Integrand size = 24, antiderivative size = 117 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=-\frac {7}{16} (775-243 x) \sqrt {2+3 x^2}-\frac {1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {18543}{32} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )+\frac {5425}{32} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 117, 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} \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\frac {18543}{32} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )+\frac {5425}{32} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}-\frac {7}{16} (775-243 x) \sqrt {3 x^2+2} \]
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Rule 212
Rule 221
Rule 739
Rule 827
Rule 829
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac {1}{8} \int \frac {(8-408 x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx \\ & = -\frac {1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac {1}{384} \int \frac {(15456-163296 x) \sqrt {2+3 x^2}}{3+2 x} \, dx \\ & = -\frac {7}{16} (775-243 x) \sqrt {2+3 x^2}-\frac {1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac {\int \frac {6620544-32042304 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{9216} \\ & = -\frac {7}{16} (775-243 x) \sqrt {2+3 x^2}-\frac {1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {55629}{32} \int \frac {1}{\sqrt {2+3 x^2}} \, dx-\frac {189875}{32} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx \\ & = -\frac {7}{16} (775-243 x) \sqrt {2+3 x^2}-\frac {1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {18543}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+\frac {189875}{32} \text {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right ) \\ & = -\frac {7}{16} (775-243 x) \sqrt {2+3 x^2}-\frac {1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {18543}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+\frac {5425}{32} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right ) \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\frac {1}{480} \left (-\frac {2 \sqrt {2+3 x^2} \left (265989+89521 x-19458 x^2+5118 x^3-1836 x^4+216 x^5\right )}{3+2 x}-162750 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )-278145 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )\right ) \]
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Time = 0.33 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-\frac {648 x^{7}-5508 x^{6}+15786 x^{5}-62046 x^{4}+278799 x^{3}+759051 x^{2}+179042 x +531978}{240 \left (3+2 x \right ) \sqrt {3 x^{2}+2}}+\frac {18543 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{32}+\frac {5425 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{32}\) | \(97\) |
trager | \(-\frac {\left (216 x^{5}-1836 x^{4}+5118 x^{3}-19458 x^{2}+89521 x +265989\right ) \sqrt {3 x^{2}+2}}{240 \left (3+2 x \right )}+\frac {5425 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{32}-\frac {21 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2339067\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2339067\right ) \sqrt {3 x^{2}+2}+2649 x \right )}{32}\) | \(121\) |
default | \(-\frac {31 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{35}+\frac {51 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{8}+\frac {1701 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{16}+\frac {18543 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{32}-\frac {155 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{12}-\frac {5425 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{32}+\frac {5425 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{32}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{70 \left (x +\frac {3}{2}\right )}+\frac {39 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{70}\) | \(164\) |
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Time = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.12 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\frac {278145 \, \sqrt {3} {\left (2 \, x + 3\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 81375 \, \sqrt {35} {\left (2 \, x + 3\right )} \log \left (\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} - 93 \, x^{2} + 36 \, x - 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, {\left (216 \, x^{5} - 1836 \, x^{4} + 5118 \, x^{3} - 19458 \, x^{2} + 89521 \, x + 265989\right )} \sqrt {3 \, x^{2} + 2}}{960 \, {\left (2 \, x + 3\right )}} \]
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Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.04 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=-\frac {1}{20} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} + \frac {51}{8} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x - \frac {155}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{4 \, {\left (2 \, x + 3\right )}} + \frac {1701}{16} \, \sqrt {3 \, x^{2} + 2} x + \frac {18543}{32} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {5425}{32} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {5425}{16} \, \sqrt {3 \, x^{2} + 2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 665 vs. \(2 (91) = 182\).
Time = 0.50 (sec) , antiderivative size = 665, normalized size of antiderivative = 5.68 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\frac {5425}{32} \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {18543}{32} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {35}}{2 \, x + 3} \right |}}{2 \, {\left (\sqrt {3} + \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {15925}{128} \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {9 \, {\left (238455 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{9} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 149045 \, \sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{8} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 697600 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{7} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 719040 \, \sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{6} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 4150566 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 2707250 \, \sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 6756120 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 4557000 \, \sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 3563595 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 2833425 \, \sqrt {35} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{320 \, {\left ({\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{2} - 3\right )}^{5}} \]
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Time = 0.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.18 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\frac {18543\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{32}-\frac {275027\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{960}-\frac {5425\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{32}+\frac {5425\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{32}-\frac {1393\,\sqrt {3}\,x^2\,\sqrt {x^2+\frac {2}{3}}}{80}+\frac {9\,\sqrt {3}\,x^3\,\sqrt {x^2+\frac {2}{3}}}{2}-\frac {9\,\sqrt {3}\,x^4\,\sqrt {x^2+\frac {2}{3}}}{20}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{128\,\left (x+\frac {3}{2}\right )}+\frac {2133\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}}}{32} \]
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