Integrand size = 24, antiderivative size = 126 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {15}{64} (859-267 x) \sqrt {2+3 x^2}+\frac {5 (178+29 x) \left (2+3 x^2\right )^{3/2}}{32 (3+2 x)}-\frac {(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac {43995}{128} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {12885}{128} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {827, 829, 858, 221, 739, 212} \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=-\frac {43995}{128} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {12885}{128} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}+\frac {5 (29 x+178) \left (3 x^2+2\right )^{3/2}}{32 (2 x+3)}+\frac {15}{64} (859-267 x) \sqrt {3 x^2+2} \]
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Rule 212
Rule 221
Rule 739
Rule 827
Rule 829
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\frac {(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac {5}{64} \int \frac {(16-348 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx \\ & = \frac {5 (178+29 x) \left (2+3 x^2\right )^{3/2}}{32 (3+2 x)}-\frac {(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}+\frac {5}{512} \int \frac {(2784-25632 x) \sqrt {2+3 x^2}}{3+2 x} \, dx \\ & = \frac {15}{64} (859-267 x) \sqrt {2+3 x^2}+\frac {5 (178+29 x) \left (2+3 x^2\right )^{3/2}}{32 (3+2 x)}-\frac {(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}+\frac {5 \int \frac {1056384-5068224 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{12288} \\ & = \frac {15}{64} (859-267 x) \sqrt {2+3 x^2}+\frac {5 (178+29 x) \left (2+3 x^2\right )^{3/2}}{32 (3+2 x)}-\frac {(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac {131985}{128} \int \frac {1}{\sqrt {2+3 x^2}} \, dx+\frac {450975}{128} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx \\ & = \frac {15}{64} (859-267 x) \sqrt {2+3 x^2}+\frac {5 (178+29 x) \left (2+3 x^2\right )^{3/2}}{32 (3+2 x)}-\frac {(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac {43995}{128} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {450975}{128} \text {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right ) \\ & = \frac {15}{64} (859-267 x) \sqrt {2+3 x^2}+\frac {5 (178+29 x) \left (2+3 x^2\right )^{3/2}}{32 (3+2 x)}-\frac {(29+2 x) \left (2+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac {43995}{128} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {12885}{128} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right ) \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.96 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {1}{128} \left (-\frac {2 \sqrt {2+3 x^2} \left (-126181-127403 x-19268 x^2+2826 x^3-696 x^4+72 x^5\right )}{(3+2 x)^2}+25770 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )+43995 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )\right ) \]
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Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.77
method | result | size |
risch | \(-\frac {216 x^{7}-2088 x^{6}+8622 x^{5}-59196 x^{4}-376557 x^{3}-417079 x^{2}-254806 x -252362}{64 \left (3+2 x \right )^{2} \sqrt {3 x^{2}+2}}-\frac {43995 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{128}-\frac {12885 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{128}\) | \(97\) |
trager | \(-\frac {\left (72 x^{5}-696 x^{4}+2826 x^{3}-19268 x^{2}-127403 x -126181\right ) \sqrt {3 x^{2}+2}}{64 \left (3+2 x \right )^{2}}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{2}-25825835\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-25825835\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-25825835\right )-30065 \sqrt {3 x^{2}+2}}{3+2 x}\right )}{128}+\frac {43995 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{128}\) | \(122\) |
default | \(\frac {421 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{4900 \left (x +\frac {3}{2}\right )}+\frac {2577 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4900}-\frac {807 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{224}-\frac {4005 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{64}-\frac {43995 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{128}+\frac {859 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{112}+\frac {12885 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{128}-\frac {12885 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{128}-\frac {1263 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4900}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{280 \left (x +\frac {3}{2}\right )^{2}}\) | \(185\) |
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Time = 0.31 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.16 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {43995 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 12885 \, \sqrt {35} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, {\left (72 \, x^{5} - 696 \, x^{4} + 2826 \, x^{3} - 19268 \, x^{2} - 127403 \, x - 126181\right )} \sqrt {3 \, x^{2} + 2}}{256 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \]
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Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.15 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {39}{280} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{70 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {807}{224} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {859}{112} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {421 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{280 \, {\left (2 \, x + 3\right )}} - \frac {4005}{64} \, \sqrt {3 \, x^{2} + 2} x - \frac {43995}{128} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {12885}{128} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {12885}{64} \, \sqrt {3 \, x^{2} + 2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (100) = 200\).
Time = 0.30 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.83 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=-\frac {1}{32} \, {\left (3 \, {\left ({\left (3 \, x - 38\right )} x + 225\right )} x - 4177\right )} \sqrt {3 \, x^{2} + 2} + \frac {43995}{128} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {12885}{128} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {35 \, {\left (11472 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 25829 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 57912 \, \sqrt {3} x + 8984 \, \sqrt {3} + 57912 \, \sqrt {3 \, x^{2} + 2}\right )}}{256 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \]
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Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.17 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx=\frac {12885\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{128}+\frac {4177\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{32}-\frac {43995\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{128}-\frac {12885\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{128}+\frac {57\,\sqrt {3}\,x^2\,\sqrt {x^2+\frac {2}{3}}}{16}-\frac {9\,\sqrt {3}\,x^3\,\sqrt {x^2+\frac {2}{3}}}{32}+\frac {39305\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{256\,\left (x+\frac {3}{2}\right )}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{512\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {675\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}}}{32} \]
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