Integrand size = 26, antiderivative size = 65 \[ \int \frac {x^3 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {7}{72} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{36} \sqrt {2-3 x^2} \left (-1+3 x^2\right )^{3/2}-\frac {7}{144} \arcsin \left (3-6 x^2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {457, 81, 52, 55, 633, 222} \[ \int \frac {x^3 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {7}{144} \arcsin \left (3-6 x^2\right )-\frac {1}{36} \sqrt {2-3 x^2} \left (3 x^2-1\right )^{3/2}-\frac {7}{72} \sqrt {2-3 x^2} \sqrt {3 x^2-1} \]
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Rule 52
Rule 55
Rule 81
Rule 222
Rule 457
Rule 633
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x \sqrt {-1+3 x}}{\sqrt {2-3 x}} \, dx,x,x^2\right ) \\ & = -\frac {1}{36} \sqrt {2-3 x^2} \left (-1+3 x^2\right )^{3/2}+\frac {7}{24} \text {Subst}\left (\int \frac {\sqrt {-1+3 x}}{\sqrt {2-3 x}} \, dx,x,x^2\right ) \\ & = -\frac {7}{72} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{36} \sqrt {2-3 x^2} \left (-1+3 x^2\right )^{3/2}+\frac {7}{48} \text {Subst}\left (\int \frac {1}{\sqrt {2-3 x} \sqrt {-1+3 x}} \, dx,x,x^2\right ) \\ & = -\frac {7}{72} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{36} \sqrt {2-3 x^2} \left (-1+3 x^2\right )^{3/2}+\frac {7}{48} \text {Subst}\left (\int \frac {1}{\sqrt {-2+9 x-9 x^2}} \, dx,x,x^2\right ) \\ & = -\frac {7}{72} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{36} \sqrt {2-3 x^2} \left (-1+3 x^2\right )^{3/2}-\frac {7}{432} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{9}}} \, dx,x,9 \left (1-2 x^2\right )\right ) \\ & = -\frac {7}{72} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{36} \sqrt {2-3 x^2} \left (-1+3 x^2\right )^{3/2}-\frac {7}{144} \sin ^{-1}\left (3-6 x^2\right ) \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.02 \[ \int \frac {x^3 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\frac {1}{72} \left (\frac {\sqrt {2-3 x^2} \left (5-9 x^2-18 x^4\right )}{\sqrt {-1+3 x^2}}-7 \arctan \left (\frac {\sqrt {2-3 x^2}}{\sqrt {-1+3 x^2}}\right )\right ) \]
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Time = 3.14 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.25
method | result | size |
default | \(\frac {\sqrt {3 x^{2}-1}\, \sqrt {-3 x^{2}+2}\, \left (-12 \sqrt {-9 x^{4}+9 x^{2}-2}\, x^{2}+7 \arcsin \left (6 x^{2}-3\right )-10 \sqrt {-9 x^{4}+9 x^{2}-2}\right )}{144 \sqrt {-9 x^{4}+9 x^{2}-2}}\) | \(81\) |
elliptic | \(\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )}\, \left (-\frac {5 \sqrt {-9 x^{4}+9 x^{2}-2}}{72}+\frac {7 \arcsin \left (6 x^{2}-3\right )}{144}-\frac {\sqrt {-9 x^{4}+9 x^{2}-2}\, x^{2}}{12}\right )}{\sqrt {-3 x^{2}+2}\, \sqrt {3 x^{2}-1}}\) | \(84\) |
risch | \(\frac {\left (6 x^{2}+5\right ) \left (3 x^{2}-2\right ) \sqrt {3 x^{2}-1}\, \sqrt {\left (3 x^{2}-1\right ) \left (-3 x^{2}+2\right )}}{72 \sqrt {-\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )}\, \sqrt {-3 x^{2}+2}}+\frac {7 \arcsin \left (6 x^{2}-3\right ) \sqrt {\left (3 x^{2}-1\right ) \left (-3 x^{2}+2\right )}}{144 \sqrt {-3 x^{2}+2}\, \sqrt {3 x^{2}-1}}\) | \(116\) |
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Time = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.11 \[ \int \frac {x^3 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {1}{72} \, {\left (6 \, x^{2} + 5\right )} \sqrt {3 \, x^{2} - 1} \sqrt {-3 \, x^{2} + 2} - \frac {7}{144} \, \arctan \left (\frac {3 \, \sqrt {3 \, x^{2} - 1} {\left (2 \, x^{2} - 1\right )} \sqrt {-3 \, x^{2} + 2}}{2 \, {\left (9 \, x^{4} - 9 \, x^{2} + 2\right )}}\right ) \]
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\[ \int \frac {x^3 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int \frac {x^{3} \sqrt {3 x^{2} - 1}}{\sqrt {2 - 3 x^{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71 \[ \int \frac {x^3 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {1}{12} \, \sqrt {-9 \, x^{4} + 9 \, x^{2} - 2} x^{2} - \frac {5}{72} \, \sqrt {-9 \, x^{4} + 9 \, x^{2} - 2} + \frac {7}{144} \, \arcsin \left (6 \, x^{2} - 3\right ) \]
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Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.62 \[ \int \frac {x^3 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {1}{72} \, {\left (6 \, x^{2} + 5\right )} \sqrt {3 \, x^{2} - 1} \sqrt {-3 \, x^{2} + 2} + \frac {7}{72} \, \arcsin \left (\sqrt {3 \, x^{2} - 1}\right ) \]
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Time = 16.95 (sec) , antiderivative size = 414, normalized size of antiderivative = 6.37 \[ \int \frac {x^3 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {7\,\mathrm {atan}\left (\frac {\sqrt {3\,x^2-1}-\mathrm {i}}{\sqrt {2}-\sqrt {2-3\,x^2}}\right )}{36}+\frac {\frac {7\,\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}{36\,\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}+\frac {143\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^3}{36\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^3}-\frac {143\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^5}{36\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^5}-\frac {7\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^7}{36\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^7}+\frac {\sqrt {2}\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^2\,4{}\mathrm {i}}{9\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^2}-\frac {\sqrt {2}\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^4\,40{}\mathrm {i}}{9\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^4}+\frac {\sqrt {2}\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^6\,4{}\mathrm {i}}{9\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^6}}{\frac {4\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^2}+\frac {6\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^4}+\frac {4\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^6}+\frac {{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^8}+1} \]
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