Integrand size = 24, antiderivative size = 82 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {26+41 x}{70 (3+2 x) \sqrt {2+3 x^2}}+\frac {19 \sqrt {2+3 x^2}}{1225 (3+2 x)}-\frac {632 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1225 \sqrt {35}} \]
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Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {837, 821, 739, 212} \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=-\frac {632 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1225 \sqrt {35}}+\frac {41 x+26}{70 (2 x+3) \sqrt {3 x^2+2}}+\frac {19 \sqrt {3 x^2+2}}{1225 (2 x+3)} \]
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Rule 212
Rule 739
Rule 821
Rule 837
Rubi steps \begin{align*} \text {integral}& = \frac {26+41 x}{70 (3+2 x) \sqrt {2+3 x^2}}-\frac {1}{210} \int \frac {-312-246 x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx \\ & = \frac {26+41 x}{70 (3+2 x) \sqrt {2+3 x^2}}+\frac {19 \sqrt {2+3 x^2}}{1225 (3+2 x)}+\frac {632 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{1225} \\ & = \frac {26+41 x}{70 (3+2 x) \sqrt {2+3 x^2}}+\frac {19 \sqrt {2+3 x^2}}{1225 (3+2 x)}-\frac {632 \text {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{1225} \\ & = \frac {26+41 x}{70 (3+2 x) \sqrt {2+3 x^2}}+\frac {19 \sqrt {2+3 x^2}}{1225 (3+2 x)}-\frac {632 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1225 \sqrt {35}} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.79 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {\frac {35 \left (986+1435 x+114 x^2\right )}{(3+2 x) \sqrt {2+3 x^2}}-1264 \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{85750} \]
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Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.73
method | result | size |
risch | \(\frac {114 x^{2}+1435 x +986}{2450 \left (3+2 x \right ) \sqrt {3 x^{2}+2}}-\frac {632 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{42875}\) | \(60\) |
default | \(\frac {316}{1225 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {57 x}{2450 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {632 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{42875}-\frac {13}{70 \left (x +\frac {3}{2}\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}\) | \(86\) |
trager | \(\frac {\left (114 x^{2}+1435 x +986\right ) \sqrt {3 x^{2}+2}}{14700 x^{3}+22050 x^{2}+9800 x +14700}+\frac {632 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{42875}\) | \(86\) |
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Time = 0.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.27 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {632 \, \sqrt {35} {\left (6 \, x^{3} + 9 \, x^{2} + 4 \, x + 6\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 35 \, {\left (114 \, x^{2} + 1435 \, x + 986\right )} \sqrt {3 \, x^{2} + 2}}{85750 \, {\left (6 \, x^{3} + 9 \, x^{2} + 4 \, x + 6\right )}} \]
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\[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=- \int \frac {x}{12 x^{4} \sqrt {3 x^{2} + 2} + 36 x^{3} \sqrt {3 x^{2} + 2} + 35 x^{2} \sqrt {3 x^{2} + 2} + 24 x \sqrt {3 x^{2} + 2} + 18 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {5}{12 x^{4} \sqrt {3 x^{2} + 2} + 36 x^{3} \sqrt {3 x^{2} + 2} + 35 x^{2} \sqrt {3 x^{2} + 2} + 24 x \sqrt {3 x^{2} + 2} + 18 \sqrt {3 x^{2} + 2}}\right )\, dx \]
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Time = 0.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {632}{42875} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {57 \, x}{2450 \, \sqrt {3 \, x^{2} + 2}} + \frac {316}{1225 \, \sqrt {3 \, x^{2} + 2}} - \frac {13}{35 \, {\left (2 \, \sqrt {3 \, x^{2} + 2} x + 3 \, \sqrt {3 \, x^{2} + 2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (67) = 134\).
Time = 0.31 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.05 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=-\frac {1}{85750} \, \sqrt {35} {\left (19 \, \sqrt {35} \sqrt {3} - 1264 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {\frac {\frac {1093}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} - \frac {1820}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {57}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2450 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3}} - \frac {632 \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right )}{42875 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \]
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Time = 0.12 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.91 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {632\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {632\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}+\frac {71\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4900\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {71\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4900\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {26\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1225\,\left (x+\frac {3}{2}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,199{}\mathrm {i}}{14700\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,199{}\mathrm {i}}{14700\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]
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