Integrand size = 18, antiderivative size = 89 \[ \int x^2 \sqrt {2 r x-x^2} \, dx=-\frac {5}{8} r^2 (r-x) \sqrt {2 r x-x^2}-\frac {5}{12} r \left (2 r x-x^2\right )^{3/2}-\frac {1}{4} x \left (2 r x-x^2\right )^{3/2}+\frac {5}{4} r^4 \arctan \left (\frac {x}{\sqrt {2 r x-x^2}}\right ) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {684, 654, 626, 634, 209} \[ \int x^2 \sqrt {2 r x-x^2} \, dx=\frac {5}{4} r^4 \arctan \left (\frac {x}{\sqrt {2 r x-x^2}}\right )-\frac {5}{8} r^2 (r-x) \sqrt {2 r x-x^2}-\frac {5}{12} r \left (2 r x-x^2\right )^{3/2}-\frac {1}{4} x \left (2 r x-x^2\right )^{3/2} \]
[In]
[Out]
Rule 209
Rule 626
Rule 634
Rule 654
Rule 684
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4} x \left (2 r x-x^2\right )^{3/2}+\frac {1}{4} (5 r) \int x \sqrt {2 r x-x^2} \, dx \\ & = -\frac {5}{12} r \left (2 r x-x^2\right )^{3/2}-\frac {1}{4} x \left (2 r x-x^2\right )^{3/2}+\frac {1}{4} \left (5 r^2\right ) \int \sqrt {2 r x-x^2} \, dx \\ & = -\frac {5}{8} r^2 (r-x) \sqrt {2 r x-x^2}-\frac {5}{12} r \left (2 r x-x^2\right )^{3/2}-\frac {1}{4} x \left (2 r x-x^2\right )^{3/2}+\frac {1}{8} \left (5 r^4\right ) \int \frac {1}{\sqrt {2 r x-x^2}} \, dx \\ & = -\frac {5}{8} r^2 (r-x) \sqrt {2 r x-x^2}-\frac {5}{12} r \left (2 r x-x^2\right )^{3/2}-\frac {1}{4} x \left (2 r x-x^2\right )^{3/2}+\frac {1}{4} \left (5 r^4\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {2 r x-x^2}}\right ) \\ & = -\frac {5}{8} r^2 (r-x) \sqrt {2 r x-x^2}-\frac {5}{12} r \left (2 r x-x^2\right )^{3/2}-\frac {1}{4} x \left (2 r x-x^2\right )^{3/2}+\frac {5}{4} r^4 \arctan \left (\frac {x}{\sqrt {2 r x-x^2}}\right ) \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.85 \[ \int x^2 \sqrt {2 r x-x^2} \, dx=\frac {1}{24} \sqrt {-x (-2 r+x)} \left (-15 r^3-5 r^2 x-2 r x^2+6 x^3+\frac {30 r^4 \log \left (-\sqrt {x}+\sqrt {-2 r+x}\right )}{\sqrt {x} \sqrt {-2 r+x}}\right ) \]
[In]
[Out]
Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(-\frac {5 \arctan \left (\frac {\sqrt {x \left (2 r -x \right )}}{x}\right ) r^{4}}{4}-\frac {5 \sqrt {x \left (2 r -x \right )}\, \left (r^{3}+\frac {1}{3} r^{2} x +\frac {2}{15} r \,x^{2}-\frac {2}{5} x^{3}\right )}{8}\) | \(57\) |
risch | \(-\frac {\left (15 r^{3}+5 r^{2} x +2 r \,x^{2}-6 x^{3}\right ) x \left (2 r -x \right )}{24 \sqrt {-x \left (-2 r +x \right )}}+\frac {5 r^{4} \arctan \left (\frac {x -r}{\sqrt {2 r x -x^{2}}}\right )}{8}\) | \(69\) |
default | \(-\frac {x \left (2 r x -x^{2}\right )^{\frac {3}{2}}}{4}+\frac {5 r \left (-\frac {\left (2 r x -x^{2}\right )^{\frac {3}{2}}}{3}+r \left (-\frac {\left (2 r -2 x \right ) \sqrt {2 r x -x^{2}}}{4}+\frac {r^{2} \arctan \left (\frac {x -r}{\sqrt {2 r x -x^{2}}}\right )}{2}\right )\right )}{4}\) | \(83\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.67 \[ \int x^2 \sqrt {2 r x-x^2} \, dx=-\frac {5}{4} \, r^{4} \arctan \left (\frac {\sqrt {2 \, r x - x^{2}}}{x}\right ) - \frac {1}{24} \, {\left (15 \, r^{3} + 5 \, r^{2} x + 2 \, r x^{2} - 6 \, x^{3}\right )} \sqrt {2 \, r x - x^{2}} \]
[In]
[Out]
Time = 0.25 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03 \[ \int x^2 \sqrt {2 r x-x^2} \, dx=\frac {5 r^{4} \left (\begin {cases} - i \log {\left (2 r - 2 x + 2 i \sqrt {2 r x - x^{2}} \right )} & \text {for}\: r^{2} \neq 0 \\\frac {\left (- r + x\right ) \log {\left (- r + x \right )}}{\sqrt {- \left (- r + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{8} + \sqrt {2 r x - x^{2}} \left (- \frac {5 r^{3}}{8} - \frac {5 r^{2} x}{24} - \frac {r x^{2}}{12} + \frac {x^{3}}{4}\right ) \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int x^2 \sqrt {2 r x-x^2} \, dx=-\frac {5}{8} \, r^{4} \arcsin \left (\frac {r - x}{r}\right ) - \frac {5}{8} \, \sqrt {2 \, r x - x^{2}} r^{3} + \frac {5}{8} \, \sqrt {2 \, r x - x^{2}} r^{2} x - \frac {5}{12} \, {\left (2 \, r x - x^{2}\right )}^{\frac {3}{2}} r - \frac {1}{4} \, {\left (2 \, r x - x^{2}\right )}^{\frac {3}{2}} x \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.61 \[ \int x^2 \sqrt {2 r x-x^2} \, dx=-\frac {5}{8} \, r^{4} \arcsin \left (\frac {r - x}{r}\right ) \mathrm {sgn}\left (r\right ) - \frac {1}{24} \, {\left (15 \, r^{3} + {\left (5 \, r^{2} + 2 \, {\left (r - 3 \, x\right )} x\right )} x\right )} \sqrt {2 \, r x - x^{2}} \]
[In]
[Out]
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.84 \[ \int x^2 \sqrt {2 r x-x^2} \, dx=-\frac {x\,{\left (2\,r\,x-x^2\right )}^{3/2}}{4}-\frac {5\,r\,\left (\frac {\sqrt {2\,r\,x-x^2}\,\left (12\,r^2+4\,r\,x-8\,x^2\right )}{24}+\frac {r^3\,\ln \left (x-r-\sqrt {x\,\left (2\,r-x\right )}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}\right )}{4} \]
[In]
[Out]