Integrand size = 27, antiderivative size = 182 \[ \int \frac {x \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=-\frac {\sqrt {b+2 c-\sqrt {b^2-4 a c}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}+\frac {\sqrt {b+2 c+\sqrt {b^2-4 a c}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \]
-1/2*arctanh(2^(1/2)*c^(1/2)*(-x^2+1)^(1/2)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/ 2))*(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2)/c^(1/2)/(-4*a*c+b^2)^(1/2)+1/ 2*arctanh(2^(1/2)*c^(1/2)*(-x^2+1)^(1/2)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)) *(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2)/c^(1/2)/(-4*a*c+b^2)^(1/2)
Time = 0.23 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.93 \[ \int \frac {x \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\frac {\sqrt {-b-2 c-\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-b-2 c-\sqrt {b^2-4 a c}}}\right )-\sqrt {-b-2 c+\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-b-2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \]
(Sqrt[-b - 2*c - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2]) /Sqrt[-b - 2*c - Sqrt[b^2 - 4*a*c]]] - Sqrt[-b - 2*c + Sqrt[b^2 - 4*a*c]]* ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[-b - 2*c + Sqrt[b^2 - 4*a*c]]] )/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])
Time = 0.35 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1576, 1148, 1450, 221}
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.
| \(\displaystyle \int \frac {x \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {1}{2} \int \frac {\sqrt {1-x^2}}{c x^4+b x^2+a}dx^2\) |
| \(\Big \downarrow \) 1148 |
| \(\displaystyle -\int \frac {x^4}{c x^8-(b+2 c) x^4+a+b+c}d\sqrt {1-x^2}\) |
| \(\Big \downarrow \) 1450 |
| \(\displaystyle -\frac {1}{2} \left (\frac {b+2 c}{\sqrt {b^2-4 a c}}+1\right ) \int \frac {1}{c x^4+\frac {1}{2} \left (-b-2 c-\sqrt {b^2-4 a c}\right )}d\sqrt {1-x^2}-\frac {1}{2} \left (1-\frac {b+2 c}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^4+\frac {1}{2} \left (-b-2 c+\sqrt {b^2-4 a c}\right )}d\sqrt {1-x^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (1-\frac {b+2 c}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}+\frac {\left (\frac {b+2 c}{\sqrt {b^2-4 a c}}+1\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}\) |
((1 - (b + 2*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2]) /Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b + 2*c - Sqrt[ b^2 - 4*a*c]]) + ((1 + (b + 2*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[ c]*Sqrt[1 - x^2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqr t[b + 2*c + Sqrt[b^2 - 4*a*c]])
3.4.78.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[Sqrt[(d_.) + (e_.)*(x_)]/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[2*e Subst[Int[x^2/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c *x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Wi th[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^2/2)*(b/q + 1) Int[(d*x)^(m - 2)/(b/ 2 + q/2 + c*x^2), x], x] - Simp[(d^2/2)*(b/q - 1) Int[(d*x)^(m - 2)/(b/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Time = 0.45 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.89
| method | result | size |
| pseudoelliptic | \(-\frac {\sqrt {2}\, \left (\frac {\left (\sqrt {-4 a c +b^{2}}-b -2 c \right ) \arctan \left (\frac {c \sqrt {-x^{2}+1}\, \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +b^{2}}-b -2 c \right ) c}}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}-b -2 c \right ) c}}-\frac {\left (b +2 c +\sqrt {-4 a c +b^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {-x^{2}+1}\, \sqrt {2}}{\sqrt {\left (b +2 c +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (b +2 c +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {-4 a c +b^{2}}}\) | \(162\) |
| default | \(2 a \left (\frac {\left (-2 \sqrt {-4 a c +b^{2}}\, a -b \sqrt {-4 a c +b^{2}}+4 a c -b^{2}\right ) \arctan \left (\frac {\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}+2 a +2 b}{2 \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}-\frac {\left (2 \sqrt {-4 a c +b^{2}}\, a +b \sqrt {-4 a c +b^{2}}+4 a c -b^{2}\right ) \arctan \left (\frac {-\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}-2 a -2 b}{2 \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )\) | \(360\) |
-1/2*2^(1/2)/(-4*a*c+b^2)^(1/2)*(((-4*a*c+b^2)^(1/2)-b-2*c)/(((-4*a*c+b^2) ^(1/2)-b-2*c)*c)^(1/2)*arctan(c*(-x^2+1)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2 )-b-2*c)*c)^(1/2))-(b+2*c+(-4*a*c+b^2)^(1/2))/((b+2*c+(-4*a*c+b^2)^(1/2))* c)^(1/2)*arctanh(c*(-x^2+1)^(1/2)*2^(1/2)/((b+2*c+(-4*a*c+b^2)^(1/2))*c)^( 1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 871 vs. \(2 (143) = 286\).
Time = 0.75 (sec) , antiderivative size = 871, normalized size of antiderivative = 4.79 \[ \int \frac {x \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=-\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {b + 2 \, c - \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {b x^{2} + \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}} + \sqrt {\frac {1}{2}} {\left ({\left (b^{2} - 4 \, a c\right )} x^{2} + \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {b + 2 \, c - \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} - 2 \, \sqrt {-x^{2} + 1} a + 2 \, a}{x^{2}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {b + 2 \, c - \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {b x^{2} + \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}} - \sqrt {\frac {1}{2}} {\left ({\left (b^{2} - 4 \, a c\right )} x^{2} + \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {b + 2 \, c - \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} - 2 \, \sqrt {-x^{2} + 1} a + 2 \, a}{x^{2}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {b + 2 \, c + \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {b x^{2} - \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}} + \sqrt {\frac {1}{2}} {\left ({\left (b^{2} - 4 \, a c\right )} x^{2} - \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {b + 2 \, c + \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} - 2 \, \sqrt {-x^{2} + 1} a + 2 \, a}{x^{2}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {b + 2 \, c + \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {b x^{2} - \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}} - \sqrt {\frac {1}{2}} {\left ({\left (b^{2} - 4 \, a c\right )} x^{2} - \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {b + 2 \, c + \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} - 2 \, \sqrt {-x^{2} + 1} a + 2 \, a}{x^{2}}\right ) \]
-1/2*sqrt(1/2)*sqrt((b + 2*c - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/ (b^2*c - 4*a*c^2))*log((b*x^2 + (b^2*c - 4*a*c^2)*x^2/sqrt(b^2*c^2 - 4*a*c ^3) + sqrt(1/2)*((b^2 - 4*a*c)*x^2 + (b^3*c - 4*a*b*c^2)*x^2/sqrt(b^2*c^2 - 4*a*c^3))*sqrt((b + 2*c - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^ 2*c - 4*a*c^2)) - 2*sqrt(-x^2 + 1)*a + 2*a)/x^2) + 1/2*sqrt(1/2)*sqrt((b + 2*c - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))*log(( b*x^2 + (b^2*c - 4*a*c^2)*x^2/sqrt(b^2*c^2 - 4*a*c^3) - sqrt(1/2)*((b^2 - 4*a*c)*x^2 + (b^3*c - 4*a*b*c^2)*x^2/sqrt(b^2*c^2 - 4*a*c^3))*sqrt((b + 2* c - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2)) - 2*sqrt (-x^2 + 1)*a + 2*a)/x^2) - 1/2*sqrt(1/2)*sqrt((b + 2*c + (b^2*c - 4*a*c^2) /sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))*log((b*x^2 - (b^2*c - 4*a*c^2 )*x^2/sqrt(b^2*c^2 - 4*a*c^3) + sqrt(1/2)*((b^2 - 4*a*c)*x^2 - (b^3*c - 4* a*b*c^2)*x^2/sqrt(b^2*c^2 - 4*a*c^3))*sqrt((b + 2*c + (b^2*c - 4*a*c^2)/sq rt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2)) - 2*sqrt(-x^2 + 1)*a + 2*a)/x^2) + 1/2*sqrt(1/2)*sqrt((b + 2*c + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3) )/(b^2*c - 4*a*c^2))*log((b*x^2 - (b^2*c - 4*a*c^2)*x^2/sqrt(b^2*c^2 - 4*a *c^3) - sqrt(1/2)*((b^2 - 4*a*c)*x^2 - (b^3*c - 4*a*b*c^2)*x^2/sqrt(b^2*c^ 2 - 4*a*c^3))*sqrt((b + 2*c + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/( b^2*c - 4*a*c^2)) - 2*sqrt(-x^2 + 1)*a + 2*a)/x^2)
\[ \int \frac {x \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\int \frac {x \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{a + b x^{2} + c x^{4}}\, dx \]
\[ \int \frac {x \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\int { \frac {\sqrt {-x^{2} + 1} x}{c x^{4} + b x^{2} + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (143) = 286\).
Time = 1.57 (sec) , antiderivative size = 591, normalized size of antiderivative = 3.25 \[ \int \frac {x \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\frac {{\left (2 \, b^{2} c^{2} - 8 \, a c^{3} - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} - \sqrt {b^{2} - 4 \, a c} c} b^{2} + 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} - \sqrt {b^{2} - 4 \, a c} c} a c - 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} - \sqrt {b^{2} - 4 \, a c} c} b c - 5 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} - \sqrt {b^{2} - 4 \, a c} c} c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-x^{2} + 1}}{\sqrt {-\frac {b + 2 \, c + \sqrt {{\left (b + 2 \, c\right )}^{2} - 4 \, {\left (a + b + c\right )} c}}{c}}}\right )}{2 \, {\left (b^{4} - 8 \, a b^{2} c + 2 \, b^{3} c + 16 \, a^{2} c^{2} - 8 \, a b c^{2} + 5 \, b^{2} c^{2} - 20 \, a c^{3}\right )} {\left | c \right |}} + \frac {{\left (2 \, b^{2} c^{2} - 8 \, a c^{3} - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} + \sqrt {b^{2} - 4 \, a c} c} b^{2} + 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} + \sqrt {b^{2} - 4 \, a c} c} a c - 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} + \sqrt {b^{2} - 4 \, a c} c} b c - 5 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - 2 \, c^{2} + \sqrt {b^{2} - 4 \, a c} c} c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-x^{2} + 1}}{\sqrt {-\frac {b + 2 \, c - \sqrt {{\left (b + 2 \, c\right )}^{2} - 4 \, {\left (a + b + c\right )} c}}{c}}}\right )}{2 \, {\left (b^{4} - 8 \, a b^{2} c + 2 \, b^{3} c + 16 \, a^{2} c^{2} - 8 \, a b c^{2} + 5 \, b^{2} c^{2} - 20 \, a c^{3}\right )} {\left | c \right |}} \]
1/2*(2*b^2*c^2 - 8*a*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - s qrt(b^2 - 4*a*c)*c)*b^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b*c - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*c^2 - 2*(b^2 - 4*a*c)*c^2)*arctan(2*sqrt(1/2)*sqrt( -x^2 + 1)/sqrt(-(b + 2*c + sqrt((b + 2*c)^2 - 4*(a + b + c)*c))/c))/((b^4 - 8*a*b^2*c + 2*b^3*c + 16*a^2*c^2 - 8*a*b*c^2 + 5*b^2*c^2 - 20*a*c^3)*abs (c)) + 1/2*(2*b^2*c^2 - 8*a*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2* c^2 + sqrt(b^2 - 4*a*c)*c)*b^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2 *c^2 + sqrt(b^2 - 4*a*c)*c)*a*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b*c - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*c^2 - 2*(b^2 - 4*a*c)*c^2)*arctan(2*sqrt(1/2 )*sqrt(-x^2 + 1)/sqrt(-(b + 2*c - sqrt((b + 2*c)^2 - 4*(a + b + c)*c))/c)) /((b^4 - 8*a*b^2*c + 2*b^3*c + 16*a^2*c^2 - 8*a*b*c^2 + 5*b^2*c^2 - 20*a*c ^3)*abs(c))
Time = 8.30 (sec) , antiderivative size = 649, normalized size of antiderivative = 3.57 \[ \int \frac {x \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}+2\,c\,\sqrt {b^2-4\,a\,c}-b^2\right )}{4\,c\,\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (4\,a\,c-b^2\right )}-\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+2\,c\,\sqrt {b^2-4\,a\,c}+b^2\right )}{4\,c\,\left (4\,a\,c-b^2\right )\,\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}+2\,c\,\sqrt {b^2-4\,a\,c}-b^2\right )}{4\,c\,\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (4\,a\,c-b^2\right )}-\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+2\,c\,\sqrt {b^2-4\,a\,c}+b^2\right )}{4\,c\,\left (4\,a\,c-b^2\right )\,\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}} \]
(log((((x*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 1)*1i)/((b - (b^2 - 4 *a*c)^(1/2))/(2*c) + 1)^(1/2) - (1 - x^2)^(1/2)*1i)/(x - (-(b - (b^2 - 4*a *c)^(1/2))/(2*c))^(1/2)))*(4*a*c + b*(b^2 - 4*a*c)^(1/2) + 2*c*(b^2 - 4*a* c)^(1/2) - b^2))/(4*c*((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2)*(4*a*c - b^2)) - (log((((x*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 1)*1i)/((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) - (1 - x^2)^(1/2)*1i)/(x - (-(b + ( b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)))*(b*(b^2 - 4*a*c)^(1/2) - 4*a*c + 2*c*(b ^2 - 4*a*c)^(1/2) + b^2))/(4*c*(4*a*c - b^2)*((b + (b^2 - 4*a*c)^(1/2))/(2 *c) + 1)^(1/2)) + (log((((x*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + 1)* 1i)/((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) + (1 - x^2)^(1/2)*1i)/(x + (-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)))*(4*a*c + b*(b^2 - 4*a*c)^(1/2) + 2*c*(b^2 - 4*a*c)^(1/2) - b^2))/(4*c*((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2)*(4*a*c - b^2)) - (log((((x*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1 /2) + 1)*1i)/((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) + (1 - x^2)^(1/2) *1i)/(x + (-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)))*(b*(b^2 - 4*a*c)^(1/2 ) - 4*a*c + 2*c*(b^2 - 4*a*c)^(1/2) + b^2))/(4*c*(4*a*c - b^2)*((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2))