Integrand size = 27, antiderivative size = 98 \[ \int \frac {5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2} \, dx=-\frac {262}{15 (3+2 x)^{3/2}}-\frac {686}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-10 \text {arctanh}\left (\sqrt {3+2 x}\right )+\frac {936}{25} \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 98, 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, 842, 840, 1180, 213} \[ \int \frac {5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2} \, dx=-10 \text {arctanh}\left (\sqrt {2 x+3}\right )+\frac {936}{25} \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )-\frac {3 (47 x+37)}{5 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )}-\frac {686}{25 \sqrt {2 x+3}}-\frac {262}{15 (2 x+3)^{3/2}} \]
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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)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-\frac {1}{5} \int \frac {730+705 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx \\ & = -\frac {262}{15 (3+2 x)^{3/2}}-\frac {3 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-\frac {1}{25} \int \frac {2090+1965 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx \\ & = -\frac {262}{15 (3+2 x)^{3/2}}-\frac {686}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-\frac {1}{125} \int \frac {5770+5145 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx \\ & = -\frac {262}{15 (3+2 x)^{3/2}}-\frac {686}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-\frac {2}{125} \text {Subst}\left (\int \frac {-3895+5145 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right ) \\ & = -\frac {262}{15 (3+2 x)^{3/2}}-\frac {686}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}+30 \text {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-\frac {2808}{25} \text {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right ) \\ & = -\frac {262}{15 (3+2 x)^{3/2}}-\frac {686}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-10 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+\frac {936}{25} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84 \[ \int \frac {5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2} \, dx=\frac {-16633-47767 x-43032 x^2-12348 x^3}{75 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-10 \text {arctanh}\left (\sqrt {3+2 x}\right )+\frac {936}{25} \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \]
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Time = 0.40 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\frac {12348 x^{3}+43032 x^{2}+47767 x +16633}{75 \left (3+2 x \right )^{\frac {3}{2}} \left (3 x^{2}+5 x +2\right )}-5 \ln \left (\sqrt {3+2 x}+1\right )+\frac {936 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{125}+5 \ln \left (\sqrt {3+2 x}-1\right )\) | \(80\) |
trager | \(-\frac {12348 x^{3}+43032 x^{2}+47767 x +16633}{75 \left (3+2 x \right )^{\frac {3}{2}} \left (3 x^{2}+5 x +2\right )}+\frac {468 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) x +15 \sqrt {3+2 x}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right )}{2+3 x}\right )}{125}-5 \ln \left (\frac {\sqrt {3+2 x}+2+x}{1+x}\right )\) | \(101\) |
derivativedivides | \(-\frac {6}{\sqrt {3+2 x}+1}-5 \ln \left (\sqrt {3+2 x}+1\right )-\frac {6}{\sqrt {3+2 x}-1}+5 \ln \left (\sqrt {3+2 x}-1\right )-\frac {104}{75 \left (3+2 x \right )^{\frac {3}{2}}}-\frac {1624}{125 \sqrt {3+2 x}}-\frac {306 \sqrt {3+2 x}}{125 \left (\frac {4}{3}+2 x \right )}+\frac {936 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{125}\) | \(104\) |
default | \(-\frac {6}{\sqrt {3+2 x}+1}-5 \ln \left (\sqrt {3+2 x}+1\right )-\frac {6}{\sqrt {3+2 x}-1}+5 \ln \left (\sqrt {3+2 x}-1\right )-\frac {104}{75 \left (3+2 x \right )^{\frac {3}{2}}}-\frac {1624}{125 \sqrt {3+2 x}}-\frac {306 \sqrt {3+2 x}}{125 \left (\frac {4}{3}+2 x \right )}+\frac {936 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{125}\) | \(104\) |
pseudoelliptic | \(\frac {16848 \sqrt {3+2 x}\, \sqrt {15}\, \left (x +\frac {2}{3}\right ) \left (1+x \right ) \left (x +\frac {3}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right )+11250 \sqrt {3+2 x}\, \left (x +\frac {2}{3}\right ) \left (1+x \right ) \left (x +\frac {3}{2}\right ) \ln \left (\sqrt {3+2 x}-1\right )-11250 \sqrt {3+2 x}\, \left (x +\frac {2}{3}\right ) \left (1+x \right ) \left (x +\frac {3}{2}\right ) \ln \left (\sqrt {3+2 x}+1\right )-61740 x^{3}-215160 x^{2}-238835 x -83165}{\left (3+2 x \right )^{\frac {3}{2}} \left (1125 x^{2}+1875 x +750\right )}\) | \(126\) |
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Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (73) = 146\).
Time = 0.31 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.72 \[ \int \frac {5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2} \, dx=\frac {1404 \, \sqrt {5} \sqrt {3} {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 1875 \, {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) + 1875 \, {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 5 \, {\left (12348 \, x^{3} + 43032 \, x^{2} + 47767 \, x + 16633\right )} \sqrt {2 \, x + 3}}{375 \, {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )}} \]
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Time = 46.72 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.41 \[ \int \frac {5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2} \, dx=- \frac {2187 \sqrt {15} \left (\log {\left (\sqrt {2 x + 3} - \frac {\sqrt {15}}{3} \right )} - \log {\left (\sqrt {2 x + 3} + \frac {\sqrt {15}}{3} \right )}\right )}{625} + \frac {1836 \left (\begin {cases} \frac {\sqrt {15} \left (- \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )}\right )}{75} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{25} + 5 \log {\left (\sqrt {2 x + 3} - 1 \right )} - 5 \log {\left (\sqrt {2 x + 3} + 1 \right )} - \frac {6}{\sqrt {2 x + 3} + 1} - \frac {6}{\sqrt {2 x + 3} - 1} - \frac {1624}{125 \sqrt {2 x + 3}} - \frac {104}{75 \left (2 x + 3\right )^{\frac {3}{2}}} \]
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Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.18 \[ \int \frac {5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2} \, dx=-\frac {468}{125} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {2 \, {\left (3087 \, {\left (2 \, x + 3\right )}^{3} - 6267 \, {\left (2 \, x + 3\right )}^{2} + 4040 \, x + 6320\right )}}{75 \, {\left (3 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 8 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 5 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}\right )}} - 5 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 5 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.18 \[ \int \frac {5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2} \, dx=-\frac {468}{125} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) - \frac {6 \, {\left (903 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 1403 \, \sqrt {2 \, x + 3}\right )}}{125 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - \frac {16 \, {\left (609 \, x + 946\right )}}{375 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}} - 5 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 5 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84 \[ \int \frac {5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2} \, dx=\frac {936\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{125}-10\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {\frac {1616\,x}{45}-\frac {4178\,{\left (2\,x+3\right )}^2}{75}+\frac {686\,{\left (2\,x+3\right )}^3}{25}+\frac {2528}{45}}{\frac {5\,{\left (2\,x+3\right )}^{3/2}}{3}-\frac {8\,{\left (2\,x+3\right )}^{5/2}}{3}+{\left (2\,x+3\right )}^{7/2}} \]
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