Integrand size = 18, antiderivative size = 113 \[ \int x^3 \sqrt {2 r x-x^2} \, dx=-\frac {7}{8} r^3 (r-x) \sqrt {2 r x-x^2}-\frac {7}{12} r^2 \left (2 r x-x^2\right )^{3/2}-\frac {7}{20} r x \left (2 r x-x^2\right )^{3/2}-\frac {1}{5} x^2 \left (2 r x-x^2\right )^{3/2}+\frac {7}{4} r^5 \arctan \left (\frac {x}{\sqrt {2 r x-x^2}}\right ) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {684, 654, 626, 634, 209} \[ \int x^3 \sqrt {2 r x-x^2} \, dx=\frac {7}{4} r^5 \arctan \left (\frac {x}{\sqrt {2 r x-x^2}}\right )-\frac {7}{8} r^3 (r-x) \sqrt {2 r x-x^2}-\frac {7}{12} r^2 \left (2 r x-x^2\right )^{3/2}-\frac {7}{20} r x \left (2 r x-x^2\right )^{3/2}-\frac {1}{5} x^2 \left (2 r x-x^2\right )^{3/2} \]
[In]
[Out]
Rule 209
Rule 626
Rule 634
Rule 654
Rule 684
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{5} x^2 \left (2 r x-x^2\right )^{3/2}+\frac {1}{5} (7 r) \int x^2 \sqrt {2 r x-x^2} \, dx \\ & = -\frac {7}{20} r x \left (2 r x-x^2\right )^{3/2}-\frac {1}{5} x^2 \left (2 r x-x^2\right )^{3/2}+\frac {1}{4} \left (7 r^2\right ) \int x \sqrt {2 r x-x^2} \, dx \\ & = -\frac {7}{12} r^2 \left (2 r x-x^2\right )^{3/2}-\frac {7}{20} r x \left (2 r x-x^2\right )^{3/2}-\frac {1}{5} x^2 \left (2 r x-x^2\right )^{3/2}+\frac {1}{4} \left (7 r^3\right ) \int \sqrt {2 r x-x^2} \, dx \\ & = -\frac {7}{8} r^3 (r-x) \sqrt {2 r x-x^2}-\frac {7}{12} r^2 \left (2 r x-x^2\right )^{3/2}-\frac {7}{20} r x \left (2 r x-x^2\right )^{3/2}-\frac {1}{5} x^2 \left (2 r x-x^2\right )^{3/2}+\frac {1}{8} \left (7 r^5\right ) \int \frac {1}{\sqrt {2 r x-x^2}} \, dx \\ & = -\frac {7}{8} r^3 (r-x) \sqrt {2 r x-x^2}-\frac {7}{12} r^2 \left (2 r x-x^2\right )^{3/2}-\frac {7}{20} r x \left (2 r x-x^2\right )^{3/2}-\frac {1}{5} x^2 \left (2 r x-x^2\right )^{3/2}+\frac {1}{4} \left (7 r^5\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {2 r x-x^2}}\right ) \\ & = -\frac {7}{8} r^3 (r-x) \sqrt {2 r x-x^2}-\frac {7}{12} r^2 \left (2 r x-x^2\right )^{3/2}-\frac {7}{20} r x \left (2 r x-x^2\right )^{3/2}-\frac {1}{5} x^2 \left (2 r x-x^2\right )^{3/2}+\frac {7}{4} r^5 \arctan \left (\frac {x}{\sqrt {2 r x-x^2}}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.74 \[ \int x^3 \sqrt {2 r x-x^2} \, dx=\frac {1}{120} \sqrt {-x (-2 r+x)} \left (-105 r^4-35 r^3 x-14 r^2 x^2-6 r x^3+24 x^4+\frac {210 r^5 \log \left (-\sqrt {x}+\sqrt {-2 r+x}\right )}{\sqrt {x} \sqrt {-2 r+x}}\right ) \]
[In]
[Out]
Time = 0.32 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.58
method | result | size |
pseudoelliptic | \(-\frac {7 \arctan \left (\frac {\sqrt {x \left (2 r -x \right )}}{x}\right ) r^{5}}{4}-\frac {7 \sqrt {x \left (2 r -x \right )}\, \left (r^{4}+\frac {1}{3} r^{3} x +\frac {2}{15} r^{2} x^{2}+\frac {2}{35} r \,x^{3}-\frac {8}{35} x^{4}\right )}{8}\) | \(65\) |
risch | \(-\frac {\left (105 r^{4}+35 r^{3} x +14 r^{2} x^{2}+6 r \,x^{3}-24 x^{4}\right ) x \left (2 r -x \right )}{120 \sqrt {-x \left (-2 r +x \right )}}+\frac {7 r^{5} \arctan \left (\frac {x -r}{\sqrt {2 r x -x^{2}}}\right )}{8}\) | \(77\) |
default | \(-\frac {x^{2} \left (2 r x -x^{2}\right )^{\frac {3}{2}}}{5}+\frac {7 r \left (-\frac {x \left (2 r x -x^{2}\right )^{\frac {3}{2}}}{4}+\frac {5 r \left (-\frac {\left (2 r x -x^{2}\right )^{\frac {3}{2}}}{3}+r \left (-\frac {\left (2 r -2 x \right ) \sqrt {2 r x -x^{2}}}{4}+\frac {r^{2} \arctan \left (\frac {x -r}{\sqrt {2 r x -x^{2}}}\right )}{2}\right )\right )}{4}\right )}{5}\) | \(104\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.60 \[ \int x^3 \sqrt {2 r x-x^2} \, dx=-\frac {7}{4} \, r^{5} \arctan \left (\frac {\sqrt {2 \, r x - x^{2}}}{x}\right ) - \frac {1}{120} \, {\left (105 \, r^{4} + 35 \, r^{3} x + 14 \, r^{2} x^{2} + 6 \, r x^{3} - 24 \, x^{4}\right )} \sqrt {2 \, r x - x^{2}} \]
[In]
[Out]
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.90 \[ \int x^3 \sqrt {2 r x-x^2} \, dx=\frac {7 r^{5} \left (\begin {cases} - i \log {\left (2 r - 2 x + 2 i \sqrt {2 r x - x^{2}} \right )} & \text {for}\: r^{2} \neq 0 \\\frac {\left (- r + x\right ) \log {\left (- r + x \right )}}{\sqrt {- \left (- r + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{8} + \sqrt {2 r x - x^{2}} \left (- \frac {7 r^{4}}{8} - \frac {7 r^{3} x}{24} - \frac {7 r^{2} x^{2}}{60} - \frac {r x^{3}}{20} + \frac {x^{4}}{5}\right ) \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89 \[ \int x^3 \sqrt {2 r x-x^2} \, dx=-\frac {7}{8} \, r^{5} \arcsin \left (\frac {r - x}{r}\right ) - \frac {7}{8} \, \sqrt {2 \, r x - x^{2}} r^{4} + \frac {7}{8} \, \sqrt {2 \, r x - x^{2}} r^{3} x - \frac {7}{12} \, {\left (2 \, r x - x^{2}\right )}^{\frac {3}{2}} r^{2} - \frac {7}{20} \, {\left (2 \, r x - x^{2}\right )}^{\frac {3}{2}} r x - \frac {1}{5} \, {\left (2 \, r x - x^{2}\right )}^{\frac {3}{2}} x^{2} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.56 \[ \int x^3 \sqrt {2 r x-x^2} \, dx=-\frac {7}{8} \, r^{5} \arcsin \left (\frac {r - x}{r}\right ) \mathrm {sgn}\left (r\right ) - \frac {1}{120} \, {\left (105 \, r^{4} + {\left (35 \, r^{3} + 2 \, {\left (7 \, r^{2} + 3 \, {\left (r - 4 \, x\right )} x\right )} x\right )} x\right )} \sqrt {2 \, r x - x^{2}} \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.85 \[ \int x^3 \sqrt {2 r x-x^2} \, dx=-\frac {7\,r\,\left (\frac {x\,{\left (2\,r\,x-x^2\right )}^{3/2}}{4}+\frac {5\,r\,\left (\frac {\sqrt {2\,r\,x-x^2}\,\left (12\,r^2+4\,r\,x-8\,x^2\right )}{24}+\frac {r^3\,\ln \left (x-r-\sqrt {x\,\left (2\,r-x\right )}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}\right )}{4}\right )}{5}-\frac {x^2\,{\left (2\,r\,x-x^2\right )}^{3/2}}{5} \]
[In]
[Out]