Integrand size = 24, antiderivative size = 123 \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {4}{9} \sqrt [4]{2-3 x^2}-\frac {2}{135} \left (2-3 x^2\right )^{5/4}+\frac {4}{27} 2^{3/4} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-\frac {4}{27} 2^{3/4} \text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{2-3 x^2}}{\sqrt {2}+\sqrt {2-3 x^2}}\right ) \] Output:
4/9*(-3*x^2+2)^(1/4)-2/135*(-3*x^2+2)^(5/4)+4/27*2^(3/4)*arctan(1/2*(2^(1/ 2)-(-3*x^2+2)^(1/2))*2^(1/4)/(-3*x^2+2)^(1/4))-4/27*2^(3/4)*arctanh(2^(3/4 )*(-3*x^2+2)^(1/4)/(2^(1/2)+(-3*x^2+2)^(1/2)))
Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.85 \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {1}{135} \left (2 \sqrt [4]{2-3 x^2} \left (28+3 x^2\right )+20\ 2^{3/4} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-20\ 2^{3/4} \text {arctanh}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )\right ) \] Input:
Integrate[x^5/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
Output:
(2*(2 - 3*x^2)^(1/4)*(28 + 3*x^2) + 20*2^(3/4)*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))] - 20*2^(3/4)*ArcTanh[(2*(4 - 6*x^2)^( 1/4))/(2 + Sqrt[4 - 6*x^2])])/135
Time = 0.38 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.41, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {352, 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.
| \(\displaystyle \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 352 |
| \(\displaystyle \int \left (-\frac {4 x}{9 \left (2-3 x^2\right )^{3/4}}+\frac {16 x}{9 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )}-\frac {x^3}{3 \left (2-3 x^2\right )^{3/4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4}{27} 2^{3/4} \arctan \left (\sqrt [4]{4-6 x^2}+1\right )+\frac {4}{27} 2^{3/4} \arctan \left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )-\frac {2}{135} \left (2-3 x^2\right )^{5/4}+\frac {4}{9} \sqrt [4]{2-3 x^2}+\frac {2}{27} 2^{3/4} \log \left (\sqrt {2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )-\frac {2}{27} 2^{3/4} \log \left (\sqrt {2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )\) |
Input:
Int[x^5/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
Output:
(4*(2 - 3*x^2)^(1/4))/9 - (2*(2 - 3*x^2)^(5/4))/135 - (4*2^(3/4)*ArcTan[1 + (4 - 6*x^2)^(1/4)])/27 + (4*2^(3/4)*ArcTan[1 - 2^(1/4)*(2 - 3*x^2)^(1/4) ])/27 + (2*2^(3/4)*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^ 2]])/27 - (2*2^(3/4)*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3* x^2]])/27
Int[(x_)^(m_)/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol ] :> Int[ExpandIntegrand[x^m/((a + b*x^2)^(3/4)*(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 = 2.75 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07
| method | result | size |
| pseudoelliptic | \(\frac {2 x^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}}{45}+\frac {56 \left (-3 x^{2}+2\right )^{\frac {1}{4}}}{135}-\frac {2 \ln \left (\frac {\sqrt {-3 x^{2}+2}+2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}}{\sqrt {-3 x^{2}+2}-2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}}\right ) 2^{\frac {3}{4}}}{27}-\frac {4 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+1\right ) 2^{\frac {3}{4}}}{27}-\frac {4 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-1\right ) 2^{\frac {3}{4}}}{27}\) | \(131\) |
| trager | \(\left (\frac {2 x^{2}}{45}+\frac {56}{135}\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-3 x^{2}+2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \left (-3 x^{2}+2\right )^{\frac {3}{4}}-3 x^{2}}{3 x^{2}-4}\right )}{27}-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-2 \left (-3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-3 x^{2}+2}+3 x^{2}}{3 x^{2}-4}\right )}{27}\) | \(207\) |
| risch | \(-\frac {2 \left (3 x^{2}+28\right ) \left (3 x^{2}-2\right )}{135 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}-\frac {\left (\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {3}{4}}-6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-27 x^{6}+54 x^{4}-36 x^{2}+8}\, x^{2}+18 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} x^{4}-27 x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-27 x^{6}+54 x^{4}-36 x^{2}+8}-24 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} x^{2}+36 x^{4}+8 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}}-12 x^{2}}{\left (3 x^{2}-4\right ) \left (3 x^{2}-2\right )^{2}}\right )}{27}-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {3}{4}}-18 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} x^{4}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-27 x^{6}+54 x^{4}-36 x^{2}+8}\, x^{2}-27 x^{6}+24 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-27 x^{6}+54 x^{4}-36 x^{2}+8}+36 x^{4}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}}-12 x^{2}}{\left (3 x^{2}-4\right ) \left (3 x^{2}-2\right )^{2}}\right )}{27}\right ) {\left (-\left (3 x^{2}-2\right )^{3}\right )}^{\frac {1}{4}}}{\left (-3 x^{2}+2\right )^{\frac {3}{4}}}\) | \(535\) |
Input:
int(x^5/(-3*x^2+2)^(3/4)/(-3*x^2+4),x,method=_RETURNVERBOSE)
Output:
2/45*x^2*(-3*x^2+2)^(1/4)+56/135*(-3*x^2+2)^(1/4)-2/27*ln(((-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^(3/4)*(-3*x^2+2)^( 1/4)+2^(1/2)))*2^(3/4)-4/27*arctan(2^(1/4)*(-3*x^2+2)^(1/4)+1)*2^(3/4)-4/2 7*arctan(2^(1/4)*(-3*x^2+2)^(1/4)-1)*2^(3/4)
Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.11 \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {2}{135} \, {\left (3 \, x^{2} + 28\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - \frac {4}{27} \cdot 8^{\frac {1}{4}} \arctan \left (\frac {1}{4} \cdot 8^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 1\right ) - \frac {4}{27} \cdot 8^{\frac {1}{4}} \arctan \left (\frac {1}{4} \cdot 8^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 1\right ) - \frac {2}{27} \cdot 8^{\frac {1}{4}} \log \left (2 \, \sqrt {2} + 2 \cdot 8^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 2 \, \sqrt {-3 \, x^{2} + 2}\right ) + \frac {2}{27} \cdot 8^{\frac {1}{4}} \log \left (2 \, \sqrt {2} - 2 \cdot 8^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 2 \, \sqrt {-3 \, x^{2} + 2}\right ) \] Input:
integrate(x^5/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="fricas")
Output:
2/135*(3*x^2 + 28)*(-3*x^2 + 2)^(1/4) - 4/27*8^(1/4)*arctan(1/4*8^(3/4)*(- 3*x^2 + 2)^(1/4) + 1) - 4/27*8^(1/4)*arctan(1/4*8^(3/4)*(-3*x^2 + 2)^(1/4) - 1) - 2/27*8^(1/4)*log(2*sqrt(2) + 2*8^(1/4)*(-3*x^2 + 2)^(1/4) + 2*sqrt (-3*x^2 + 2)) + 2/27*8^(1/4)*log(2*sqrt(2) - 2*8^(1/4)*(-3*x^2 + 2)^(1/4) + 2*sqrt(-3*x^2 + 2))
\[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{5}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \] Input:
integrate(x**5/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)
Output:
-Integral(x**5/(3*x**2*(2 - 3*x**2)**(3/4) - 4*(2 - 3*x**2)**(3/4)), x)
Time = 0.12 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.14 \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {4}{27} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {4}{27} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {2}{27} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {2}{27} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {2}{135} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {5}{4}} + \frac {4}{9} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} \] Input:
integrate(x^5/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="maxima")
Output:
-4/27*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 4/27* 2^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) - 2/27*2^(3/ 4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/27*2^( 3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 2/135 *(-3*x^2 + 2)^(5/4) + 4/9*(-3*x^2 + 2)^(1/4)
Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.14 \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {4}{27} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {4}{27} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {2}{27} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {2}{27} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {2}{135} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {5}{4}} + \frac {4}{9} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} \] Input:
integrate(x^5/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="giac")
Output:
-4/27*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 4/27* 2^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) - 2/27*2^(3/ 4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/27*2^( 3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 2/135 *(-3*x^2 + 2)^(5/4) + 4/9*(-3*x^2 + 2)^(1/4)
Time = 1.47 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.58 \[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {4\,{\left (2-3\,x^2\right )}^{1/4}}{9}-\frac {2\,{\left (2-3\,x^2\right )}^{5/4}}{135}+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {4}{27}-\frac {4}{27}{}\mathrm {i}\right )+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {4}{27}+\frac {4}{27}{}\mathrm {i}\right ) \] Input:
int(-x^5/((2 - 3*x^2)^(3/4)*(3*x^2 - 4)),x)
Output:
(4*(2 - 3*x^2)^(1/4))/9 - 2^(3/4)*atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 + 1i /2))*(4/27 - 4i/27) - 2^(3/4)*atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 - 1i/2)) *(4/27 + 4i/27) - (2*(2 - 3*x^2)^(5/4))/135
\[ \int \frac {x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int \frac {x^{5}}{\left (-3 x^{2}+2\right )^{\frac {3}{4}} \left (-3 x^{2}+4\right )}d x \] Input:
int(x^5/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)
Output:
int(x^5/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)