Integrand size = 22, antiderivative size = 109 \[ \int \frac {x^5}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {3}{2} \left (1-x^2\right )^{2/3}+\frac {3}{10} \left (1-x^2\right )^{5/3}+\frac {9 \sqrt {3} \arctan \left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{2\ 2^{2/3}}-\frac {9 \log \left (3+x^2\right )}{4\ 2^{2/3}}+\frac {27 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}} \]
3/2*(-x^2+1)^(2/3)+3/10*(-x^2+1)^(5/3)-9/8*ln(x^2+3)*2^(1/3)+27/8*ln(2^(2/ 3)-(-x^2+1)^(1/3))*2^(1/3)+9/4*arctan(1/3*(1+(-2*x^2+2)^(1/3))*3^(1/2))*3^ (1/2)*2^(1/3)
Time = 0.17 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.11 \[ \int \frac {x^5}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {1}{40} \left (72 \left (1-x^2\right )^{2/3}-12 x^2 \left (1-x^2\right )^{2/3}+90 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )+90 \sqrt [3]{2} \log \left (-2+\sqrt [3]{2-2 x^2}\right )-45 \sqrt [3]{2} \log \left (4+2 \sqrt [3]{2-2 x^2}+\left (2-2 x^2\right )^{2/3}\right )\right ) \]
(72*(1 - x^2)^(2/3) - 12*x^2*(1 - x^2)^(2/3) + 90*2^(1/3)*Sqrt[3]*ArcTan[( 1 + (2 - 2*x^2)^(1/3))/Sqrt[3]] + 90*2^(1/3)*Log[-2 + (2 - 2*x^2)^(1/3)] - 45*2^(1/3)*Log[4 + 2*(2 - 2*x^2)^(1/3) + (2 - 2*x^2)^(2/3)])/40
Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {354, 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.
| \(\displaystyle \int \frac {x^5}{\sqrt [3]{1-x^2} \left (x^2+3\right )} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4}{\sqrt [3]{1-x^2} \left (x^2+3\right )}dx^2\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {1}{2} \int \left (-\left (1-x^2\right )^{2/3}+\frac {9}{\left (x^2+3\right ) \sqrt [3]{1-x^2}}-\frac {2}{\sqrt [3]{1-x^2}}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {9 \sqrt {3} \arctan \left (\frac {\sqrt [3]{2-2 x^2}+1}{\sqrt {3}}\right )}{2^{2/3}}+\frac {3}{5} \left (1-x^2\right )^{5/3}+3 \left (1-x^2\right )^{2/3}-\frac {9 \log \left (x^2+3\right )}{2\ 2^{2/3}}+\frac {27 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{2\ 2^{2/3}}\right )\) |
(3*(1 - x^2)^(2/3) + (3*(1 - x^2)^(5/3))/5 + (9*Sqrt[3]*ArcTan[(1 + (2 - 2 *x^2)^(1/3))/Sqrt[3]])/2^(2/3) - (9*Log[3 + x^2])/(2*2^(2/3)) + (27*Log[2^ (2/3) - (1 - x^2)^(1/3)])/(2*2^(2/3)))/2
3.11.8.3.1 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 = 9.01 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (-x^{2}+1\right )^{\frac {2}{3}} x^{2}}{10}+\frac {9 \left (-x^{2}+1\right )^{\frac {2}{3}}}{5}+\frac {9 \,2^{\frac {1}{3}} \ln \left (\left (-x^{2}+1\right )^{\frac {1}{3}}-2^{\frac {2}{3}}\right )}{4}-\frac {9 \,2^{\frac {1}{3}} \ln \left (\left (-x^{2}+1\right )^{\frac {2}{3}}+2^{\frac {2}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}+2 \,2^{\frac {1}{3}}\right )}{8}+\frac {9 \sqrt {3}\, 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (1+2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}\right )}{3}\right )}{4}\) | \(111\) |
| trager | \(\left (-\frac {3 x^{2}}{10}+\frac {9}{5}\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}+27 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+144 \textit {\_Z}^{2}\right ) \ln \left (\frac {24 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+144 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{3} x^{2}+192 {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+144 \textit {\_Z}^{2}\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2} x^{2}+168 \left (-x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+144 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+5 \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right ) x^{2}+40 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+144 \textit {\_Z}^{2}\right ) x^{2}+14 \left (-x^{2}+1\right )^{\frac {2}{3}}-21 \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )-168 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+144 \textit {\_Z}^{2}\right )}{x^{2}+3}\right )+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right ) \ln \left (-\frac {24 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+144 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{3} x^{2}+432 {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+144 \textit {\_Z}^{2}\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2} x^{2}-252 \left (-x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+144 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )-\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right ) x^{2}-18 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+144 \textit {\_Z}^{2}\right ) x^{2}-21 \left (-x^{2}+1\right )^{\frac {2}{3}}+21 \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+378 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )^{2}+12 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )+144 \textit {\_Z}^{2}\right )}{x^{2}+3}\right )}{4}\) | \(478\) |
| risch | \(\text {Expression too large to display}\) | \(761\) |
-3/10*(-x^2+1)^(2/3)*x^2+9/5*(-x^2+1)^(2/3)+9/4*2^(1/3)*ln((-x^2+1)^(1/3)- 2^(2/3))-9/8*2^(1/3)*ln((-x^2+1)^(2/3)+2^(2/3)*(-x^2+1)^(1/3)+2*2^(1/3))+9 /4*3^(1/2)*2^(1/3)*arctan(1/3*3^(1/2)*(1+2^(1/3)*(-x^2+1)^(1/3)))
Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int \frac {x^5}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {3}{10} \, {\left (x^{2} - 6\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + \frac {9}{4} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {9}{16} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + \frac {9}{8} \cdot 4^{\frac {2}{3}} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) \]
-3/10*(x^2 - 6)*(-x^2 + 1)^(2/3) + 9/4*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)* sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^(1/3))) - 9/16*4^(2/3)*log(4^(2/3) + 4^(1/ 3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) + 9/8*4^(2/3)*log(-4^(1/3) + (-x^2 + 1)^(1/3))
\[ \int \frac {x^5}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {x^{5}}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]
Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.99 \[ \int \frac {x^5}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {9}{8} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {3}{10} \, {\left (-x^{2} + 1\right )}^{\frac {5}{3}} - \frac {9}{16} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + \frac {9}{8} \cdot 4^{\frac {2}{3}} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) + \frac {3}{2} \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} \]
9/8*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^(1 /3))) + 3/10*(-x^2 + 1)^(5/3) - 9/16*4^(2/3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) + 9/8*4^(2/3)*log(-4^(1/3) + (-x^2 + 1)^(1/3 )) + 3/2*(-x^2 + 1)^(2/3)
Time = 0.32 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.99 \[ \int \frac {x^5}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {9}{8} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {3}{10} \, {\left (-x^{2} + 1\right )}^{\frac {5}{3}} - \frac {9}{16} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + \frac {9}{8} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) + \frac {3}{2} \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}} \]
9/8*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^(1 /3))) + 3/10*(-x^2 + 1)^(5/3) - 9/16*4^(2/3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) + 9/8*4^(2/3)*log(4^(1/3) - (-x^2 + 1)^(1/3) ) + 3/2*(-x^2 + 1)^(2/3)
Time = 5.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.17 \[ \int \frac {x^5}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {9\,2^{1/3}\,\ln \left (\frac {729\,{\left (1-x^2\right )}^{1/3}}{4}-\frac {729\,2^{2/3}}{4}\right )}{4}+\frac {3\,{\left (1-x^2\right )}^{2/3}}{2}+\frac {3\,{\left (1-x^2\right )}^{5/3}}{10}+\frac {9\,2^{1/3}\,\ln \left (\frac {729\,{\left (1-x^2\right )}^{1/3}}{4}-\frac {729\,2^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{8}-\frac {9\,2^{1/3}\,\ln \left (\frac {729\,{\left (1-x^2\right )}^{1/3}}{4}-\frac {729\,2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{8} \]
(9*2^(1/3)*log((729*(1 - x^2)^(1/3))/4 - (729*2^(2/3))/4))/4 + (3*(1 - x^2 )^(2/3))/2 + (3*(1 - x^2)^(5/3))/10 + (9*2^(1/3)*log((729*(1 - x^2)^(1/3)) /4 - (729*2^(2/3)*(3^(1/2)*1i - 1)^2)/16)*(3^(1/2)*1i - 1))/8 - (9*2^(1/3) *log((729*(1 - x^2)^(1/3))/4 - (729*2^(2/3)*(3^(1/2)*1i + 1)^2)/16)*(3^(1/ 2)*1i + 1))/8