Integrand size = 27, antiderivative size = 72 \[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=-\frac {\sqrt {3+2 x} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}-106 \text {arctanh}\left (\sqrt {3+2 x}\right )+\frac {248}{3} \sqrt {\frac {5}{3}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {832, 840, 1180, 213} \[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=-106 \text {arctanh}\left (\sqrt {2 x+3}\right )+\frac {248}{3} \sqrt {\frac {5}{3}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )-\frac {\sqrt {2 x+3} (139 x+121)}{3 \left (3 x^2+5 x+2\right )} \]
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Rule 213
Rule 832
Rule 840
Rule 1180
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {3+2 x} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {1}{3} \int \frac {-302-143 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx \\ & = -\frac {\sqrt {3+2 x} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {2}{3} \text {Subst}\left (\int \frac {-175-143 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right ) \\ & = -\frac {\sqrt {3+2 x} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+318 \text {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-\frac {1240}{3} \text {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right ) \\ & = -\frac {\sqrt {3+2 x} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}-106 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+\frac {248}{3} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=-\frac {2}{9} \left (\frac {3 \sqrt {3+2 x} (121+139 x)}{4+10 x+6 x^2}+477 \text {arctanh}\left (\sqrt {3+2 x}\right )-124 \sqrt {15} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\right ) \]
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Time = 0.38 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97
method | result | size |
risch | \(-\frac {\left (121+139 x \right ) \sqrt {3+2 x}}{3 \left (3 x^{2}+5 x +2\right )}-53 \ln \left (\sqrt {3+2 x}+1\right )+\frac {248 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{9}+53 \ln \left (\sqrt {3+2 x}-1\right )\) | \(70\) |
derivativedivides | \(-\frac {6}{\sqrt {3+2 x}+1}-53 \ln \left (\sqrt {3+2 x}+1\right )-\frac {170 \sqrt {3+2 x}}{9 \left (\frac {4}{3}+2 x \right )}+\frac {248 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{9}-\frac {6}{\sqrt {3+2 x}-1}+53 \ln \left (\sqrt {3+2 x}-1\right )\) | \(86\) |
default | \(-\frac {6}{\sqrt {3+2 x}+1}-53 \ln \left (\sqrt {3+2 x}+1\right )-\frac {170 \sqrt {3+2 x}}{9 \left (\frac {4}{3}+2 x \right )}+\frac {248 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{9}-\frac {6}{\sqrt {3+2 x}-1}+53 \ln \left (\sqrt {3+2 x}-1\right )\) | \(86\) |
trager | \(-\frac {\left (121+139 x \right ) \sqrt {3+2 x}}{3 \left (3 x^{2}+5 x +2\right )}-53 \ln \left (\frac {\sqrt {3+2 x}+2+x}{1+x}\right )-\frac {124 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) x +15 \sqrt {3+2 x}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right )}{2+3 x}\right )}{9}\) | \(91\) |
pseudoelliptic | \(\frac {744 \sqrt {15}\, \left (x +\frac {2}{3}\right ) \left (1+x \right ) \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right )+\left (1431 x^{2}+2385 x +954\right ) \ln \left (\sqrt {3+2 x}-1\right )+\left (-1431 x^{2}-2385 x -954\right ) \ln \left (\sqrt {3+2 x}+1\right )+\left (-417 x -363\right ) \sqrt {3+2 x}}{27 x^{2}+45 x +18}\) | \(94\) |
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Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (55) = 110\).
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.65 \[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=\frac {124 \, \sqrt {5} \sqrt {3} {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 477 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) + 477 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 3 \, {\left (139 \, x + 121\right )} \sqrt {2 \, x + 3}}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \]
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Time = 40.67 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.94 \[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=- \frac {107 \sqrt {15} \left (\log {\left (\sqrt {2 x + 3} - \frac {\sqrt {15}}{3} \right )} - \log {\left (\sqrt {2 x + 3} + \frac {\sqrt {15}}{3} \right )}\right )}{9} + \frac {1700 \left (\begin {cases} \frac {\sqrt {15} \left (- \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )}\right )}{75} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{3} + 53 \log {\left (\sqrt {2 x + 3} - 1 \right )} - 53 \log {\left (\sqrt {2 x + 3} + 1 \right )} - \frac {6}{\sqrt {2 x + 3} + 1} - \frac {6}{\sqrt {2 x + 3} - 1} \]
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Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.36 \[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=-\frac {124}{9} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {2 \, {\left (139 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 175 \, \sqrt {2 \, x + 3}\right )}}{3 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 53 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 53 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.42 \[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=-\frac {124}{9} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) - \frac {2 \, {\left (139 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 175 \, \sqrt {2 \, x + 3}\right )}}{3 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 53 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 53 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \]
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Time = 11.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.92 \[ \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=\frac {248\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{9}-\frac {\frac {350\,\sqrt {2\,x+3}}{9}-\frac {278\,{\left (2\,x+3\right )}^{3/2}}{9}}{\frac {16\,x}{3}-{\left (2\,x+3\right )}^2+\frac {19}{3}}-106\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right ) \]
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