Integrand size = 27, antiderivative size = 92 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {2 (3+2 x)^2 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {2}{9} (1239+554 x) \sqrt {2+5 x+3 x^2}+\frac {247 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{9 \sqrt {3}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {832, 793, 635, 212} \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {247 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{9 \sqrt {3}}-\frac {2 (139 x+121) (2 x+3)^2}{3 \sqrt {3 x^2+5 x+2}}+\frac {2}{9} (554 x+1239) \sqrt {3 x^2+5 x+2} \]
[In]
[Out]
Rule 212
Rule 635
Rule 793
Rule 832
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (3+2 x)^2 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {2}{3} \int \frac {(3+2 x) (481+554 x)}{\sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (3+2 x)^2 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {2}{9} (1239+554 x) \sqrt {2+5 x+3 x^2}+\frac {247}{9} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (3+2 x)^2 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {2}{9} (1239+554 x) \sqrt {2+5 x+3 x^2}+\frac {494}{9} \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)^2 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {2}{9} (1239+554 x) \sqrt {2+5 x+3 x^2}+\frac {247 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{9 \sqrt {3}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.85 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {2+5 x+3 x^2} \left (789+806 x-31 x^2+6 x^3\right )}{9 (1+x) (2+3 x)}+\frac {494 \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{9 \sqrt {3}} \]
[In]
[Out]
Time = 0.35 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.65
method | result | size |
risch | \(-\frac {2 \left (6 x^{3}-31 x^{2}+806 x +789\right )}{9 \sqrt {3 x^{2}+5 x +2}}+\frac {247 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{27}\) | \(60\) |
trager | \(-\frac {2 \left (6 x^{3}-31 x^{2}+806 x +789\right )}{9 \sqrt {3 x^{2}+5 x +2}}+\frac {247 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{27}\) | \(72\) |
default | \(-\frac {455 \left (5+6 x \right )}{18 \sqrt {3 x^{2}+5 x +2}}-\frac {881}{18 \sqrt {3 x^{2}+5 x +2}}-\frac {4 x^{3}}{3 \sqrt {3 x^{2}+5 x +2}}+\frac {62 x^{2}}{9 \sqrt {3 x^{2}+5 x +2}}-\frac {247 x}{9 \sqrt {3 x^{2}+5 x +2}}+\frac {247 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{27}\) | \(113\) |
[In]
[Out]
none
Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {247 \, \sqrt {3} {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 12 \, {\left (6 \, x^{3} - 31 \, x^{2} + 806 \, x + 789\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{54 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \]
[In]
[Out]
\[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=- \int \left (- \frac {243 x}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {126 x^{2}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {4 x^{3}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {8 x^{4}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {135}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {4 \, x^{3}}{3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {62 \, x^{2}}{9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {247}{27} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {1612 \, x}{9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {526}{3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.67 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {247}{27} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {2 \, {\left ({\left ({\left (6 \, x - 31\right )} x + 806\right )} x + 789\right )}}{9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} \]
[In]
[Out]
Timed out. \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\int \frac {{\left (2\,x+3\right )}^3\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \]
[In]
[Out]