Optimal. Leaf size=104 \[ \frac {3 (37+12 x) \sqrt {2+3 x^2}}{4 (3+2 x)}-\frac {(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {111}{8} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {1143 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{8 \sqrt {35}} \]
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Rubi [A]
time = 0.04, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {827, 858, 221,
739, 212} \begin {gather*} -\frac {(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}+\frac {3 (12 x+37) \sqrt {3 x^2+2}}{4 (2 x+3)}-\frac {1143 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{8 \sqrt {35}}-\frac {111}{8} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 221
Rule 739
Rule 827
Rule 858
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx &=-\frac {(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {3}{32} \int \frac {(16-192 x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx\\ &=\frac {3 (37+12 x) \sqrt {2+3 x^2}}{4 (3+2 x)}-\frac {(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}+\frac {3}{256} \int \frac {1536-7104 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=\frac {3 (37+12 x) \sqrt {2+3 x^2}}{4 (3+2 x)}-\frac {(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {333}{8} \int \frac {1}{\sqrt {2+3 x^2}} \, dx+\frac {1143}{8} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=\frac {3 (37+12 x) \sqrt {2+3 x^2}}{4 (3+2 x)}-\frac {(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {111}{8} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {1143}{8} \text {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=\frac {3 (37+12 x) \sqrt {2+3 x^2}}{4 (3+2 x)}-\frac {(8+x) \left (2+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {111}{8} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {1143 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{8 \sqrt {35}}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 113, normalized size = 1.09 \begin {gather*} -\frac {\sqrt {2+3 x^2} \left (-317-328 x-48 x^2+3 x^3\right )}{4 (3+2 x)^2}+\frac {1143 \tanh ^{-1}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{4 \sqrt {35}}+\frac {111}{8} \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 152, normalized size = 1.46
method | result | size |
risch | \(-\frac {9 x^{5}-144 x^{4}-978 x^{3}-1047 x^{2}-656 x -634}{4 \left (2 x +3\right )^{2} \sqrt {3 x^{2}+2}}-\frac {111 \arcsinh \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{8}-\frac {1143 \sqrt {35}\, \arctanh \left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{280}\) | \(87\) |
trager | \(-\frac {\left (3 x^{3}-48 x^{2}-328 x -317\right ) \sqrt {3 x^{2}+2}}{4 \left (2 x +3\right )^{2}}-\frac {1143 \RootOf \left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \RootOf \left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \RootOf \left (\textit {\_Z}^{2}-35\right )}{2 x +3}\right )}{280}+\frac {111 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{8}\) | \(111\) |
default | \(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{280 \left (x +\frac {3}{2}\right )^{2}}+\frac {187 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4900 \left (x +\frac {3}{2}\right )}+\frac {381 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{1225}-\frac {171 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{70}-\frac {111 \arcsinh \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{8}+\frac {1143 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{280}-\frac {1143 \sqrt {35}\, \arctanh \left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{280}-\frac {561 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{4900}\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 122, normalized size = 1.17 \begin {gather*} \frac {39}{280} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{70 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {171}{70} \, \sqrt {3 \, x^{2} + 2} x - \frac {111}{8} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {1143}{280} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {1143}{140} \, \sqrt {3 \, x^{2} + 2} + \frac {187 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{280 \, {\left (2 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.09, size = 136, normalized size = 1.31 \begin {gather*} \frac {3885 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 1143 \, \sqrt {35} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 140 \, {\left (3 \, x^{3} - 48 \, x^{2} - 328 \, x - 317\right )} \sqrt {3 \, x^{2} + 2}}{560 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {10 \sqrt {3 x^{2} + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac {2 x \sqrt {3 x^{2} + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx - \int \left (- \frac {15 x^{2} \sqrt {3 x^{2} + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 219 vs.
\(2 (82) = 164\).
time = 1.33, size = 219, normalized size = 2.11 \begin {gather*} -\frac {3}{16} \, \sqrt {3 \, x^{2} + 2} {\left (x - 19\right )} + \frac {111}{8} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {1143}{280} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {5 \, {\left (1452 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 3013 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 6528 \, \sqrt {3} x + 1048 \, \sqrt {3} + 6528 \, \sqrt {3 \, x^{2} + 2}\right )}}{64 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.82, size = 117, normalized size = 1.12 \begin {gather*} \frac {1143\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{280}+\frac {57\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{16}-\frac {111\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{8}-\frac {1143\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{280}+\frac {655\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{64\,\left (x+\frac {3}{2}\right )}-\frac {455\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{128\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {3\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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