Integrand size = 27, antiderivative size = 102 \[ \int \frac {(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=-\frac {(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {25 \sqrt {3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}+1250 \text {arctanh}\left (\sqrt {3+2 x}\right )-\frac {2905}{3} \sqrt {\frac {5}{3}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {832, 834, 840, 1180, 213} \[ \int \frac {(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=1250 \text {arctanh}\left (\sqrt {2 x+3}\right )-\frac {2905}{3} \sqrt {\frac {5}{3}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )-\frac {(139 x+121) (2 x+3)^{3/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac {25 (131 x+112) \sqrt {2 x+3}}{6 \left (3 x^2+5 x+2\right )} \]
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Rule 213
Rule 832
Rule 834
Rule 840
Rule 1180
Rubi steps \begin{align*} \text {integral}& = -\frac {(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {1}{6} \int \frac {(-900-425 x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx \\ & = -\frac {(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {25 \sqrt {3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}-\frac {1}{6} \int \frac {-7025-3275 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx \\ & = -\frac {(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {25 \sqrt {3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}-\frac {1}{3} \text {Subst}\left (\int \frac {-4225-3275 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right ) \\ & = -\frac {(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {25 \sqrt {3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}-3750 \text {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {14525}{3} \text {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right ) \\ & = -\frac {(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac {25 \sqrt {3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}+1250 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {2905}{3} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.80 \[ \int \frac {(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {\sqrt {3+2 x} \left (5237+19891 x+24497 x^2+9825 x^3\right )}{6 \left (2+5 x+3 x^2\right )^2}+1250 \text {arctanh}\left (\sqrt {3+2 x}\right )-\frac {2905}{3} \sqrt {\frac {5}{3}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \]
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Time = 0.47 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {\left (9825 x^{3}+24497 x^{2}+19891 x +5237\right ) \sqrt {3+2 x}}{6 \left (3 x^{2}+5 x +2\right )^{2}}+625 \ln \left (\sqrt {3+2 x}+1\right )-\frac {2905 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{9}-625 \ln \left (\sqrt {3+2 x}-1\right )\) | \(80\) |
trager | \(\frac {\left (9825 x^{3}+24497 x^{2}+19891 x +5237\right ) \sqrt {3+2 x}}{6 \left (3 x^{2}+5 x +2\right )^{2}}-625 \ln \left (\frac {-2-x +\sqrt {3+2 x}}{1+x}\right )+\frac {2905 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) x +15 \sqrt {3+2 x}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right )}{2+3 x}\right )}{18}\) | \(103\) |
derivativedivides | \(\frac {1835 \left (3+2 x \right )^{\frac {3}{2}}-\frac {10025 \sqrt {3+2 x}}{3}}{\left (6 x +4\right )^{2}}-\frac {2905 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{9}-\frac {3}{\left (\sqrt {3+2 x}+1\right )^{2}}+\frac {80}{\sqrt {3+2 x}+1}+625 \ln \left (\sqrt {3+2 x}+1\right )+\frac {3}{\left (\sqrt {3+2 x}-1\right )^{2}}+\frac {80}{\sqrt {3+2 x}-1}-625 \ln \left (\sqrt {3+2 x}-1\right )\) | \(124\) |
default | \(\frac {1835 \left (3+2 x \right )^{\frac {3}{2}}-\frac {10025 \sqrt {3+2 x}}{3}}{\left (6 x +4\right )^{2}}-\frac {2905 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{9}-\frac {3}{\left (\sqrt {3+2 x}+1\right )^{2}}+\frac {80}{\sqrt {3+2 x}+1}+625 \ln \left (\sqrt {3+2 x}+1\right )+\frac {3}{\left (\sqrt {3+2 x}-1\right )^{2}}+\frac {80}{\sqrt {3+2 x}-1}-625 \ln \left (\sqrt {3+2 x}-1\right )\) | \(124\) |
pseudoelliptic | \(\frac {-11620 \sqrt {15}\, \left (x +\frac {2}{3}\right )^{2} \left (1+x \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right )-22500 \left (x +\frac {2}{3}\right )^{2} \left (1+x \right )^{2} \ln \left (\sqrt {3+2 x}-1\right )+22500 \left (x +\frac {2}{3}\right )^{2} \left (1+x \right )^{2} \ln \left (\sqrt {3+2 x}+1\right )+6550 \left (x^{3}+\frac {187}{75} x^{2}+\frac {19891}{9825} x +\frac {5237}{9825}\right ) \sqrt {3+2 x}}{\left (2+3 x \right )^{2} \left (\sqrt {3+2 x}+1\right )^{2} \left (\sqrt {3+2 x}-1\right )^{2}}\) | \(127\) |
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Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (81) = 162\).
Time = 0.25 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.67 \[ \int \frac {(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {2905 \, \sqrt {5} \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 11250 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 11250 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) + 3 \, {\left (9825 \, x^{3} + 24497 \, x^{2} + 19891 \, x + 5237\right )} \sqrt {2 \, x + 3}}{18 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
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Time = 102.47 (sec) , antiderivative size = 406, normalized size of antiderivative = 3.98 \[ \int \frac {(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=141 \sqrt {15} \left (\log {\left (\sqrt {2 x + 3} - \frac {\sqrt {15}}{3} \right )} - \log {\left (\sqrt {2 x + 3} + \frac {\sqrt {15}}{3} \right )}\right ) - \frac {15800 \left (\begin {cases} \frac {\sqrt {15} \left (- \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )}\right )}{75} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{3} + \frac {17000 \left (\begin {cases} \frac {\sqrt {15} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )^{2}}\right )}{375} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{3} - 625 \log {\left (\sqrt {2 x + 3} - 1 \right )} + 625 \log {\left (\sqrt {2 x + 3} + 1 \right )} + \frac {80}{\sqrt {2 x + 3} + 1} - \frac {3}{\left (\sqrt {2 x + 3} + 1\right )^{2}} + \frac {80}{\sqrt {2 x + 3} - 1} + \frac {3}{\left (\sqrt {2 x + 3} - 1\right )^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.31 \[ \int \frac {(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {2905}{18} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {9825 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 39431 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 50875 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 21125 \, \sqrt {2 \, x + 3}}{3 \, {\left (9 \, {\left (2 \, x + 3\right )}^{4} - 48 \, {\left (2 \, x + 3\right )}^{3} + 94 \, {\left (2 \, x + 3\right )}^{2} - 160 \, x - 215\right )}} + 625 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 625 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.18 \[ \int \frac {(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {2905}{18} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) + \frac {9825 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 39431 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 50875 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 21125 \, \sqrt {2 \, x + 3}}{3 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 625 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 625 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.99 \[ \int \frac {(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=1250\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )+\frac {\frac {21125\,\sqrt {2\,x+3}}{27}-\frac {50875\,{\left (2\,x+3\right )}^{3/2}}{27}+\frac {39431\,{\left (2\,x+3\right )}^{5/2}}{27}-\frac {3275\,{\left (2\,x+3\right )}^{7/2}}{9}}{\frac {160\,x}{9}-\frac {94\,{\left (2\,x+3\right )}^2}{9}+\frac {16\,{\left (2\,x+3\right )}^3}{3}-{\left (2\,x+3\right )}^4+\frac {215}{9}}-\frac {2905\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{9} \]
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