Integrand size = 27, antiderivative size = 115 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {6848}{9} \sqrt {2+5 x+3 x^2}+\frac {152 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{27 \sqrt {3}} \]
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Time = 0.04 (sec) , antiderivative size = 115, 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, 654, 635, 212} \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {152 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{27 \sqrt {3}}-\frac {2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 (7976 x+6809) (2 x+3)}{27 \sqrt {3 x^2+5 x+2}}-\frac {6848}{9} \sqrt {3 x^2+5 x+2} \]
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Rule 212
Rule 635
Rule 654
Rule 832
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {(3+2 x)^2 (-117+272 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx \\ & = -\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {4}{27} \int \frac {-12802-15408 x}{\sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {6848}{9} \sqrt {2+5 x+3 x^2}+\frac {152}{27} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {6848}{9} \sqrt {2+5 x+3 x^2}+\frac {304}{27} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right ) \\ & = -\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {6848}{9} \sqrt {2+5 x+3 x^2}+\frac {152 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{27 \sqrt {3}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.72 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2}{81} \left (\frac {3 \sqrt {2+5 x+3 x^2} \left (-30819-118153 x-146180 x^2-58720 x^3+72 x^4\right )}{(1+x)^2 (2+3 x)^2}-152 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \]
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Time = 0.36 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.57
method | result | size |
risch | \(-\frac {2 \left (72 x^{4}-58720 x^{3}-146180 x^{2}-118153 x -30819\right )}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {152 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{81}\) | \(65\) |
trager | \(-\frac {2 \left (72 x^{4}-58720 x^{3}-146180 x^{2}-118153 x -30819\right )}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {152 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{81}\) | \(77\) |
default | \(-\frac {16181 \left (5+6 x \right )}{1458 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {\frac {295120}{243}+\frac {118048 x}{81}}{\sqrt {3 x^{2}+5 x +2}}-\frac {145763}{1458 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {16 x^{4}}{3 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {152 x^{3}}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {2380 x^{2}}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {14639 x}{81 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {152 x}{27 \sqrt {3 x^{2}+5 x +2}}+\frac {380}{81 \sqrt {3 x^{2}+5 x +2}}+\frac {152 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{81}\) | \(178\) |
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Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.02 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (38 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 3 \, {\left (72 \, x^{4} - 58720 \, x^{3} - 146180 \, x^{2} - 118153 \, x - 30819\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{81 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
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\[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {999 x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {864 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {264 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {16 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \frac {16 x^{5}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {405}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (93) = 186\).
Time = 0.26 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.86 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {16 \, x^{4}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {152}{81} \, x {\left (\frac {1410 \, x}{\sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {9 \, x^{2}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} + \frac {1175}{\sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {55 \, x}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {46}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}\right )} + \frac {152}{81} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {71440}{81} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {60704 \, x}{81 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {920 \, x^{2}}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {15680}{27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {13066 \, x}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {6766}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
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Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.59 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {152}{81} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {2 \, {\left ({\left (4 \, {\left (2 \, {\left (9 \, x - 7340\right )} x - 36545\right )} x - 118153\right )} x - 30819\right )}}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {{\left (2\,x+3\right )}^4\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]
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