Integrand size = 31, antiderivative size = 131 \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=-\frac {2}{9} \arctan \left (\frac {x^2}{\sqrt {x-x^4}}\right )-\frac {1}{18} \sqrt {8+7 i \sqrt {2}} \arctan \left (\frac {\sqrt {2-\frac {i}{\sqrt {2}}} x \sqrt {x-x^4}}{-1+x^3}\right )-\frac {1}{18} \sqrt {8-7 i \sqrt {2}} \arctan \left (\frac {\sqrt {2+\frac {i}{\sqrt {2}}} x \sqrt {x-x^4}}{-1+x^3}\right ) \]
-2/9*arctan(x^2/(-x^4+x)^(1/2))-1/18*(8+7*I*2^(1/2))^(1/2)*arctan(1/2*(8-2 *I*2^(1/2))^(1/2)*x*(-x^4+x)^(1/2)/(x^3-1))-1/18*(8-7*I*2^(1/2))^(1/2)*arc tan(1/2*(8+2*I*2^(1/2))^(1/2)*x*(-x^4+x)^(1/2)/(x^3-1))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.33 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.06 \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=\frac {\sqrt {x-x^4} \left (8 \text {arctanh}\left (\frac {1+x^{3/2}}{\sqrt {-1+x^3}}\right )-\text {RootSum}\left [9+4 \text {$\#$1}^2+22 \text {$\#$1}^4+4 \text {$\#$1}^6+9 \text {$\#$1}^8\&,\frac {9 \log \left (-1+x^{3/2}\right )-9 \log \left (\sqrt {-1+x^3}-\text {$\#$1}+x^{3/2} \text {$\#$1}\right )-11 \log \left (-1+x^{3/2}\right ) \text {$\#$1}^2+11 \log \left (\sqrt {-1+x^3}-\text {$\#$1}+x^{3/2} \text {$\#$1}\right ) \text {$\#$1}^2+11 \log \left (-1+x^{3/2}\right ) \text {$\#$1}^4-11 \log \left (\sqrt {-1+x^3}-\text {$\#$1}+x^{3/2} \text {$\#$1}\right ) \text {$\#$1}^4-9 \log \left (-1+x^{3/2}\right ) \text {$\#$1}^6+9 \log \left (\sqrt {-1+x^3}-\text {$\#$1}+x^{3/2} \text {$\#$1}\right ) \text {$\#$1}^6}{\text {$\#$1}+11 \text {$\#$1}^3+3 \text {$\#$1}^5+9 \text {$\#$1}^7}\&\right ]\right )}{18 \sqrt {x} \sqrt {-1+x^3}} \]
(Sqrt[x - x^4]*(8*ArcTanh[(1 + x^(3/2))/Sqrt[-1 + x^3]] - RootSum[9 + 4*#1 ^2 + 22*#1^4 + 4*#1^6 + 9*#1^8 & , (9*Log[-1 + x^(3/2)] - 9*Log[Sqrt[-1 + x^3] - #1 + x^(3/2)*#1] - 11*Log[-1 + x^(3/2)]*#1^2 + 11*Log[Sqrt[-1 + x^3 ] - #1 + x^(3/2)*#1]*#1^2 + 11*Log[-1 + x^(3/2)]*#1^4 - 11*Log[Sqrt[-1 + x ^3] - #1 + x^(3/2)*#1]*#1^4 - 9*Log[-1 + x^(3/2)]*#1^6 + 9*Log[Sqrt[-1 + x ^3] - #1 + x^(3/2)*#1]*#1^6)/(#1 + 11*#1^3 + 3*#1^5 + 9*#1^7) & ]))/(18*Sq rt[x]*Sqrt[-1 + x^3])
Time = 1.03 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.79, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2467, 2035, 7266, 2256, 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.
| \(\displaystyle \int \frac {\left (x^3+1\right ) \sqrt {x-x^4}}{3 x^6+4 x^3+2} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x-x^4} \int \frac {\sqrt {x} \sqrt {1-x^3} \left (x^3+1\right )}{3 x^6+4 x^3+2}dx}{\sqrt {x} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt {x-x^4} \int \frac {x \sqrt {1-x^3} \left (x^3+1\right )}{3 x^6+4 x^3+2}d\sqrt {x}}{\sqrt {x} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle \frac {2 \sqrt {x-x^4} \int \frac {\sqrt {1-x} (x+1)}{3 x^2+4 x+2}dx^{3/2}}{3 \sqrt {x} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 2256 |
| \(\displaystyle \frac {2 \sqrt {x-x^4} \int \left (\frac {\left (1-\frac {i}{\sqrt {2}}\right ) \sqrt {1-x}}{6 x-2 i \sqrt {2}+4}+\frac {\left (1+\frac {i}{\sqrt {2}}\right ) \sqrt {1-x}}{6 x+2 i \sqrt {2}+4}\right )dx^{3/2}}{3 \sqrt {x} \sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {x-x^4} \left (-\frac {1}{12} \left (2+i \sqrt {2}\right ) \arcsin \left (x^{3/2}\right )-\frac {1}{12} \left (2-i \sqrt {2}\right ) \arcsin \left (x^{3/2}\right )+\frac {\left (\sqrt {2}+i\right ) \arctan \left (\frac {x^{3/2}}{\sqrt {\frac {-\sqrt {2}+2 i}{-\sqrt {2}+5 i}} \sqrt {1-x}}\right )}{6 \sqrt {\frac {2 \left (-\sqrt {2}+2 i\right )}{-\sqrt {2}+5 i}}}+\frac {\left (7+4 i \sqrt {2}\right ) \arctan \left (\frac {x^{3/2}}{\sqrt {\frac {\sqrt {2}+2 i}{\sqrt {2}+5 i}} \sqrt {1-x}}\right )}{6 \sqrt {2 \left (-8+7 i \sqrt {2}\right )}}\right )}{3 \sqrt {x} \sqrt {1-x^3}}\) |
(2*Sqrt[x - x^4]*(-1/12*((2 - I*Sqrt[2])*ArcSin[x^(3/2)]) - ((2 + I*Sqrt[2 ])*ArcSin[x^(3/2)])/12 + ((I + Sqrt[2])*ArcTan[x^(3/2)/(Sqrt[(2*I - Sqrt[2 ])/(5*I - Sqrt[2])]*Sqrt[1 - x])])/(6*Sqrt[(2*(2*I - Sqrt[2]))/(5*I - Sqrt [2])]) + ((7 + (4*I)*Sqrt[2])*ArcTan[x^(3/2)/(Sqrt[(2*I + Sqrt[2])/(5*I + Sqrt[2])]*Sqrt[1 - x])])/(6*Sqrt[2*(-8 + (7*I)*Sqrt[2])])))/(3*Sqrt[x]*Sqr t[1 - x^3])
3.19.95.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4 )^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(367\) vs. \(2(101)=202\).
Time = 4.40 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.81
| method | result | size |
| default | \(\frac {\left (-\sqrt {2}-3\right ) \ln \left (\frac {-\sqrt {-x^{4}+x}\, \sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x +3 \sqrt {2}\, x^{3}-2 x^{3}+2}{x^{3}}\right )+6 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}-4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right ) \sqrt {2}-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}+4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right ) \sqrt {2}+\sqrt {2}\, \ln \left (\frac {\sqrt {-x^{4}+x}\, \sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x +3 \sqrt {2}\, x^{3}-2 x^{3}+2}{x^{3}}\right )+8 \arctan \left (\frac {\sqrt {-x^{4}+x}}{x^{2}}\right ) \sqrt {4+3 \sqrt {2}}+10 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}-4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right )-10 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}+4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right )+3 \ln \left (\frac {\sqrt {-x^{4}+x}\, \sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x +3 \sqrt {2}\, x^{3}-2 x^{3}+2}{x^{3}}\right )}{36 \sqrt {4+3 \sqrt {2}}}\) | \(368\) |
| pseudoelliptic | \(\frac {\left (-\sqrt {2}-3\right ) \ln \left (\frac {-\sqrt {-x^{4}+x}\, \sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x +3 \sqrt {2}\, x^{3}-2 x^{3}+2}{x^{3}}\right )+6 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}-4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right ) \sqrt {2}-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}+4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right ) \sqrt {2}+\sqrt {2}\, \ln \left (\frac {\sqrt {-x^{4}+x}\, \sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x +3 \sqrt {2}\, x^{3}-2 x^{3}+2}{x^{3}}\right )+8 \arctan \left (\frac {\sqrt {-x^{4}+x}}{x^{2}}\right ) \sqrt {4+3 \sqrt {2}}+10 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}-4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right )-10 \arctan \left (\frac {\sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x^{2}+4 \sqrt {-x^{4}+x}}{2 x^{2} \sqrt {4+3 \sqrt {2}}}\right )+3 \ln \left (\frac {\sqrt {-x^{4}+x}\, \sqrt {2}\, \sqrt {6 \sqrt {2}-8}\, x +3 \sqrt {2}\, x^{3}-2 x^{3}+2}{x^{3}}\right )}{36 \sqrt {4+3 \sqrt {2}}}\) | \(368\) |
| elliptic | \(\text {Expression too large to display}\) | \(667\) |
1/36/(4+3*2^(1/2))^(1/2)*((-2^(1/2)-3)*ln((-(-x^4+x)^(1/2)*2^(1/2)*(6*2^(1 /2)-8)^(1/2)*x+3*2^(1/2)*x^3-2*x^3+2)/x^3)+6*arctan(1/2*(2^(1/2)*(6*2^(1/2 )-8)^(1/2)*x^2-4*(-x^4+x)^(1/2))/x^2/(4+3*2^(1/2))^(1/2))*2^(1/2)-6*arctan (1/2*(2^(1/2)*(6*2^(1/2)-8)^(1/2)*x^2+4*(-x^4+x)^(1/2))/x^2/(4+3*2^(1/2))^ (1/2))*2^(1/2)+2^(1/2)*ln(((-x^4+x)^(1/2)*2^(1/2)*(6*2^(1/2)-8)^(1/2)*x+3* 2^(1/2)*x^3-2*x^3+2)/x^3)+8*arctan(1/x^2*(-x^4+x)^(1/2))*(4+3*2^(1/2))^(1/ 2)+10*arctan(1/2*(2^(1/2)*(6*2^(1/2)-8)^(1/2)*x^2-4*(-x^4+x)^(1/2))/x^2/(4 +3*2^(1/2))^(1/2))-10*arctan(1/2*(2^(1/2)*(6*2^(1/2)-8)^(1/2)*x^2+4*(-x^4+ x)^(1/2))/x^2/(4+3*2^(1/2))^(1/2))+3*ln(((-x^4+x)^(1/2)*2^(1/2)*(6*2^(1/2) -8)^(1/2)*x+3*2^(1/2)*x^3-2*x^3+2)/x^3))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (97) = 194\).
Time = 0.48 (sec) , antiderivative size = 431, normalized size of antiderivative = 3.29 \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=-\frac {1}{72} \, \sqrt {7 i \, \sqrt {2} - 8} \log \left (-\frac {2 \, {\left (4160 \, x^{4} + \sqrt {2} {\left (6971 i \, x^{4} + 6034 i \, x\right )} - 1874 \, x\right )} \sqrt {-x^{4} + x} - {\left (683 \, x^{6} + 3684 \, x^{3} - 2 \, \sqrt {2} {\left (2231 i \, x^{6} + 651 i \, x^{3} - 524 i\right )} - 794\right )} \sqrt {7 i \, \sqrt {2} - 8}}{3 \, x^{6} + 4 \, x^{3} + 2}\right ) + \frac {1}{72} \, \sqrt {7 i \, \sqrt {2} - 8} \log \left (-\frac {2 \, {\left (4160 \, x^{4} + \sqrt {2} {\left (6971 i \, x^{4} + 6034 i \, x\right )} - 1874 \, x\right )} \sqrt {-x^{4} + x} + {\left (683 \, x^{6} + 3684 \, x^{3} + 2 \, \sqrt {2} {\left (-2231 i \, x^{6} - 651 i \, x^{3} + 524 i\right )} - 794\right )} \sqrt {7 i \, \sqrt {2} - 8}}{3 \, x^{6} + 4 \, x^{3} + 2}\right ) + \frac {1}{72} \, \sqrt {-7 i \, \sqrt {2} - 8} \log \left (-\frac {2 \, {\left (4160 \, x^{4} + \sqrt {2} {\left (-6971 i \, x^{4} - 6034 i \, x\right )} - 1874 \, x\right )} \sqrt {-x^{4} + x} + {\left (683 \, x^{6} + 3684 \, x^{3} + 2 \, \sqrt {2} {\left (2231 i \, x^{6} + 651 i \, x^{3} - 524 i\right )} - 794\right )} \sqrt {-7 i \, \sqrt {2} - 8}}{3 \, x^{6} + 4 \, x^{3} + 2}\right ) - \frac {1}{72} \, \sqrt {-7 i \, \sqrt {2} - 8} \log \left (-\frac {2 \, {\left (4160 \, x^{4} + \sqrt {2} {\left (-6971 i \, x^{4} - 6034 i \, x\right )} - 1874 \, x\right )} \sqrt {-x^{4} + x} - {\left (683 \, x^{6} + 3684 \, x^{3} - 2 \, \sqrt {2} {\left (-2231 i \, x^{6} - 651 i \, x^{3} + 524 i\right )} - 794\right )} \sqrt {-7 i \, \sqrt {2} - 8}}{3 \, x^{6} + 4 \, x^{3} + 2}\right ) + \frac {1}{9} \, \arctan \left (\frac {2 \, \sqrt {-x^{4} + x} x}{2 \, x^{3} - 1}\right ) \]
-1/72*sqrt(7*I*sqrt(2) - 8)*log(-(2*(4160*x^4 + sqrt(2)*(6971*I*x^4 + 6034 *I*x) - 1874*x)*sqrt(-x^4 + x) - (683*x^6 + 3684*x^3 - 2*sqrt(2)*(2231*I*x ^6 + 651*I*x^3 - 524*I) - 794)*sqrt(7*I*sqrt(2) - 8))/(3*x^6 + 4*x^3 + 2)) + 1/72*sqrt(7*I*sqrt(2) - 8)*log(-(2*(4160*x^4 + sqrt(2)*(6971*I*x^4 + 60 34*I*x) - 1874*x)*sqrt(-x^4 + x) + (683*x^6 + 3684*x^3 + 2*sqrt(2)*(-2231* I*x^6 - 651*I*x^3 + 524*I) - 794)*sqrt(7*I*sqrt(2) - 8))/(3*x^6 + 4*x^3 + 2)) + 1/72*sqrt(-7*I*sqrt(2) - 8)*log(-(2*(4160*x^4 + sqrt(2)*(-6971*I*x^4 - 6034*I*x) - 1874*x)*sqrt(-x^4 + x) + (683*x^6 + 3684*x^3 + 2*sqrt(2)*(2 231*I*x^6 + 651*I*x^3 - 524*I) - 794)*sqrt(-7*I*sqrt(2) - 8))/(3*x^6 + 4*x ^3 + 2)) - 1/72*sqrt(-7*I*sqrt(2) - 8)*log(-(2*(4160*x^4 + sqrt(2)*(-6971* I*x^4 - 6034*I*x) - 1874*x)*sqrt(-x^4 + x) - (683*x^6 + 3684*x^3 - 2*sqrt( 2)*(-2231*I*x^6 - 651*I*x^3 + 524*I) - 794)*sqrt(-7*I*sqrt(2) - 8))/(3*x^6 + 4*x^3 + 2)) + 1/9*arctan(2*sqrt(-x^4 + x)*x/(2*x^3 - 1))
Timed out. \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=\text {Timed out} \]
\[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=\int { \frac {\sqrt {-x^{4} + x} {\left (x^{3} + 1\right )}}{3 \, x^{6} + 4 \, x^{3} + 2} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (97) = 194\).
Time = 0.42 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.58 \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=-\frac {1}{36} \, \sqrt {18 \, \sqrt {2} + 16} \arctan \left (\frac {2 \, \left (\frac {9}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {9}{2}\right )^{\frac {1}{4}} {\left (\sqrt {6} - 2 \, \sqrt {3}\right )} + 6 \, \sqrt {\frac {1}{x^{3}} - 1}\right )}}{9 \, {\left (\sqrt {6} + 2 \, \sqrt {3}\right )}}\right ) + \frac {1}{36} \, \sqrt {18 \, \sqrt {2} + 16} \arctan \left (\frac {2 \, \left (\frac {9}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {9}{2}\right )^{\frac {1}{4}} {\left (\sqrt {6} - 2 \, \sqrt {3}\right )} - 6 \, \sqrt {\frac {1}{x^{3}} - 1}\right )}}{9 \, {\left (\sqrt {6} + 2 \, \sqrt {3}\right )}}\right ) - \frac {1}{72} \, \sqrt {18 \, \sqrt {2} - 16} \log \left (\frac {1}{3} \, {\left (\sqrt {6} \left (\frac {9}{2}\right )^{\frac {1}{4}} - 2 \, \left (\frac {9}{2}\right )^{\frac {1}{4}} \sqrt {3}\right )} \sqrt {\frac {1}{x^{3}} - 1} + 3 \, \sqrt {\frac {1}{2}} + \frac {1}{x^{3}} - 1\right ) + \frac {1}{72} \, \sqrt {18 \, \sqrt {2} - 16} \log \left (-\frac {1}{3} \, {\left (\sqrt {6} \left (\frac {9}{2}\right )^{\frac {1}{4}} - 2 \, \left (\frac {9}{2}\right )^{\frac {1}{4}} \sqrt {3}\right )} \sqrt {\frac {1}{x^{3}} - 1} + 3 \, \sqrt {\frac {1}{2}} + \frac {1}{x^{3}} - 1\right ) + \frac {2}{9} \, \arctan \left (\sqrt {\frac {1}{x^{3}} - 1}\right ) \]
-1/36*sqrt(18*sqrt(2) + 16)*arctan(2/9*(9/2)^(3/4)*((9/2)^(1/4)*(sqrt(6) - 2*sqrt(3)) + 6*sqrt(1/x^3 - 1))/(sqrt(6) + 2*sqrt(3))) + 1/36*sqrt(18*sqr t(2) + 16)*arctan(2/9*(9/2)^(3/4)*((9/2)^(1/4)*(sqrt(6) - 2*sqrt(3)) - 6*s qrt(1/x^3 - 1))/(sqrt(6) + 2*sqrt(3))) - 1/72*sqrt(18*sqrt(2) - 16)*log(1/ 3*(sqrt(6)*(9/2)^(1/4) - 2*(9/2)^(1/4)*sqrt(3))*sqrt(1/x^3 - 1) + 3*sqrt(1 /2) + 1/x^3 - 1) + 1/72*sqrt(18*sqrt(2) - 16)*log(-1/3*(sqrt(6)*(9/2)^(1/4 ) - 2*(9/2)^(1/4)*sqrt(3))*sqrt(1/x^3 - 1) + 3*sqrt(1/2) + 1/x^3 - 1) + 2/ 9*arctan(sqrt(1/x^3 - 1))
Timed out. \[ \int \frac {\left (1+x^3\right ) \sqrt {x-x^4}}{2+4 x^3+3 x^6} \, dx=\int \frac {\sqrt {x-x^4}\,\left (x^3+1\right )}{3\,x^6+4\,x^3+2} \,d x \]