Integrand size = 24, antiderivative size = 165 \[ \int \frac {1}{x^3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {\sqrt [4]{2-3 x^2}}{16 x^2}-\frac {15 \arctan \left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}+\frac {3 \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{32 \sqrt [4]{2}}-\frac {15 \text {arctanh}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac {3 \text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{2-3 x^2}}{\sqrt {2}+\sqrt {2-3 x^2}}\right )}{32 \sqrt [4]{2}} \] Output:
-1/16*(-3*x^2+2)^(1/4)/x^2-15/64*arctan(1/2*2^(3/4)*(-3*x^2+2)^(1/4))*2^(1 /4)+3/64*2^(3/4)*arctan(1/2*(2^(1/2)-(-3*x^2+2)^(1/2))*2^(1/4)/(-3*x^2+2)^ (1/4))-15/64*arctanh(1/2*2^(3/4)*(-3*x^2+2)^(1/4))*2^(1/4)-3/64*2^(3/4)*ar ctanh(2^(3/4)*(-3*x^2+2)^(1/4)/(2^(1/2)+(-3*x^2+2)^(1/2)))
Time = 0.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {4 \sqrt [4]{2-3 x^2}+15 \sqrt [4]{2} x^2 \arctan \left (\sqrt [4]{1-\frac {3 x^2}{2}}\right )-3\ 2^{3/4} x^2 \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+15 \sqrt [4]{2} x^2 \text {arctanh}\left (\sqrt [4]{1-\frac {3 x^2}{2}}\right )+3\ 2^{3/4} x^2 \text {arctanh}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )}{64 x^2} \] Input:
Integrate[1/(x^3*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
Output:
-1/64*(4*(2 - 3*x^2)^(1/4) + 15*2^(1/4)*x^2*ArcTan[(1 - (3*x^2)/2)^(1/4)] - 3*2^(3/4)*x^2*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1 /4))] + 15*2^(1/4)*x^2*ArcTanh[(1 - (3*x^2)/2)^(1/4)] + 3*2^(3/4)*x^2*ArcT anh[(2*(4 - 6*x^2)^(1/4))/(2 + Sqrt[4 - 6*x^2])])/x^2
Time = 0.47 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.30, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 352 |
| \(\displaystyle \int \left (-\frac {9 x}{16 \left (2-3 x^2\right )^{3/4} \left (3 x^2-4\right )}+\frac {3}{16 \left (2-3 x^2\right )^{3/4} x}+\frac {1}{4 \left (2-3 x^2\right )^{3/4} x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {15 \arctan \left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac {3 \arctan \left (\sqrt [4]{4-6 x^2}+1\right )}{32 \sqrt [4]{2}}+\frac {3 \arctan \left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{32 \sqrt [4]{2}}-\frac {15 \text {arctanh}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac {\sqrt [4]{2-3 x^2}}{16 x^2}+\frac {3 \log \left (\sqrt {2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{64 \sqrt [4]{2}}-\frac {3 \log \left (\sqrt {2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{64 \sqrt [4]{2}}\) |
Input:
Int[1/(x^3*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
Output:
-1/16*(2 - 3*x^2)^(1/4)/x^2 - (15*ArcTan[(2 - 3*x^2)^(1/4)/2^(1/4)])/(32*2 ^(3/4)) - (3*ArcTan[1 + (4 - 6*x^2)^(1/4)])/(32*2^(1/4)) + (3*ArcTan[1 - 2 ^(1/4)*(2 - 3*x^2)^(1/4)])/(32*2^(1/4)) - (15*ArcTanh[(2 - 3*x^2)^(1/4)/2^ (1/4)])/(32*2^(3/4)) + (3*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/(64*2^(1/4)) - (3*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sq rt[2 - 3*x^2]])/(64*2^(1/4))
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 = 25.66 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.20
| method | result | size |
| pseudoelliptic | \(\frac {-3 \ln \left (\frac {\sqrt {-3 x^{2}+2}+2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}}{\sqrt {-3 x^{2}+2}-2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}}\right ) 2^{\frac {3}{4}} x^{2}-6 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+1\right ) 2^{\frac {3}{4}} x^{2}-6 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-1\right ) 2^{\frac {3}{4}} x^{2}-15 \ln \left (\frac {-\left (-3 x^{2}+2\right )^{\frac {1}{4}}-2^{\frac {1}{4}}}{-\left (-3 x^{2}+2\right )^{\frac {1}{4}}+2^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}} x^{2}-30 \arctan \left (\frac {2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}}{2}\right ) 2^{\frac {1}{4}} x^{2}-8 \left (-3 x^{2}+2\right )^{\frac {1}{4}}}{128 x^{2}}\) | \(198\) |
| trager | \(-\frac {\left (-3 x^{2}+2\right )^{\frac {1}{4}}}{16 x^{2}}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \left (-3 x^{2}+2\right )^{\frac {3}{4}}+4 \sqrt {-3 x^{2}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}}{x^{2}}\right )}{128}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}+4 \left (-3 x^{2}+2\right )^{\frac {3}{4}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {-3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )}{x^{2}}\right )}{128}+\frac {3 \ln \left (\frac {-4 \sqrt {-3 x^{2}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}+4 \left (-3 x^{2}+2\right )^{\frac {3}{4}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )}{3 x^{2}-4}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}}{128}+\frac {3 \ln \left (\frac {-4 \sqrt {-3 x^{2}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}+4 \left (-3 x^{2}+2\right )^{\frac {3}{4}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )}{3 x^{2}-4}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )}{128}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {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 \left (-3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )-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 )^{3} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{2}-4 \sqrt {-3 x^{2}+2}}{3 x^{2}-4}\right )}{64}\) | \(582\) |
| risch | \(\text {Expression too large to display}\) | \(1540\) |
Input:
int(1/x^3/(-3*x^2+2)^(3/4)/(-3*x^2+4),x,method=_RETURNVERBOSE)
Output:
1/128*(-3*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)*x^2-6*arctan(2^(1/4)*( -3*x^2+2)^(1/4)+1)*2^(3/4)*x^2-6*arctan(2^(1/4)*(-3*x^2+2)^(1/4)-1)*2^(3/4 )*x^2-15*ln((-(-3*x^2+2)^(1/4)-2^(1/4))/(-(-3*x^2+2)^(1/4)+2^(1/4)))*2^(1/ 4)*x^2-30*arctan(1/2*2^(3/4)*(-3*x^2+2)^(1/4))*2^(1/4)*x^2-8*(-3*x^2+2)^(1 /4))/x^2
Time = 0.10 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x^3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {30 \cdot 8^{\frac {3}{4}} x^{2} \arctan \left (\frac {1}{2} \cdot 8^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + 24 \cdot 2^{\frac {3}{4}} x^{2} \arctan \left (2^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 1\right ) + 24 \cdot 2^{\frac {3}{4}} x^{2} \arctan \left (2^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 1\right ) + 12 \cdot 2^{\frac {3}{4}} x^{2} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - 12 \cdot 2^{\frac {3}{4}} x^{2} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + 15 \cdot 8^{\frac {3}{4}} x^{2} \log \left (8^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - 15 \cdot 8^{\frac {3}{4}} x^{2} \log \left (-8^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + 32 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{512 \, x^{2}} \] Input:
integrate(1/x^3/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="fricas")
Output:
-1/512*(30*8^(3/4)*x^2*arctan(1/2*8^(1/4)*(-3*x^2 + 2)^(1/4)) + 24*2^(3/4) *x^2*arctan(2^(1/4)*(-3*x^2 + 2)^(1/4) + 1) + 24*2^(3/4)*x^2*arctan(2^(1/4 )*(-3*x^2 + 2)^(1/4) - 1) + 12*2^(3/4)*x^2*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 12*2^(3/4)*x^2*log(-2^(3/4)*(-3*x^2 + 2)^( 1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 15*8^(3/4)*x^2*log(8^(3/4) + 4*(-3*x^ 2 + 2)^(1/4)) - 15*8^(3/4)*x^2*log(-8^(3/4) + 4*(-3*x^2 + 2)^(1/4)) + 32*( -3*x^2 + 2)^(1/4))/x^2
\[ \int \frac {1}{x^3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=- \int \frac {1}{3 x^{5} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 x^{3} \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \] Input:
integrate(1/x**3/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)
Output:
-Integral(1/(3*x**5*(2 - 3*x**2)**(3/4) - 4*x**3*(2 - 3*x**2)**(3/4)), x)
\[ \int \frac {1}{x^3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int { -\frac {1}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} x^{3}} \,d x } \] Input:
integrate(1/x^3/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="maxima")
Output:
-integrate(1/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)*x^3), x)
Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {3}{64} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {3}{64} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {3}{128} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {3}{128} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {15}{64} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \frac {15}{128} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {15}{128} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} - {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \frac {{\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{16 \, x^{2}} \] Input:
integrate(1/x^3/(-3*x^2+2)^(3/4)/(-3*x^2+4),x, algorithm="giac")
Output:
-3/64*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 3/64* 2^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) - 3/128*2^(3 /4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 3/128*2 ^(3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 15/ 64*2^(1/4)*arctan(1/2*2^(3/4)*(-3*x^2 + 2)^(1/4)) - 15/128*2^(1/4)*log(2^( 1/4) + (-3*x^2 + 2)^(1/4)) + 15/128*2^(1/4)*log(2^(1/4) - (-3*x^2 + 2)^(1/ 4)) - 1/16*(-3*x^2 + 2)^(1/4)/x^2
Time = 1.57 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {{\left (2-3\,x^2\right )}^{1/4}}{16\,x^2}-\frac {15\,2^{1/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,{\left (2-3\,x^2\right )}^{1/4}}{2}\right )}{64}+\frac {2^{1/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,{\left (2-3\,x^2\right )}^{1/4}\,1{}\mathrm {i}}{2}\right )\,15{}\mathrm {i}}{64}+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {3}{64}-\frac {3}{64}{}\mathrm {i}\right )+\frac {{\left (-1\right )}^{1/4}\,2^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,2^{3/4}\,{\left (2-3\,x^2\right )}^{1/4}}{2}\right )\,3{}\mathrm {i}}{32} \] Input:
int(-1/(x^3*(2 - 3*x^2)^(3/4)*(3*x^2 - 4)),x)
Output:
(2^(1/4)*atan((2^(3/4)*(2 - 3*x^2)^(1/4)*1i)/2)*15i)/64 - (15*2^(1/4)*atan ((2^(3/4)*(2 - 3*x^2)^(1/4))/2))/64 - (2 - 3*x^2)^(1/4)/(16*x^2) - 2^(3/4) *atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 - 1i/2))*(3/64 + 3i/64) + ((-1)^(1/4) *2^(1/4)*atan(((-1)^(1/4)*2^(3/4)*(2 - 3*x^2)^(1/4))/2)*3i)/32
\[ \int \frac {1}{x^3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\left (\int \frac {1}{3 \left (-3 x^{2}+2\right )^{\frac {3}{4}} x^{5}-4 \left (-3 x^{2}+2\right )^{\frac {3}{4}} x^{3}}d x \right ) \] Input:
int(1/x^3/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)
Output:
- int(1/(3*( - 3*x**2 + 2)**(3/4)*x**5 - 4*( - 3*x**2 + 2)**(3/4)*x**3),x )