Integrand size = 24, antiderivative size = 121 \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {4}{27} \left (2-3 x^2\right )^{3/4}-\frac {2}{189} \left (2-3 x^2\right )^{7/4}+\frac {8}{27} \sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {8}{27} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right ) \]
4/27*(-3*x^2+2)^(3/4)-2/189*(-3*x^2+2)^(7/4)+8/27*2^(1/4)*arctan(1/2*(2^(1 /2)-(-3*x^2+2)^(1/2))*2^(1/4)/(-3*x^2+2)^(1/4))+8/27*2^(1/4)*arctanh(1/2*( 2^(1/2)+(-3*x^2+2)^(1/2))*2^(1/4)/(-3*x^2+2)^(1/4))
Time = 0.13 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.85 \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {1}{189} \left (6 \left (2-3 x^2\right )^{3/4} \left (4+x^2\right )+56 \sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+56 \sqrt [4]{2} \text {arctanh}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )\right ) \]
(6*(2 - 3*x^2)^(3/4)*(4 + x^2) + 56*2^(1/4)*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x ^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))] + 56*2^(1/4)*ArcTanh[(2*(4 - 6*x^2)^(1/4 ))/(2 + Sqrt[4 - 6*x^2])])/189
Time = 0.23 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {349, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 349 |
| \(\displaystyle \int \left (-\frac {4 x}{9 \sqrt [4]{2-3 x^2}}+\frac {16 x}{9 \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )}-\frac {x^3}{3 \sqrt [4]{2-3 x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8}{27} \sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {8}{27} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt {2-3 x^2}+\sqrt {2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-\frac {2}{189} \left (2-3 x^2\right )^{7/4}+\frac {4}{27} \left (2-3 x^2\right )^{3/4}\) |
(4*(2 - 3*x^2)^(3/4))/27 - (2*(2 - 3*x^2)^(7/4))/189 + (8*2^(1/4)*ArcTan[( Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))])/27 + (8*2^(1/4)*A rcTanh[(Sqrt[2] + Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))])/27
3.11.32.3.1 Defintions of rubi rules used
Int[(x_)^(m_)/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol ] :> Int[ExpandIntegrand[x^m/((a + b*x^2)^(1/4)*(c + d*x^2)), x], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && IntegerQ[m] && (PosQ[a] || In tegerQ[m/2])
Time = 4.94 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.08
| method | result | size |
| pseudoelliptic | \(\frac {2 x^{2} \left (-3 x^{2}+2\right )^{\frac {3}{4}}}{63}+\frac {8 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}{63}-\frac {4 \ln \left (\frac {-2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}+\sqrt {-3 x^{2}+2}}{2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}+\sqrt {-3 x^{2}+2}}\right ) 2^{\frac {1}{4}}}{27}-\frac {8 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+1\right ) 2^{\frac {1}{4}}}{27}-\frac {8 \arctan \left (-1+2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}\right ) 2^{\frac {1}{4}}}{27}\) | \(131\) |
| trager | \(\left (\frac {2 x^{2}}{63}+\frac {8}{63}\right ) \left (-3 x^{2}+2\right )^{\frac {3}{4}}+\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} \left (-3 x^{2}+2\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}+6 x^{2}}{3 x^{2}-4}\right )}{27}-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \left (-3 x^{2}+2\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}+6 x^{2}}{3 x^{2}-4}\right )}{27}\) | \(206\) |
| risch | \(-\frac {2 \left (3 x^{2}-2\right ) \left (x^{2}+4\right )}{63 \left (-3 x^{2}+2\right )^{\frac {1}{4}}}-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} \left (-3 x^{2}+2\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{27}+\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \left (-3 x^{2}+2\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{27}\) | \(210\) |
2/63*x^2*(-3*x^2+2)^(3/4)+8/63*(-3*x^2+2)^(3/4)-4/27*ln((-2^(3/4)*(-3*x^2+ 2)^(1/4)+2^(1/2)+(-3*x^2+2)^(1/2))/(2^(3/4)*(-3*x^2+2)^(1/4)+2^(1/2)+(-3*x ^2+2)^(1/2)))*2^(1/4)-8/27*arctan(2^(1/4)*(-3*x^2+2)^(1/4)+1)*2^(1/4)-8/27 *arctan(-1+2^(1/4)*(-3*x^2+2)^(1/4))*2^(1/4)
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.88 \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {2}{63} \, {\left (x^{2} + 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} - \frac {4}{27} \, \left (-8\right )^{\frac {1}{4}} \log \left (\left (-8\right )^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {4}{27} i \, \left (-8\right )^{\frac {1}{4}} \log \left (i \, \left (-8\right )^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \frac {4}{27} i \, \left (-8\right )^{\frac {1}{4}} \log \left (-i \, \left (-8\right )^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {4}{27} \, \left (-8\right )^{\frac {1}{4}} \log \left (-\left (-8\right )^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) \]
2/63*(x^2 + 4)*(-3*x^2 + 2)^(3/4) - 4/27*(-8)^(1/4)*log((-8)^(3/4) + 4*(-3 *x^2 + 2)^(1/4)) + 4/27*I*(-8)^(1/4)*log(I*(-8)^(3/4) + 4*(-3*x^2 + 2)^(1/ 4)) - 4/27*I*(-8)^(1/4)*log(-I*(-8)^(3/4) + 4*(-3*x^2 + 2)^(1/4)) + 4/27*( -8)^(1/4)*log(-(-8)^(3/4) + 4*(-3*x^2 + 2)^(1/4))
\[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{5}}{3 x^{2} \sqrt [4]{2 - 3 x^{2}} - 4 \sqrt [4]{2 - 3 x^{2}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16 \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\frac {2}{189} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {7}{4}} - \frac {8}{27} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {8}{27} \cdot 2^{\frac {1}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {4}{27} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {4}{27} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {4}{27} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} \]
-2/189*(-3*x^2 + 2)^(7/4) - 8/27*2^(1/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*( -3*x^2 + 2)^(1/4))) - 8/27*2^(1/4)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^ 2 + 2)^(1/4))) + 4/27*2^(1/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + s qrt(-3*x^2 + 2)) - 4/27*2^(1/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 4/27*(-3*x^2 + 2)^(3/4)
Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16 \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\frac {2}{189} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {7}{4}} - \frac {2}{27} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {2}{27} \cdot 8^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {4}{27} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {4}{27} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {4}{27} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} \]
-2/189*(-3*x^2 + 2)^(7/4) - 2/27*8^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*( -3*x^2 + 2)^(1/4))) - 2/27*8^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^ 2 + 2)^(1/4))) + 4/27*2^(1/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + s qrt(-3*x^2 + 2)) - 4/27*2^(1/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 4/27*(-3*x^2 + 2)^(3/4)
Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.59 \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {4\,{\left (2-3\,x^2\right )}^{3/4}}{27}-\frac {2\,{\left (2-3\,x^2\right )}^{7/4}}{189}+2^{1/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {8}{27}+\frac {8}{27}{}\mathrm {i}\right )+2^{1/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {8}{27}-\frac {8}{27}{}\mathrm {i}\right ) \]