Integrand size = 20, antiderivative size = 47 \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {280 (5+6 x)}{3 \sqrt {2+5 x+3 x^2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {652, 627} \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {280 (6 x+5)}{3 \sqrt {3 x^2+5 x+2}}-\frac {2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]
[In]
[Out]
Rule 627
Rule 652
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {140}{3} \int \frac {1}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx \\ & = -\frac {2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {280 (5+6 x)}{3 \sqrt {2+5 x+3 x^2}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {2+5 x+3 x^2} \left (457+1715 x+2100 x^2+840 x^3\right )}{(1+x)^2 (2+3 x)^2} \]
[In]
[Out]
Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.64
method | result | size |
trager | \(\frac {1680 x^{3}+4200 x^{2}+3430 x +914}{\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\) | \(30\) |
risch | \(\frac {1680 x^{3}+4200 x^{2}+3430 x +914}{\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\) | \(30\) |
gosper | \(\frac {2 \left (840 x^{3}+2100 x^{2}+1715 x +457\right ) \left (1+x \right ) \left (2+3 x \right )}{\left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}\) | \(38\) |
default | \(-\frac {35 \left (5+6 x \right )}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {\frac {1400}{3}+560 x}{\sqrt {3 x^{2}+5 x +2}}+\frac {1}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}\) | \(54\) |
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.09 \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (840 \, x^{3} + 2100 \, x^{2} + 1715 \, x + 457\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4} \]
[In]
[Out]
\[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \frac {x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.26 \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {560 \, x}{\sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {1400}{3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {70 \, x}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {58}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.62 \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (35 \, {\left (12 \, {\left (2 \, x + 5\right )} x + 49\right )} x + 457\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
[In]
[Out]
Time = 11.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77 \[ \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2310\,x+560\,x\,\left (3\,x^2+5\,x+2\right )+1400\,x^2+914}{{\left (3\,x^2+5\,x+2\right )}^{3/2}} \]
[In]
[Out]