Integrand size = 27, antiderivative size = 55 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx=-\frac {26}{5 \sqrt {3+2 x}}+12 \text {arctanh}\left (\sqrt {3+2 x}\right )-\frac {34}{5} \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 55, 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 = {842, 840, 1180, 213} \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx=12 \text {arctanh}\left (\sqrt {2 x+3}\right )-\frac {34}{5} \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )-\frac {26}{5 \sqrt {2 x+3}} \]
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Rule 213
Rule 840
Rule 842
Rule 1180
Rubi steps \begin{align*} \text {integral}& = -\frac {26}{5 \sqrt {3+2 x}}+\frac {1}{5} \int \frac {-9-39 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx \\ & = -\frac {26}{5 \sqrt {3+2 x}}+\frac {2}{5} \text {Subst}\left (\int \frac {99-39 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right ) \\ & = -\frac {26}{5 \sqrt {3+2 x}}+\frac {102}{5} \text {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-36 \text {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right ) \\ & = -\frac {26}{5 \sqrt {3+2 x}}+12 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {34}{5} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx=-\frac {26}{5 \sqrt {3+2 x}}+12 \text {arctanh}\left (\sqrt {3+2 x}\right )-\frac {34}{5} \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \]
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Time = 0.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(-\frac {26}{5 \sqrt {3+2 x}}-6 \ln \left (\sqrt {3+2 x}-1\right )+6 \ln \left (\sqrt {3+2 x}+1\right )-\frac {34 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{25}\) | \(53\) |
default | \(-\frac {26}{5 \sqrt {3+2 x}}-6 \ln \left (\sqrt {3+2 x}-1\right )+6 \ln \left (\sqrt {3+2 x}+1\right )-\frac {34 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{25}\) | \(53\) |
risch | \(-\frac {26}{5 \sqrt {3+2 x}}-6 \ln \left (\sqrt {3+2 x}-1\right )+6 \ln \left (\sqrt {3+2 x}+1\right )-\frac {34 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{25}\) | \(53\) |
pseudoelliptic | \(-\frac {2 \left (17 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}\, \sqrt {3+2 x}+75 \ln \left (\sqrt {3+2 x}-1\right ) \sqrt {3+2 x}-75 \ln \left (\sqrt {3+2 x}+1\right ) \sqrt {3+2 x}+65\right )}{25 \sqrt {3+2 x}}\) | \(75\) |
trager | \(-\frac {26}{5 \sqrt {3+2 x}}-6 \ln \left (\frac {-2-x +\sqrt {3+2 x}}{1+x}\right )-\frac {17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) x +15 \sqrt {3+2 x}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right )}{2+3 x}\right )}{25}\) | \(76\) |
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (38) = 76\).
Time = 0.37 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.73 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx=\frac {17 \, \sqrt {5} \sqrt {3} {\left (2 \, x + 3\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 150 \, {\left (2 \, x + 3\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 150 \, {\left (2 \, x + 3\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 130 \, \sqrt {2 \, x + 3}}{25 \, {\left (2 \, x + 3\right )}} \]
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Time = 3.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.45 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx=\frac {17 \sqrt {15} \left (\log {\left (\sqrt {2 x + 3} - \frac {\sqrt {15}}{3} \right )} - \log {\left (\sqrt {2 x + 3} + \frac {\sqrt {15}}{3} \right )}\right )}{25} - 6 \log {\left (\sqrt {2 x + 3} - 1 \right )} + 6 \log {\left (\sqrt {2 x + 3} + 1 \right )} - \frac {26}{5 \sqrt {2 x + 3}} \]
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Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx=\frac {17}{25} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {26}{5 \, \sqrt {2 \, x + 3}} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.35 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx=\frac {17}{25} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) - \frac {26}{5 \, \sqrt {2 \, x + 3}} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \]
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Time = 11.13 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.69 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx=12\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {34\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{25}-\frac {26}{5\,\sqrt {2\,x+3}} \]
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