Integrand size = 27, antiderivative size = 54 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (3+2 x)^2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {376 (7+8 x)}{3 \sqrt {2+5 x+3 x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {818, 650} \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {376 (8 x+7)}{3 \sqrt {3 x^2+5 x+2}}-\frac {2 (2 x+3)^2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]
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Rule 650
Rule 818
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (3+2 x)^2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {188}{3} \int \frac {3+2 x}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx \\ & = -\frac {2 (3+2 x)^2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {376 (7+8 x)}{3 \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {2+5 x+3 x^2} \left (2371+8925 x+10932 x^2+4372 x^3\right )}{3 (1+x)^2 (2+3 x)^2} \]
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Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.56
method | result | size |
trager | \(\frac {\frac {8744}{3} x^{3}+7288 x^{2}+5950 x +\frac {4742}{3}}{\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\) | \(30\) |
risch | \(\frac {\frac {8744}{3} x^{3}+7288 x^{2}+5950 x +\frac {4742}{3}}{\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\) | \(30\) |
gosper | \(\frac {2 \left (4372 x^{3}+10932 x^{2}+8925 x +2371\right ) \left (1+x \right ) \left (2+3 x \right )}{3 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}\) | \(38\) |
default | \(-\frac {1093 \left (5+6 x \right )}{162 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {\frac {21860}{27}+\frac {8744 x}{9}}{\sqrt {3 x^{2}+5 x +2}}-\frac {787}{162 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {4 x^{2}}{3 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {7 x}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\) | \(86\) |
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Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (4372 \, x^{3} + 10932 \, x^{2} + 8925 \, x + 2371\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{3 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
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\[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {51 x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {8 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {4 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {45}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
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Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.41 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {8744 \, x}{9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {4 \, x^{2}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} + \frac {21860}{27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {1114 \, x}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {1042}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.52 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (1093 \, x + 2733\right )} x + 8925\right )} x + 2371\right )}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
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Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {36062\,x+8744\,x\,\left (3\,x^2+5\,x+2\right )+21872\,x^2+14226}{\sqrt {3\,x^2+5\,x+2}\,\left (27\,x^2+45\,x+18\right )} \]
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