Integrand size = 27, antiderivative size = 169 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac {25458 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}+\frac {82039 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{2500 \sqrt {5}} \]
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Time = 0.08 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {836, 848, 820, 738, 212} \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {82039 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2500 \sqrt {5}}-\frac {6 (47 x+37)}{5 (2 x+3)^4 \sqrt {3 x^2+5 x+2}}-\frac {25458 \sqrt {3 x^2+5 x+2}}{625 (2 x+3)}-\frac {973 \sqrt {3 x^2+5 x+2}}{30 (2 x+3)^2}-\frac {11596 \sqrt {3 x^2+5 x+2}}{375 (2 x+3)^3}-\frac {817 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^4} \]
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Rule 212
Rule 738
Rule 820
Rule 836
Rule 848
Rubi steps \begin{align*} \text {integral}& = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {2}{5} \int \frac {875+1128 x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}+\frac {1}{50} \int \frac {-10463-14706 x}{(3+2 x)^4 \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {1}{750} \int \frac {87103+139152 x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}+\frac {\int \frac {-330885-729750 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx}{7500} \\ & = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac {25458 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}+\frac {82039 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{2500} \\ & = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac {25458 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}-\frac {82039 \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{1250} \\ & = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac {25458 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}+\frac {82039 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{2500 \sqrt {5}} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.56 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {-\frac {5 \sqrt {2+5 x+3 x^2} \left (11545002+48537379 x+78737669 x^2+62190544 x^3+24066204 x^4+3665952 x^5\right )}{(1+x) (3+2 x)^4 (2+3 x)}+246117 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{18750} \]
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Time = 0.39 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.46
method | result | size |
risch | \(-\frac {3665952 x^{5}+24066204 x^{4}+62190544 x^{3}+78737669 x^{2}+48537379 x +11545002}{3750 \left (3+2 x \right )^{4} \sqrt {3 x^{2}+5 x +2}}-\frac {82039 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{12500}\) | \(78\) |
trager | \(-\frac {3665952 x^{5}+24066204 x^{4}+62190544 x^{3}+78737669 x^{2}+48537379 x +11545002}{3750 \left (3+2 x \right )^{4} \sqrt {3 x^{2}+5 x +2}}-\frac {82039 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{12500}\) | \(97\) |
default | \(-\frac {14}{75 \left (x +\frac {3}{2}\right )^{3} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {9619}{12000 \left (x +\frac {3}{2}\right )^{2} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {6931}{1500 \left (x +\frac {3}{2}\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}+\frac {82039}{5000 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {12729 \left (5+6 x \right )}{1250 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {82039 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{12500}-\frac {13}{320 \left (x +\frac {3}{2}\right )^{4} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}\) | \(153\) |
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Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.92 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {246117 \, \sqrt {5} {\left (48 \, x^{6} + 368 \, x^{5} + 1160 \, x^{4} + 1920 \, x^{3} + 1755 \, x^{2} + 837 \, x + 162\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (3665952 \, x^{5} + 24066204 \, x^{4} + 62190544 \, x^{3} + 78737669 \, x^{2} + 48537379 \, x + 11545002\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{75000 \, {\left (48 \, x^{6} + 368 \, x^{5} + 1160 \, x^{4} + 1920 \, x^{3} + 1755 \, x^{2} + 837 \, x + 162\right )}} \]
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\[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=- \int \frac {x}{96 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 880 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 3424 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 7320 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 9270 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 6939 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 2835 x \sqrt {3 x^{2} + 5 x + 2} + 486 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{96 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 880 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 3424 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 7320 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 9270 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 6939 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 2835 x \sqrt {3 x^{2} + 5 x + 2} + 486 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (139) = 278\).
Time = 0.27 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.83 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {82039}{12500} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {38187 \, x}{625 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {172541}{5000 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {13}{20 \, {\left (16 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{4} + 96 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{3} + 216 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 216 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 81 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {112}{75 \, {\left (8 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{3} + 36 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 54 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {9619}{3000 \, {\left (4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 12 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {6931}{750 \, {\left (2 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.39 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {1}{12500} \, \sqrt {5} {\left (50916 \, \sqrt {5} \sqrt {3} + 82039 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {\frac {\frac {5 \, {\left (\frac {\frac {10 \, {\left (\frac {448}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {195}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} + \frac {9619}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {27724}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} - \frac {857109}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {458244}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{7500 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3}} - \frac {82039 \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right )}{12500 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \]
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Timed out. \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\int \frac {x-5}{{\left (2\,x+3\right )}^5\,{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \]
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