Integrand size = 27, antiderivative size = 141 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+266 \text {arctanh}\left (\sqrt {3+2 x}\right )-\frac {806841 \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )}{3125} \]
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Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {836, 842, 840, 1180, 213} \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3} \, dx=266 \text {arctanh}\left (\sqrt {2 x+3}\right )-\frac {806841 \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{3125}-\frac {3 (47 x+37)}{10 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^2}+\frac {9957 x+8852}{50 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac {24409}{3125 \sqrt {2 x+3}}+\frac {102697}{1875 (2 x+3)^{3/2}}+\frac {56399}{625 (2 x+3)^{5/2}} \]
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Rule 213
Rule 836
Rule 840
Rule 842
Rule 1180
Rubi steps \begin{align*} \text {integral}& = -\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}-\frac {1}{10} \int \frac {1772+1551 x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2} \, dx \\ & = -\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {1}{50} \int \frac {76349+69699 x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )} \, dx \\ & = \frac {56399}{625 (3+2 x)^{5/2}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {1}{250} \int \frac {202447+169197 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx \\ & = \frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {\int \frac {474341+308091 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx}{1250} \\ & = \frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {\int \frac {758023-73227 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx}{6250} \\ & = \frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {\text {Subst}\left (\int \frac {1735727-73227 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )}{3125} \\ & = \frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {2420523 \text {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )}{3125}-798 \text {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right ) \\ & = \frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+266 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {806841 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )}{3125} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.69 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {20250051+94082723 x+160041829 x^2+114099329 x^3+18312714 x^4-14906052 x^5-5272344 x^6}{18750 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+266 \text {arctanh}\left (\sqrt {3+2 x}\right )-\frac {806841 \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )}{3125} \]
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Time = 0.42 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.67
method | result | size |
risch | \(-\frac {5272344 x^{6}+14906052 x^{5}-18312714 x^{4}-114099329 x^{3}-160041829 x^{2}-94082723 x -20250051}{18750 \left (3+2 x \right )^{\frac {5}{2}} \left (3 x^{2}+5 x +2\right )^{2}}+133 \ln \left (\sqrt {3+2 x}+1\right )-\frac {806841 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{15625}-133 \ln \left (\sqrt {3+2 x}-1\right )\) | \(95\) |
trager | \(-\frac {5272344 x^{6}+14906052 x^{5}-18312714 x^{4}-114099329 x^{3}-160041829 x^{2}-94082723 x -20250051}{18750 \left (3+2 x \right )^{\frac {5}{2}} \left (3 x^{2}+5 x +2\right )^{2}}+133 \ln \left (\frac {\sqrt {3+2 x}+2+x}{1+x}\right )+\frac {567 \operatorname {RootOf}\left (\textit {\_Z}^{2}-30373935\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-30373935\right ) x +7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-30373935\right )-21345 \sqrt {3+2 x}}{2+3 x}\right )}{31250}\) | \(117\) |
derivativedivides | \(\frac {3}{\left (\sqrt {3+2 x}-1\right )^{2}}+\frac {8}{\sqrt {3+2 x}-1}-133 \ln \left (\sqrt {3+2 x}-1\right )-\frac {3}{\left (\sqrt {3+2 x}+1\right )^{2}}+\frac {8}{\sqrt {3+2 x}+1}+133 \ln \left (\sqrt {3+2 x}+1\right )-\frac {416}{625 \left (3+2 x \right )^{\frac {5}{2}}}-\frac {9824}{1875 \left (3+2 x \right )^{\frac {3}{2}}}-\frac {137184}{3125 \sqrt {3+2 x}}+\frac {\frac {22599 \left (3+2 x \right )^{\frac {3}{2}}}{125}-\frac {196587 \sqrt {3+2 x}}{625}}{\left (6 x +4\right )^{2}}-\frac {806841 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{15625}\) | \(151\) |
default | \(\frac {3}{\left (\sqrt {3+2 x}-1\right )^{2}}+\frac {8}{\sqrt {3+2 x}-1}-133 \ln \left (\sqrt {3+2 x}-1\right )-\frac {3}{\left (\sqrt {3+2 x}+1\right )^{2}}+\frac {8}{\sqrt {3+2 x}+1}+133 \ln \left (\sqrt {3+2 x}+1\right )-\frac {416}{625 \left (3+2 x \right )^{\frac {5}{2}}}-\frac {9824}{1875 \left (3+2 x \right )^{\frac {3}{2}}}-\frac {137184}{3125 \sqrt {3+2 x}}+\frac {\frac {22599 \left (3+2 x \right )^{\frac {3}{2}}}{125}-\frac {196587 \sqrt {3+2 x}}{625}}{\left (6 x +4\right )^{2}}-\frac {806841 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{15625}\) | \(151\) |
pseudoelliptic | \(-\frac {12909456 \left (\sqrt {3+2 x}\, \sqrt {15}\, \left (x +\frac {2}{3}\right )^{2} \left (1+x \right )^{2} \left (x +\frac {3}{2}\right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right )+\frac {296875 \sqrt {3+2 x}\, \left (x +\frac {2}{3}\right )^{2} \left (1+x \right )^{2} \left (x +\frac {3}{2}\right )^{2} \ln \left (\sqrt {3+2 x}-1\right )}{115263}-\frac {296875 \sqrt {3+2 x}\, \left (x +\frac {2}{3}\right )^{2} \left (1+x \right )^{2} \left (x +\frac {3}{2}\right )^{2} \ln \left (\sqrt {3+2 x}+1\right )}{115263}-\frac {800209145 x^{2}}{174277656}-\frac {570496645 x^{3}}{174277656}-\frac {726695 x^{4}}{1383156}+\frac {98585 x^{5}}{230526}+\frac {17435 x^{6}}{115263}-\frac {67201945 x}{24896808}-\frac {33750085}{58092552}\right )}{15625 \left (3+2 x \right )^{\frac {5}{2}} \left (\sqrt {3+2 x}+1\right )^{2} \left (x +\frac {2}{3}\right )^{2} \left (\sqrt {3+2 x}-1\right )^{2}}\) | \(174\) |
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Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (108) = 216\).
Time = 0.30 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.74 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {2420523 \, \sqrt {5} \sqrt {3} {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 12468750 \, {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 12468750 \, {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 5 \, {\left (5272344 \, x^{6} + 14906052 \, x^{5} - 18312714 \, x^{4} - 114099329 \, x^{3} - 160041829 \, x^{2} - 94082723 \, x - 20250051\right )} \sqrt {2 \, x + 3}}{93750 \, {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )}} \]
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Time = 117.03 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.15 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {372033 \sqrt {15} \left (\log {\left (\sqrt {2 x + 3} - \frac {\sqrt {15}}{3} \right )} - \log {\left (\sqrt {2 x + 3} + \frac {\sqrt {15}}{3} \right )}\right )}{15625} - \frac {351864 \left (\begin {cases} \frac {\sqrt {15} \left (- \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )}\right )}{75} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{625} + \frac {33048 \left (\begin {cases} \frac {\sqrt {15} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )^{2}}\right )}{375} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{125} - 133 \log {\left (\sqrt {2 x + 3} - 1 \right )} + 133 \log {\left (\sqrt {2 x + 3} + 1 \right )} + \frac {8}{\sqrt {2 x + 3} + 1} - \frac {3}{\left (\sqrt {2 x + 3} + 1\right )^{2}} + \frac {8}{\sqrt {2 x + 3} - 1} + \frac {3}{\left (\sqrt {2 x + 3} - 1\right )^{2}} - \frac {137184}{3125 \sqrt {2 x + 3}} - \frac {9824}{1875 \left (2 x + 3\right )^{\frac {3}{2}}} - \frac {416}{625 \left (2 x + 3\right )^{\frac {5}{2}}} \]
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Time = 0.28 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.14 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {806841}{31250} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {659043 \, {\left (2 \, x + 3\right )}^{6} - 8136261 \, {\left (2 \, x + 3\right )}^{5} + 23916753 \, {\left (2 \, x + 3\right )}^{4} - 24720095 \, {\left (2 \, x + 3\right )}^{3} + 6945760 \, {\left (2 \, x + 3\right )}^{2} + 1457600 \, x + 2342400}{9375 \, {\left (9 \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - 48 \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + 94 \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - 80 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + 25 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}\right )}} + 133 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 133 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.01 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {806841}{31250} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) + \frac {202995 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 745077 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 831169 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 259087 \, \sqrt {2 \, x + 3}}{625 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} - \frac {32 \, {\left (12861 \, {\left (2 \, x + 3\right )}^{2} + 3070 \, x + 4800\right )}}{9375 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}} + 133 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 133 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.90 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3} \, dx=266\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {806841\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{15625}-\frac {\frac {58304\,x}{3375}+\frac {1389152\,{\left (2\,x+3\right )}^2}{16875}-\frac {4944019\,{\left (2\,x+3\right )}^3}{16875}+\frac {2657417\,{\left (2\,x+3\right )}^4}{9375}-\frac {301343\,{\left (2\,x+3\right )}^5}{3125}+\frac {24409\,{\left (2\,x+3\right )}^6}{3125}+\frac {31232}{1125}}{\frac {25\,{\left (2\,x+3\right )}^{5/2}}{9}-\frac {80\,{\left (2\,x+3\right )}^{7/2}}{9}+\frac {94\,{\left (2\,x+3\right )}^{9/2}}{9}-\frac {16\,{\left (2\,x+3\right )}^{11/2}}{3}+{\left (2\,x+3\right )}^{13/2}} \]
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