Integrand size = 24, antiderivative size = 138 \[ \int \frac {x^7}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {56}{81} \sqrt [4]{2-3 x^2}-\frac {16}{405} \left (2-3 x^2\right )^{5/4}+\frac {2}{729} \left (2-3 x^2\right )^{9/4}+\frac {16}{81} 2^{3/4} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-\frac {16}{81} 2^{3/4} \text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{2-3 x^2}}{\sqrt {2}+\sqrt {2-3 x^2}}\right ) \] Output:
56/81*(-3*x^2+2)^(1/4)-16/405*(-3*x^2+2)^(5/4)+2/729*(-3*x^2+2)^(9/4)+16/8 1*2^(3/4)*arctan(1/2*(2^(1/2)-(-3*x^2+2)^(1/2))*2^(1/4)/(-3*x^2+2)^(1/4))- 16/81*2^(3/4)*arctanh(2^(3/4)*(-3*x^2+2)^(1/4)/(2^(1/2)+(-3*x^2+2)^(1/2)))
Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.80 \[ \int \frac {x^7}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {2 \sqrt [4]{2-3 x^2} \left (1136+156 x^2+45 x^4\right )+720\ 2^{3/4} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-720\ 2^{3/4} \text {arctanh}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )}{3645} \] Input:
Integrate[x^7/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
Output:
(2*(2 - 3*x^2)^(1/4)*(1136 + 156*x^2 + 45*x^4) + 720*2^(3/4)*ArcTan[(Sqrt[ 2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4))] - 720*2^(3/4)*ArcTanh[( 2*(4 - 6*x^2)^(1/4))/(2 + Sqrt[4 - 6*x^2])])/3645
Time = 0.44 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.36, 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.
| \(\displaystyle \int \frac {x^7}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 352 |
| \(\displaystyle \int \left (-\frac {16 x}{27 \left (2-3 x^2\right )^{3/4}}+\frac {64 x}{27 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )}-\frac {x^5}{3 \left (2-3 x^2\right )^{3/4}}-\frac {4 x^3}{9 \left (2-3 x^2\right )^{3/4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {16}{81} 2^{3/4} \arctan \left (\sqrt [4]{4-6 x^2}+1\right )+\frac {16}{81} 2^{3/4} \arctan \left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )+\frac {2}{729} \left (2-3 x^2\right )^{9/4}-\frac {16}{405} \left (2-3 x^2\right )^{5/4}+\frac {56}{81} \sqrt [4]{2-3 x^2}+\frac {8}{81} 2^{3/4} \log \left (\sqrt {2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )-\frac {8}{81} 2^{3/4} \log \left (\sqrt {2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )\) |
Input:
Int[x^7/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
Output:
(56*(2 - 3*x^2)^(1/4))/81 - (16*(2 - 3*x^2)^(5/4))/405 + (2*(2 - 3*x^2)^(9 /4))/729 - (16*2^(3/4)*ArcTan[1 + (4 - 6*x^2)^(1/4)])/81 + (16*2^(3/4)*Arc Tan[1 - 2^(1/4)*(2 - 3*x^2)^(1/4)])/81 + (8*2^(3/4)*Log[Sqrt[2] - 2^(3/4)* (2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/81 - (8*2^(3/4)*Log[Sqrt[2] + 2^(3/4 )*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/81
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 = 3.06 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91
| method | result | size |
| pseudoelliptic | \(\frac {8 \left (-\ln \left (\frac {\sqrt {-3 x^{2}+2}+2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}}{\sqrt {-3 x^{2}+2}-2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}}\right )-2 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-1\right )-2 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+1\right )\right ) 2^{\frac {3}{4}}}{81}+\frac {2 \left (-3 x^{2}+2\right )^{\frac {1}{4}} \left (45 x^{4}+156 x^{2}+1136\right )}{3645}\) | \(126\) |
| trager | \(\left (\frac {2}{81} x^{4}+\frac {104}{1215} x^{2}+\frac {2272}{3645}\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-3 x^{2}+2}-2 \left (-3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right )-3 x^{2}}{3 x^{2}-4}\right )}{81}-\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-3 x^{2}+2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \left (-3 x^{2}+2\right )^{\frac {3}{4}}-3 x^{2}}{3 x^{2}-4}\right )}{81}\) | \(211\) |
| risch | \(-\frac {2 \left (45 x^{4}+156 x^{2}+1136\right ) \left (3 x^{2}-2\right )}{3645 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}-\frac {\left (\frac {16 \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 )}{81}+\frac {16 \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}}-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}^{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}+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}^{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}-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 )}{81}\right ) {\left (-\left (3 x^{2}-2\right )^{3}\right )}^{\frac {1}{4}}}{\left (-3 x^{2}+2\right )^{\frac {3}{4}}}\) | \(541\) |
Input:
int(x^7/(-3*x^2+2)^(3/4)/(-3*x^2+4),x,method=_RETURNVERBOSE)
Output:
8/81*(-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*arctan(2^(1/4)*(-3*x^2+2)^(1/4) -1)-2*arctan(2^(1/4)*(-3*x^2+2)^(1/4)+1))*2^(3/4)+2/3645*(-3*x^2+2)^(1/4)* (45*x^4+156*x^2+1136)
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.03 \[ \int \frac {x^7}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {2}{3645} \, {\left (45 \, x^{4} + 156 \, x^{2} + 1136\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - \frac {16}{81} \cdot 8^{\frac {1}{4}} \arctan \left (\frac {1}{4} \cdot 8^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 1\right ) - \frac {16}{81} \cdot 8^{\frac {1}{4}} \arctan \left (\frac {1}{4} \cdot 8^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 1\right ) - \frac {8}{81} \cdot 8^{\frac {1}{4}} \log \left (2 \, \sqrt {2} + 2 \cdot 8^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 2 \, \sqrt {-3 \, x^{2} + 2}\right ) + \frac {8}{81} \cdot 8^{\frac {1}{4}} \log \left (2 \, \sqrt {2} - 2 \cdot 8^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 2 \, \sqrt {-3 \, x^{2} + 2}\right ) \] Input:
integrate(x^7/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="fricas")
Output:
2/3645*(45*x^4 + 156*x^2 + 1136)*(-3*x^2 + 2)^(1/4) - 16/81*8^(1/4)*arctan (1/4*8^(3/4)*(-3*x^2 + 2)^(1/4) + 1) - 16/81*8^(1/4)*arctan(1/4*8^(3/4)*(- 3*x^2 + 2)^(1/4) - 1) - 8/81*8^(1/4)*log(2*sqrt(2) + 2*8^(1/4)*(-3*x^2 + 2 )^(1/4) + 2*sqrt(-3*x^2 + 2)) + 8/81*8^(1/4)*log(2*sqrt(2) - 2*8^(1/4)*(-3 *x^2 + 2)^(1/4) + 2*sqrt(-3*x^2 + 2))
\[ \int \frac {x^7}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{7}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \] Input:
integrate(x**7/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)
Output:
-Integral(x**7/(3*x**2*(2 - 3*x**2)**(3/4) - 4*(2 - 3*x**2)**(3/4)), x)
Time = 0.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.09 \[ \int \frac {x^7}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {2}{729} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {9}{4}} - \frac {16}{81} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {16}{81} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {8}{81} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {8}{81} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {16}{405} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {5}{4}} + \frac {56}{81} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} \] Input:
integrate(x^7/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="maxima")
Output:
2/729*(-3*x^2 + 2)^(9/4) - 16/81*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*( -3*x^2 + 2)^(1/4))) - 16/81*2^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x ^2 + 2)^(1/4))) - 8/81*2^(3/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 8/81*2^(3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 16/405*(-3*x^2 + 2)^(5/4) + 56/81*(-3*x^2 + 2)^(1/4 )
Time = 0.14 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.16 \[ \int \frac {x^7}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {2}{729} \, {\left (3 \, x^{2} - 2\right )}^{2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - \frac {16}{81} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {16}{81} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {8}{81} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {8}{81} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {16}{405} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {5}{4}} + \frac {56}{81} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} \] Input:
integrate(x^7/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="giac")
Output:
2/729*(3*x^2 - 2)^2*(-3*x^2 + 2)^(1/4) - 16/81*2^(3/4)*arctan(1/2*2^(1/4)* (2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 16/81*2^(3/4)*arctan(-1/2*2^(1/4)*(2^( 3/4) - 2*(-3*x^2 + 2)^(1/4))) - 8/81*2^(3/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4 ) + sqrt(2) + sqrt(-3*x^2 + 2)) + 8/81*2^(3/4)*log(-2^(3/4)*(-3*x^2 + 2)^( 1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 16/405*(-3*x^2 + 2)^(5/4) + 56/81*(-3 *x^2 + 2)^(1/4)
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.59 \[ \int \frac {x^7}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {56\,{\left (2-3\,x^2\right )}^{1/4}}{81}-\frac {16\,{\left (2-3\,x^2\right )}^{5/4}}{405}+\frac {2\,{\left (2-3\,x^2\right )}^{9/4}}{729}+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {16}{81}-\frac {16}{81}{}\mathrm {i}\right )+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {16}{81}+\frac {16}{81}{}\mathrm {i}\right ) \] Input:
int(-x^7/((2 - 3*x^2)^(3/4)*(3*x^2 - 4)),x)
Output:
(56*(2 - 3*x^2)^(1/4))/81 - 2^(3/4)*atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 + 1i/2))*(16/81 - 16i/81) - 2^(3/4)*atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 - 1i /2))*(16/81 + 16i/81) - (16*(2 - 3*x^2)^(5/4))/405 + (2*(2 - 3*x^2)^(9/4)) /729
\[ \int \frac {x^7}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int \frac {x^{7}}{\left (-3 x^{2}+2\right )^{\frac {3}{4}} \left (-3 x^{2}+4\right )}d x \] Input:
int(x^7/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)
Output:
int(x^7/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)