Integrand size = 24, antiderivative size = 138 \[ \int \frac {x^7}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {56}{243} \left (2-3 x^2\right )^{3/4}-\frac {16}{567} \left (2-3 x^2\right )^{7/4}+\frac {2}{891} \left (2-3 x^2\right )^{11/4}+\frac {32}{81} \sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {32}{81} \sqrt [4]{2} \text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{2-3 x^2}}{\sqrt {2}+\sqrt {2-3 x^2}}\right ) \] Output:
56/243*(-3*x^2+2)^(3/4)-16/567*(-3*x^2+2)^(7/4)+2/891*(-3*x^2+2)^(11/4)+32 /81*2^(1/4)*arctan(1/2*(2^(1/2)-(-3*x^2+2)^(1/2))*2^(1/4)/(-3*x^2+2)^(1/4) )+32/81*2^(1/4)*arctanh(2^(3/4)*(-3*x^2+2)^(1/4)/(2^(1/2)+(-3*x^2+2)^(1/2) ))
Time = 0.17 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.80 \[ \int \frac {x^7}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {2 \left (2-3 x^2\right )^{3/4} \left (1712+540 x^2+189 x^4\right )+7392 \sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+7392 \sqrt [4]{2} \text {arctanh}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )}{18711} \] Input:
Integrate[x^7/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]
Output:
(2*(2 - 3*x^2)^(3/4)*(1712 + 540*x^2 + 189*x^4) + 7392*2^(1/4)*ArcTan[(Sqr t[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))] + 7392*2^(1/4)*ArcTan h[(2*(4 - 6*x^2)^(1/4))/(2 + Sqrt[4 - 6*x^2])])/18711
Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.99, 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.
| \(\displaystyle \int \frac {x^7}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 349 |
| \(\displaystyle \int \left (-\frac {16 x}{27 \sqrt [4]{2-3 x^2}}+\frac {64 x}{27 \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )}-\frac {x^5}{3 \sqrt [4]{2-3 x^2}}-\frac {4 x^3}{9 \sqrt [4]{2-3 x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {32}{81} \sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {32}{81} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt {2-3 x^2}+\sqrt {2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {2}{891} \left (2-3 x^2\right )^{11/4}-\frac {16}{567} \left (2-3 x^2\right )^{7/4}+\frac {56}{243} \left (2-3 x^2\right )^{3/4}\) |
Input:
Int[x^7/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]
Output:
(56*(2 - 3*x^2)^(3/4))/243 - (16*(2 - 3*x^2)^(7/4))/567 + (2*(2 - 3*x^2)^( 11/4))/891 + (32*2^(1/4)*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))])/81 + (32*2^(1/4)*ArcTanh[(Sqrt[2] + Sqrt[2 - 3*x^2])/(2^(3 /4)*(2 - 3*x^2)^(1/4))])/81
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 = 3.42 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91
| method | result | size |
| pseudoelliptic | \(\frac {16 \left (-2 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-1\right )-2 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+1\right )-\ln \left (\frac {\sqrt {-3 x^{2}+2}-2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}}{\sqrt {-3 x^{2}+2}+2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}}\right )\right ) 2^{\frac {1}{4}}}{81}+\frac {2 \left (-3 x^{2}+2\right )^{\frac {3}{4}} \left (189 x^{4}+540 x^{2}+1712\right )}{18711}\) | \(126\) |
| trager | \(\left (\frac {2}{99} x^{4}+\frac {40}{693} x^{2}+\frac {3424}{18711}\right ) \left (-3 x^{2}+2\right )^{\frac {3}{4}}-\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \left (-3 x^{2}+2\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}+6 x^{2}}{3 x^{2}-4}\right )}{81}-\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} \left (-3 x^{2}+2\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{81}\) | \(210\) |
| risch | \(-\frac {2 \left (189 x^{4}+540 x^{2}+1712\right ) \left (3 x^{2}-2\right )}{18711 \left (-3 x^{2}+2\right )^{\frac {1}{4}}}+\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \left (-3 x^{2}+2\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{81}-\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} \left (-3 x^{2}+2\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{81}\) | \(217\) |
Input:
int(x^7/(-3*x^2+2)^(1/4)/(-3*x^2+4),x,method=_RETURNVERBOSE)
Output:
16/81*(-2*arctan(2^(1/4)*(-3*x^2+2)^(1/4)-1)-2*arctan(2^(1/4)*(-3*x^2+2)^( 1/4)+1)-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^(1/4)+2/18711*(-3*x^2+2)^(3/4 )*(189*x^4+540*x^2+1712)
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.95 \[ \int \frac {x^7}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {2}{18711} \, {\left (189 \, x^{4} + 540 \, x^{2} + 1712\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} - \frac {32}{81} \cdot 2^{\frac {1}{4}} \arctan \left (2^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 1\right ) - \frac {32}{81} \cdot 2^{\frac {1}{4}} \arctan \left (2^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 1\right ) + \frac {16}{81} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {16}{81} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) \] Input:
integrate(x^7/(-3*x^2+2)^(1/4)/(-3*x^2+4),x, algorithm="fricas")
Output:
2/18711*(189*x^4 + 540*x^2 + 1712)*(-3*x^2 + 2)^(3/4) - 32/81*2^(1/4)*arct an(2^(1/4)*(-3*x^2 + 2)^(1/4) + 1) - 32/81*2^(1/4)*arctan(2^(1/4)*(-3*x^2 + 2)^(1/4) - 1) + 16/81*2^(1/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 16/81*2^(1/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt( 2) + sqrt(-3*x^2 + 2))
\[ \int \frac {x^7}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{7}}{3 x^{2} \sqrt [4]{2 - 3 x^{2}} - 4 \sqrt [4]{2 - 3 x^{2}}}\, dx \] Input:
integrate(x**7/(-3*x**2+2)**(1/4)/(-3*x**2+4),x)
Output:
-Integral(x**7/(3*x**2*(2 - 3*x**2)**(1/4) - 4*(2 - 3*x**2)**(1/4)), x)
Time = 0.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.09 \[ \int \frac {x^7}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {2}{891} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {11}{4}} - \frac {16}{567} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {7}{4}} - \frac {32}{81} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {32}{81} \cdot 2^{\frac {1}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {16}{81} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {16}{81} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {56}{243} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} \] Input:
integrate(x^7/(-3*x^2+2)^(1/4)/(-3*x^2+4),x, algorithm="maxima")
Output:
2/891*(-3*x^2 + 2)^(11/4) - 16/567*(-3*x^2 + 2)^(7/4) - 32/81*2^(1/4)*arct an(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 32/81*2^(1/4)*arctan(-1 /2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) + 16/81*2^(1/4)*log(2^(3/4)*( -3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 16/81*2^(1/4)*log(-2^(3/ 4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 56/243*(-3*x^2 + 2)^ (3/4)
Time = 0.13 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.16 \[ \int \frac {x^7}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {2}{891} \, {\left (3 \, x^{2} - 2\right )}^{2} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} - \frac {16}{567} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {7}{4}} - \frac {32}{81} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {32}{81} \cdot 2^{\frac {1}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {16}{81} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {16}{81} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {56}{243} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} \] Input:
integrate(x^7/(-3*x^2+2)^(1/4)/(-3*x^2+4),x, algorithm="giac")
Output:
2/891*(3*x^2 - 2)^2*(-3*x^2 + 2)^(3/4) - 16/567*(-3*x^2 + 2)^(7/4) - 32/81 *2^(1/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 32/81*2^(1 /4)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) + 16/81*2^(1/4)* log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 16/81*2^(1/ 4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 56/243* (-3*x^2 + 2)^(3/4)
Time = 1.68 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.59 \[ \int \frac {x^7}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {56\,{\left (2-3\,x^2\right )}^{3/4}}{243}-\frac {16\,{\left (2-3\,x^2\right )}^{7/4}}{567}+\frac {2\,{\left (2-3\,x^2\right )}^{11/4}}{891}+2^{1/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {32}{81}+\frac {32}{81}{}\mathrm {i}\right )+2^{1/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {32}{81}-\frac {32}{81}{}\mathrm {i}\right ) \] Input:
int(-x^7/((2 - 3*x^2)^(1/4)*(3*x^2 - 4)),x)
Output:
(56*(2 - 3*x^2)^(3/4))/243 - 2^(1/4)*atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 + 1i/2))*(32/81 + 32i/81) - 2^(1/4)*atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 - 1 i/2))*(32/81 - 32i/81) - (16*(2 - 3*x^2)^(7/4))/567 + (2*(2 - 3*x^2)^(11/4 ))/891
\[ \int \frac {x^7}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\int \frac {x^{7}}{\left (-3 x^{2}+2\right )^{\frac {1}{4}} \left (-3 x^{2}+4\right )}d x \] Input:
int(x^7/(-3*x^2+2)^(1/4)/(-3*x^2+4),x)
Output:
int(x^7/(-3*x^2+2)^(1/4)/(-3*x^2+4),x)