Integrand size = 27, antiderivative size = 115 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+402 \text {arctanh}\left (\sqrt {3+2 x}\right )-\frac {12717}{25} \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 115, 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} \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=402 \text {arctanh}\left (\sqrt {2 x+3}\right )-\frac {12717}{25} \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )-\frac {3 (47 x+37)}{10 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^2}+\frac {2229 x+1888}{10 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )}+\frac {2667}{25 \sqrt {2 x+3}} \]
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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 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}-\frac {1}{10} \int \frac {1328+987 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2} \, dx \\ & = -\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+\frac {1}{50} \int \frac {43485+33435 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx \\ & = \frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+\frac {1}{250} \int \frac {90255+40005 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx \\ & = \frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+\frac {1}{125} \text {Subst}\left (\int \frac {60495+40005 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right ) \\ & = \frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}-1206 \text {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {38151}{25} \text {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right ) \\ & = \frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+402 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {12717}{25} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=402 \text {arctanh}\left (\sqrt {3+2 x}\right )+\frac {1}{250} \left (\frac {5 \left (39661+175465 x+281403 x^2+193455 x^3+48006 x^4\right )}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}-25434 \sqrt {15} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\right ) \]
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Time = 0.41 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {48006 x^{4}+193455 x^{3}+281403 x^{2}+175465 x +39661}{50 \left (3 x^{2}+5 x +2\right )^{2} \sqrt {3+2 x}}+201 \ln \left (\sqrt {3+2 x}+1\right )-\frac {12717 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{125}-201 \ln \left (\sqrt {3+2 x}-1\right )\) | \(85\) |
trager | \(\frac {48006 x^{4}+193455 x^{3}+281403 x^{2}+175465 x +39661}{50 \left (3 x^{2}+5 x +2\right )^{2} \sqrt {3+2 x}}+201 \ln \left (\frac {\sqrt {3+2 x}+2+x}{1+x}\right )-\frac {81 \operatorname {RootOf}\left (\textit {\_Z}^{2}-369735\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-369735\right ) x +7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-369735\right )+2355 \sqrt {3+2 x}}{2+3 x}\right )}{250}\) | \(106\) |
derivativedivides | \(\frac {32}{\sqrt {3+2 x}-1}+\frac {3}{\left (\sqrt {3+2 x}-1\right )^{2}}-201 \ln \left (\sqrt {3+2 x}-1\right )+\frac {\frac {51759 \left (3+2 x \right )^{\frac {3}{2}}}{125}-\frac {18171 \sqrt {3+2 x}}{25}}{\left (6 x +4\right )^{2}}-\frac {12717 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{125}+\frac {32}{\sqrt {3+2 x}+1}-\frac {3}{\left (\sqrt {3+2 x}+1\right )^{2}}+201 \ln \left (\sqrt {3+2 x}+1\right )-\frac {416}{125 \sqrt {3+2 x}}\) | \(133\) |
default | \(\frac {32}{\sqrt {3+2 x}-1}+\frac {3}{\left (\sqrt {3+2 x}-1\right )^{2}}-201 \ln \left (\sqrt {3+2 x}-1\right )+\frac {\frac {51759 \left (3+2 x \right )^{\frac {3}{2}}}{125}-\frac {18171 \sqrt {3+2 x}}{25}}{\left (6 x +4\right )^{2}}-\frac {12717 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{125}+\frac {32}{\sqrt {3+2 x}+1}-\frac {3}{\left (\sqrt {3+2 x}+1\right )^{2}}+201 \ln \left (\sqrt {3+2 x}+1\right )-\frac {416}{125 \sqrt {3+2 x}}\) | \(133\) |
pseudoelliptic | \(-\frac {457812 \left (\sqrt {3+2 x}\, \sqrt {15}\, \left (x +\frac {2}{3}\right )^{2} \left (1+x \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right )+\frac {8375 \left (x +\frac {2}{3}\right )^{2} \left (1+x \right )^{2} \sqrt {3+2 x}\, \ln \left (\sqrt {3+2 x}-1\right )}{4239}-\frac {8375 \left (x +\frac {2}{3}\right )^{2} \left (1+x \right )^{2} \sqrt {3+2 x}\, \ln \left (\sqrt {3+2 x}+1\right )}{4239}-\frac {4445 x^{4}}{4239}-\frac {35825 x^{3}}{8478}-\frac {156335 x^{2}}{25434}-\frac {877325 x}{228906}-\frac {198305}{228906}\right )}{125 \sqrt {3+2 x}\, \left (\sqrt {3+2 x}-1\right )^{2} \left (2+3 x \right )^{2} \left (\sqrt {3+2 x}+1\right )^{2}}\) | \(151\) |
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Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (90) = 180\).
Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.70 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {12717 \, \sqrt {5} \sqrt {3} {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 50250 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 50250 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) + 5 \, {\left (48006 \, x^{4} + 193455 \, x^{3} + 281403 \, x^{2} + 175465 \, x + 39661\right )} \sqrt {2 \, x + 3}}{250 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )}} \]
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Time = 115.77 (sec) , antiderivative size = 420, normalized size of antiderivative = 3.65 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {28917 \sqrt {15} \left (\log {\left (\sqrt {2 x + 3} - \frac {\sqrt {15}}{3} \right )} - \log {\left (\sqrt {2 x + 3} + \frac {\sqrt {15}}{3} \right )}\right )}{625} - \frac {31752 \left (\begin {cases} \frac {\sqrt {15} \left (- \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )}\right )}{75} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{25} + \frac {3672 \left (\begin {cases} \frac {\sqrt {15} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )^{2}}\right )}{375} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{5} - 201 \log {\left (\sqrt {2 x + 3} - 1 \right )} + 201 \log {\left (\sqrt {2 x + 3} + 1 \right )} + \frac {32}{\sqrt {2 x + 3} + 1} - \frac {3}{\left (\sqrt {2 x + 3} + 1\right )^{2}} + \frac {32}{\sqrt {2 x + 3} - 1} + \frac {3}{\left (\sqrt {2 x + 3} - 1\right )^{2}} - \frac {416}{125 \sqrt {2 x + 3}} \]
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Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.24 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {12717}{250} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {24003 \, {\left (2 \, x + 3\right )}^{4} - 94581 \, {\left (2 \, x + 3\right )}^{3} + 117873 \, {\left (2 \, x + 3\right )}^{2} - 88030 \, x - 134125}{25 \, {\left (9 \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - 48 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + 94 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - 80 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + 25 \, \sqrt {2 \, x + 3}\right )}} + 201 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 201 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.12 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {12717}{250} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) - \frac {416}{125 \, \sqrt {2 \, x + 3}} + \frac {123759 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 492873 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 628469 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 253355 \, \sqrt {2 \, x + 3}}{125 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 201 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 201 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \]
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Time = 11.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=402\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {12717\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{125}-\frac {\frac {17606\,x}{45}-\frac {13097\,{\left (2\,x+3\right )}^2}{25}+\frac {10509\,{\left (2\,x+3\right )}^3}{25}-\frac {2667\,{\left (2\,x+3\right )}^4}{25}+\frac {5365}{9}}{\frac {25\,\sqrt {2\,x+3}}{9}-\frac {80\,{\left (2\,x+3\right )}^{3/2}}{9}+\frac {94\,{\left (2\,x+3\right )}^{5/2}}{9}-\frac {16\,{\left (2\,x+3\right )}^{7/2}}{3}+{\left (2\,x+3\right )}^{9/2}} \]
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