Integrand size = 24, antiderivative size = 120 \[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {2-\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}{2+\sqrt {2} \sqrt {2-3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt {3}} \] Output:
1/18*arctan(1/6*(2-2^(1/2)*(-3*x^2+2)^(1/2))*2^(3/4)*3^(1/2)/x/(-3*x^2+2)^ (1/4))*2^(3/4)*3^(1/2)-1/18*arctanh(2^(1/4)*3^(1/2)*x*(-3*x^2+2)^(1/4)/(2+ 2^(1/2)*(-3*x^2+2)^(1/2)))*2^(3/4)*3^(1/2)
Time = 1.70 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {\arctan \left (\frac {-3 \sqrt {2} x^2+4 \sqrt {2-3 x^2}}{2\ 2^{3/4} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )+\text {arctanh}\left (\frac {2 \sqrt {3} x \sqrt [4]{4-6 x^2}}{3 x^2+2 \sqrt {4-6 x^2}}\right )}{6 \sqrt [4]{2} \sqrt {3}} \] Input:
Integrate[x^2/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
Output:
-1/6*(ArcTan[(-3*Sqrt[2]*x^2 + 4*Sqrt[2 - 3*x^2])/(2*2^(3/4)*Sqrt[3]*x*(2 - 3*x^2)^(1/4))] + ArcTanh[(2*Sqrt[3]*x*(4 - 6*x^2)^(1/4))/(3*x^2 + 2*Sqrt [4 - 6*x^2])])/(2^(1/4)*Sqrt[3])
Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {350}
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.
\(\displaystyle \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\) |
\(\Big \downarrow \) 350 |
\(\displaystyle \frac {\arctan \left (\frac {2-\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {2-3 x^2}+2}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}\) |
Input:
Int[x^2/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
Output:
ArcTan[(2 - Sqrt[2]*Sqrt[2 - 3*x^2])/(2^(1/4)*Sqrt[3]*x*(2 - 3*x^2)^(1/4)) ]/(3*2^(1/4)*Sqrt[3]) - ArcTanh[(2 + Sqrt[2]*Sqrt[2 - 3*x^2])/(2^(1/4)*Sqr t[3]*x*(2 - 3*x^2)^(1/4))]/(3*2^(1/4)*Sqrt[3])
Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] : > Simp[(-b/(a*d*Rt[b^2/a, 4]^3))*ArcTan[(b + Rt[b^2/a, 4]^2*Sqrt[a + b*x^2] )/(Rt[b^2/a, 4]^3*x*(a + b*x^2)^(1/4))], x] + Simp[(b/(a*d*Rt[b^2/a, 4]^3)) *ArcTanh[(b - Rt[b^2/a, 4]^2*Sqrt[a + b*x^2])/(Rt[b^2/a, 4]^3*x*(a + b*x^2) ^(1/4))], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[b^2/a ]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.49 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.54
method | result | size |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right ) \ln \left (\frac {\left (-3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{3}+9 \sqrt {-3 x^{2}+2}\, x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2} x -6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}}{3 x^{2}-4}\right )}{18}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}\right ) \ln \left (\frac {\left (-3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}\right )+9 \sqrt {-3 x^{2}+2}\, x -3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2} x +6 \left (-3 x^{2}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}\right )}{3 x^{2}-4}\right )}{18}\) | \(185\) |
Input:
int(x^2/(-3*x^2+2)^(3/4)/(-3*x^2+4),x,method=_RETURNVERBOSE)
Output:
1/18*RootOf(_Z^4+18)*ln(((-3*x^2+2)^(3/4)*RootOf(_Z^4+18)^3+9*(-3*x^2+2)^( 1/2)*x+3*RootOf(_Z^4+18)^2*x-6*RootOf(_Z^4+18)*(-3*x^2+2)^(1/4))/(3*x^2-4) )-1/18*RootOf(_Z^2+RootOf(_Z^4+18)^2)*ln(((-3*x^2+2)^(3/4)*RootOf(_Z^4+18) ^2*RootOf(_Z^2+RootOf(_Z^4+18)^2)+9*(-3*x^2+2)^(1/2)*x-3*RootOf(_Z^4+18)^2 *x+6*(-3*x^2+2)^(1/4)*RootOf(_Z^2+RootOf(_Z^4+18)^2))/(3*x^2-4))
Time = 0.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.24 \[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {1}{108} \cdot 18^{\frac {3}{4}} \arctan \left (\frac {3 \, x + 2 \cdot 18^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{3 \, x}\right ) - \frac {1}{108} \cdot 18^{\frac {3}{4}} \arctan \left (-\frac {3 \, x - 2 \cdot 18^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{3 \, x}\right ) - \frac {1}{216} \cdot 18^{\frac {3}{4}} \log \left (\frac {9 \, \sqrt {2} x^{2} + 2 \cdot 18^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 12 \, \sqrt {-3 \, x^{2} + 2}}{x^{2}}\right ) + \frac {1}{216} \cdot 18^{\frac {3}{4}} \log \left (\frac {9 \, \sqrt {2} x^{2} - 2 \cdot 18^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 12 \, \sqrt {-3 \, x^{2} + 2}}{x^{2}}\right ) \] Input:
integrate(x^2/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="fricas")
Output:
-1/108*18^(3/4)*arctan(1/3*(3*x + 2*18^(1/4)*(-3*x^2 + 2)^(1/4))/x) - 1/10 8*18^(3/4)*arctan(-1/3*(3*x - 2*18^(1/4)*(-3*x^2 + 2)^(1/4))/x) - 1/216*18 ^(3/4)*log((9*sqrt(2)*x^2 + 2*18^(3/4)*(-3*x^2 + 2)^(1/4)*x + 12*sqrt(-3*x ^2 + 2))/x^2) + 1/216*18^(3/4)*log((9*sqrt(2)*x^2 - 2*18^(3/4)*(-3*x^2 + 2 )^(1/4)*x + 12*sqrt(-3*x^2 + 2))/x^2)
\[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{2}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \] Input:
integrate(x**2/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)
Output:
-Integral(x**2/(3*x**2*(2 - 3*x**2)**(3/4) - 4*(2 - 3*x**2)**(3/4)), x)
\[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{2}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^2/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="maxima")
Output:
-integrate(x^2/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)), x)
\[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{2}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^2/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="giac")
Output:
integrate(-x^2/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)), x)
Timed out. \[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\int \frac {x^2}{{\left (2-3\,x^2\right )}^{3/4}\,\left (3\,x^2-4\right )} \,d x \] Input:
int(-x^2/((2 - 3*x^2)^(3/4)*(3*x^2 - 4)),x)
Output:
-int(x^2/((2 - 3*x^2)^(3/4)*(3*x^2 - 4)), x)
\[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\left (\int \frac {x^{2}}{3 \left (-3 x^{2}+2\right )^{\frac {3}{4}} x^{2}-4 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}d x \right ) \] Input:
int(x^2/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)
Output:
- int(x**2/(3*( - 3*x**2 + 2)**(3/4)*x**2 - 4*( - 3*x**2 + 2)**(3/4)),x)