
risch | \(-\frac {\left (1296 x^{8}-2916 x^{7}-18900 x^{6}-27378 x^{5}-41256 x^{4}-50571 x^{3}-28272 x^{2}-40365 x -6368\right ) \sqrt {3 x^{2}+2}}{324}+\frac {665 \arcsinh \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{18}\) | \(65\) |
trager | \(\left (-4 x^{8}+9 x^{7}+\frac {175}{3} x^{6}+\frac {169}{2} x^{5}+\frac {382}{3} x^{4}+\frac {1873}{12} x^{3}+\frac {2356}{27} x^{2}+\frac {1495}{12} x +\frac {1592}{81}\right ) \sqrt {3 x^{2}+2}-\frac {665 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{18}\) | \(82\) |
default | \(-\frac {4 x^{2} \left (3 x^{2}+2\right )^{\frac {7}{2}}}{27}+\frac {199 \left (3 x^{2}+2\right )^{\frac {7}{2}}}{81}+\frac {x \left (3 x^{2}+2\right )^{\frac {7}{2}}}{3}+\frac {133 x \left (3 x^{2}+2\right )^{\frac {5}{2}}}{18}+\frac {665 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{36}+\frac {665 x \sqrt {3 x^{2}+2}}{12}+\frac {665 \arcsinh \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{18}\) | \(87\) |
meijerg | \(-\frac {225 \sqrt {3}\, \left (-\frac {8 \sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (\frac {3}{8} x^{4}+\frac {13}{16} x^{2}+\frac {11}{16}\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{15}-\frac {\sqrt {\pi }\, \arcsinh \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{3}\right )}{2 \sqrt {\pi }}-\frac {40 \sqrt {3}\, \left (-\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (162 x^{6}+306 x^{4}+177 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{720}+\frac {\sqrt {\pi }\, \arcsinh \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{24}\right )}{3 \sqrt {\pi }}-\frac {255 \sqrt {2}\, \left (\frac {16 \sqrt {\pi }}{105}-\frac {8 \sqrt {\pi }\, \left (\frac {27}{4} x^{6}+\frac {27}{2} x^{4}+9 x^{2}+2\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{105}\right )}{2 \sqrt {\pi }}+\frac {20 \sqrt {2}\, \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-\frac {567}{4} x^{8}-\frac {513}{2} x^{6}-135 x^{4}-6 x^{2}+8\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{945}\right )}{3 \sqrt {\pi }}\) | \(213\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 86, normalized size = 0.78 \begin {gather*} -\frac {4}{27} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{2} + \frac {1}{3} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x + \frac {199}{81} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} + \frac {133}{18} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {665}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {665}{12} \, \sqrt {3 \, x^{2} + 2} x + \frac {665}{18} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.71, size = 80, normalized size = 0.73 \begin {gather*} -\frac {1}{324} \, {\left (1296 \, x^{8} - 2916 \, x^{7} - 18900 \, x^{6} - 27378 \, x^{5} - 41256 \, x^{4} - 50571 \, x^{3} - 28272 \, x^{2} - 40365 \, x - 6368\right )} \sqrt {3 \, x^{2} + 2} + \frac {665}{36} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.40, size = 162, normalized size = 1.47 \begin {gather*} - 4 x^{8} \sqrt {3 x^{2} + 2} + 9 x^{7} \sqrt {3 x^{2} + 2} + \frac {175 x^{6} \sqrt {3 x^{2} + 2}}{3} + \frac {169 x^{5} \sqrt {3 x^{2} + 2}}{2} + \frac {382 x^{4} \sqrt {3 x^{2} + 2}}{3} + \frac {1873 x^{3} \sqrt {3 x^{2} + 2}}{12} + \frac {2356 x^{2} \sqrt {3 x^{2} + 2}}{27} + \frac {1495 x \sqrt {3 x^{2} + 2}}{12} + \frac {1592 \sqrt {3 x^{2} + 2}}{81} + \frac {665 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.20, size = 72, normalized size = 0.65 \begin {gather*} -\frac {1}{324} \, {\left (3 \, {\left ({\left (9 \, {\left (2 \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 9\right )} x - 175\right )} x - 507\right )} x - 764\right )} x - 1873\right )} x - 9424\right )} x - 13455\right )} x - 6368\right )} \sqrt {3 \, x^{2} + 2} - \frac {665}{18} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 65, normalized size = 0.59 \begin {gather*} \frac {665\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{18}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-12\,x^8+27\,x^7+175\,x^6+\frac {507\,x^5}{2}+382\,x^4+\frac {1873\,x^3}{4}+\frac {2356\,x^2}{9}+\frac {1495\,x}{4}+\frac {1592}{27}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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