Integrand size = 24, antiderivative size = 63 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{9} \sqrt [4]{-1+3 x^2}+\frac {2}{135} \left (-1+3 x^2\right )^{5/4}-\frac {4}{27} \arctan \left (\sqrt [4]{-1+3 x^2}\right )-\frac {4}{27} \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right ) \]
2/9*(3*x^2-1)^(1/4)+2/135*(3*x^2-1)^(5/4)-4/27*arctan((3*x^2-1)^(1/4))-4/2 7*arctanh((3*x^2-1)^(1/4))
Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{135} \left (\sqrt [4]{-1+3 x^2} \left (14+3 x^2\right )-10 \arctan \left (\sqrt [4]{-1+3 x^2}\right )-10 \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right )\right ) \]
(2*((-1 + 3*x^2)^(1/4)*(14 + 3*x^2) - 10*ArcTan[(-1 + 3*x^2)^(1/4)] - 10*A rcTanh[(-1 + 3*x^2)^(1/4)]))/135
Time = 0.20 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {354, 25, 99, 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.
\(\displaystyle \int \frac {x^5}{\left (3 x^2-2\right ) \left (3 x^2-1\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int -\frac {x^4}{\left (2-3 x^2\right ) \left (3 x^2-1\right )^{3/4}}dx^2\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {x^4}{\left (2-3 x^2\right ) \left (3 x^2-1\right )^{3/4}}dx^2\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\frac {1}{2} \int \left (-\frac {1}{9} \sqrt [4]{3 x^2-1}+\frac {4}{9 \left (2-3 x^2\right ) \left (3 x^2-1\right )^{3/4}}-\frac {1}{3 \left (3 x^2-1\right )^{3/4}}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {8}{27} \arctan \left (\sqrt [4]{3 x^2-1}\right )-\frac {8}{27} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )+\frac {4}{135} \left (3 x^2-1\right )^{5/4}+\frac {4}{9} \sqrt [4]{3 x^2-1}\right )\) |
((4*(-1 + 3*x^2)^(1/4))/9 + (4*(-1 + 3*x^2)^(5/4))/135 - (8*ArcTan[(-1 + 3 *x^2)^(1/4)])/27 - (8*ArcTanh[(-1 + 3*x^2)^(1/4)])/27)/2
3.11.82.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 4.00 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.06
method | result | size |
pseudoelliptic | \(\frac {2 x^{2} \left (3 x^{2}-1\right )^{\frac {1}{4}}}{45}+\frac {28 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{135}+\frac {2 \ln \left (-1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{27}-\frac {2 \ln \left (1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{27}-\frac {4 \arctan \left (\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{27}\) | \(67\) |
trager | \(\left (\frac {2 x^{2}}{45}+\frac {28}{135}\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}-2 \sqrt {3 x^{2}-1}+3 x^{2}}{3 x^{2}-2}\right )}{27}+\frac {2 \ln \left (\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {3 x^{2}-1}-3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{27}\) | \(142\) |
risch | \(\frac {2 \left (3 x^{2}+14\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}}{135}+\frac {\left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-18 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{4}+27 x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {3}{4}}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{2}-6 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}\, x^{2}-18 x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}}+2 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}+3 x^{2}}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{2}}\right )}{27}+\frac {2 \ln \left (\frac {-27 x^{6}+18 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{4}-6 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}\, x^{2}+18 x^{4}+2 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {3}{4}}-12 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{2}+2 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}-3 x^{2}+2 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}}}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{2}}\right )}{27}\right ) {\left (\left (3 x^{2}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (3 x^{2}-1\right )^{\frac {3}{4}}}\) | \(419\) |
2/45*x^2*(3*x^2-1)^(1/4)+28/135*(3*x^2-1)^(1/4)+2/27*ln(-1+(3*x^2-1)^(1/4) )-2/27*ln(1+(3*x^2-1)^(1/4))-4/27*arctan((3*x^2-1)^(1/4))
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{135} \, {\left (3 \, x^{2} + 14\right )} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - \frac {4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \]
2/135*(3*x^2 + 14)*(3*x^2 - 1)^(1/4) - 4/27*arctan((3*x^2 - 1)^(1/4)) - 2/ 27*log((3*x^2 - 1)^(1/4) + 1) + 2/27*log((3*x^2 - 1)^(1/4) - 1)
Time = 5.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.19 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2 \left (3 x^{2} - 1\right )^{\frac {5}{4}}}{135} + \frac {2 \sqrt [4]{3 x^{2} - 1}}{9} + \frac {2 \log {\left (\sqrt [4]{3 x^{2} - 1} - 1 \right )}}{27} - \frac {2 \log {\left (\sqrt [4]{3 x^{2} - 1} + 1 \right )}}{27} - \frac {4 \operatorname {atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{27} \]
2*(3*x**2 - 1)**(5/4)/135 + 2*(3*x**2 - 1)**(1/4)/9 + 2*log((3*x**2 - 1)** (1/4) - 1)/27 - 2*log((3*x**2 - 1)**(1/4) + 1)/27 - 4*atan((3*x**2 - 1)**( 1/4))/27
Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{135} \, {\left (3 \, x^{2} - 1\right )}^{\frac {5}{4}} + \frac {2}{9} \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - \frac {4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \]
2/135*(3*x^2 - 1)^(5/4) + 2/9*(3*x^2 - 1)^(1/4) - 4/27*arctan((3*x^2 - 1)^ (1/4)) - 2/27*log((3*x^2 - 1)^(1/4) + 1) + 2/27*log((3*x^2 - 1)^(1/4) - 1)
Time = 0.38 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{135} \, {\left (3 \, x^{2} - 1\right )}^{\frac {5}{4}} + \frac {2}{9} \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - \frac {4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {2}{27} \, \log \left ({\left | {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
2/135*(3*x^2 - 1)^(5/4) + 2/9*(3*x^2 - 1)^(1/4) - 4/27*arctan((3*x^2 - 1)^ (1/4)) - 2/27*log((3*x^2 - 1)^(1/4) + 1) + 2/27*log(abs((3*x^2 - 1)^(1/4) - 1))
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2\,{\left (3\,x^2-1\right )}^{1/4}}{9}-\frac {4\,\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{27}+\frac {2\,{\left (3\,x^2-1\right )}^{5/4}}{135}+\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{27} \]