Integrand size = 29, antiderivative size = 281 \[ \int \frac {x^5 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=-\frac {b \sqrt {1-x^2}}{c^2}-\frac {\left (1-x^2\right )^{3/2}}{3 c}+\frac {\left (b^2-a c+b c-\frac {b^3-3 a b c+b^2 c-2 a c^2}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{5/2} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}+\frac {\left (b^2-a c+b c+\frac {b^3-3 a b c+b^2 c-2 a c^2}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{5/2} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]
-1/3*(-x^2+1)^(3/2)/c-b*(-x^2+1)^(1/2)/c^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-a*c+b*c+(3*a*b*c+2*a*c^ 2-b^3-b^2*c)/(-4*a*c+b^2)^(1/2))/c^(5/2)*2^(1/2)/(b+2*c-(-4*a*c+b^2)^(1/2) )^(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-a*c+b*c+(-3*a*b*c-2*a*c^2+b^3+b^2*c)/(-4*a*c+b^2)^(1/2))/c ^(5/2)*2^(1/2)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.79 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.23 \[ \int \frac {x^5 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\frac {2 \sqrt {c} \sqrt {1-x^2} \left (-3 b+c \left (-1+x^2\right )\right )-\frac {3 \sqrt {2} \left (b^3+b c \left (-3 a+\sqrt {b^2-4 a c}\right )+b^2 \left (c+\sqrt {b^2-4 a c}\right )-a c \left (2 c+\sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-b-2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b-2 c-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {2} \left (-b^3+a c \left (2 c-\sqrt {b^2-4 a c}\right )+b c \left (3 a+\sqrt {b^2-4 a c}\right )+b^2 \left (-c+\sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-b-2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b-2 c+\sqrt {b^2-4 a c}}}}{6 c^{5/2}} \]
(2*Sqrt[c]*Sqrt[1 - x^2]*(-3*b + c*(-1 + x^2)) - (3*Sqrt[2]*(b^3 + b*c*(-3 *a + Sqrt[b^2 - 4*a*c]) + b^2*(c + Sqrt[b^2 - 4*a*c]) - a*c*(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 - 4*a*c]*Sqrt[-b - 2*c - Sqrt[b^2 - 4*a*c]]) - (3 *Sqrt[2]*(-b^3 + a*c*(2*c - Sqrt[b^2 - 4*a*c]) + b*c*(3*a + Sqrt[b^2 - 4*a *c]) + 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 - 4*a*c]*Sqrt[-b - 2*c + Sqrt[b^2 - 4*a*c]]))/(6*c^(5/2))
Time = 3.18 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1578, 1199, 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.
| \(\displaystyle \int \frac {x^5 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4 \sqrt {1-x^2}}{c x^4+b x^2+a}dx^2\) |
| \(\Big \downarrow \) 1199 |
| \(\displaystyle -\int \left (\frac {x^4}{c}-\frac {b (a+b+c)-\left (b^2+c b-a c\right ) x^4}{c^2 \left (c x^8-(b+2 c) x^4+a+b+c\right )}+\frac {b}{c^2}\right )d\sqrt {1-x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (-\frac {-3 a b c-2 a c^2+b^3+b^2 c}{\sqrt {b^2-4 a c}}-a c+b^2+b c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {2} c^{5/2} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}+\frac {\left (\frac {-3 a b c-2 a c^2+b^3+b^2 c}{\sqrt {b^2-4 a c}}-a c+b^2+b c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {2} c^{5/2} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}-\frac {b \sqrt {1-x^2}}{c^2}-\frac {x^6}{3 c}\) |
-1/3*x^6/c - (b*Sqrt[1 - x^2])/c^2 + ((b^2 - a*c + b*c - (b^3 - 3*a*b*c + b^2*c - 2*a*c^2)/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]*c^(5/2)*Sqrt[b + 2*c - Sqrt [b^2 - 4*a*c]]) + ((b^2 - a*c + b*c + (b^3 - 3*a*b*c + b^2*c - 2*a*c^2)/Sq rt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[b + 2*c + Sq rt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(5/2)*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])
3.4.76.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e Subs t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer Q[n] && FractionQ[m]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(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] && Int egerQ[(m - 1)/2]
Time = 1.06 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.12
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (\left (\frac {\left (\left (-a +b \right ) c +b^{2}\right ) \sqrt {-4 a c +b^{2}}}{3}+\frac {2 a \,c^{2}}{3}+b \left (a -\frac {b}{3}\right ) c -\frac {b^{3}}{3}\right ) \sqrt {2}\, \sqrt {\left (b +2 c +\sqrt {-4 a c +b^{2}}\right ) c}\, \arctan \left (\frac {c \sqrt {-x^{2}+1}\, \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +b^{2}}-b -2 c \right ) c}}\right )+\sqrt {\left (\sqrt {-4 a c +b^{2}}-b -2 c \right ) c}\, \left (\sqrt {2}\, \left (\frac {\left (\left (a -b \right ) c -b^{2}\right ) \sqrt {-4 a c +b^{2}}}{3}+\frac {2 a \,c^{2}}{3}+b \left (a -\frac {b}{3}\right ) c -\frac {b^{3}}{3}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {-x^{2}+1}\, \sqrt {2}}{\sqrt {\left (b +2 c +\sqrt {-4 a c +b^{2}}\right ) c}}\right )+\frac {2 \left (b +\frac {1}{3} c -\frac {1}{3} c \,x^{2}\right ) \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +2 c +\sqrt {-4 a c +b^{2}}\right ) c}\, \sqrt {-x^{2}+1}}{3}\right )\right )}{2 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (\sqrt {-4 a c +b^{2}}-b -2 c \right ) c}\, \sqrt {\left (b +2 c +\sqrt {-4 a c +b^{2}}\right ) c}\, c^{2}}\) | \(316\) |
| risch | \(\frac {\left (-c \,x^{2}+3 b +c \right ) \left (x^{2}-1\right )}{3 c^{2} \sqrt {-x^{2}+1}}-\frac {2 a \left (\frac {\left (-2 a c \sqrt {-4 a c +b^{2}}+b^{2} \sqrt {-4 a c +b^{2}}+\sqrt {-4 a c +b^{2}}\, b c +4 a b c +4 a \,c^{2}-b^{3}-b^{2} c \right ) \arctan \left (\frac {\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}+2 a +2 b}{2 \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{\left (8 a c -2 b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}-\frac {\left (2 a c \sqrt {-4 a c +b^{2}}-b^{2} \sqrt {-4 a c +b^{2}}-\sqrt {-4 a c +b^{2}}\, b c +4 a b c +4 a \,c^{2}-b^{3}-b^{2} c \right ) \arctan \left (\frac {-\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}-2 a -2 b}{2 \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{\left (8 a c -2 b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{c^{2}}\) | \(446\) |
| default | \(-\frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{3 c}-\frac {\frac {2 a \left (\frac {\left (-2 a c \sqrt {-4 a c +b^{2}}+b^{2} \sqrt {-4 a c +b^{2}}+\sqrt {-4 a c +b^{2}}\, b c +4 a b c +4 a \,c^{2}-b^{3}-b^{2} c \right ) \arctan \left (\frac {\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}+2 a +2 b}{2 \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{\left (8 a c -2 b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}-\frac {\left (2 a c \sqrt {-4 a c +b^{2}}-b^{2} \sqrt {-4 a c +b^{2}}-\sqrt {-4 a c +b^{2}}\, b c +4 a b c +4 a \,c^{2}-b^{3}-b^{2} c \right ) \arctan \left (\frac {-\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}-2 a -2 b}{2 \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{\left (8 a c -2 b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{c}+\frac {2 b}{c \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+1\right )}}{c}\) | \(463\) |
-3/2/(-4*a*c+b^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))*c)^(1/2)*((1/3*((-a+b)*c+b^2)*(-4*a*c+b^2)^(1/2)+2/3*a*c^2+ b*(a-1/3*b)*c-1/3*b^3)*2^(1/2)*((b+2*c+(-4*a*c+b^2)^(1/2))*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))+(((-4*a*c+ b^2)^(1/2)-b-2*c)*c)^(1/2)*(2^(1/2)*(1/3*((a-b)*c-b^2)*(-4*a*c+b^2)^(1/2)+ 2/3*a*c^2+b*(a-1/3*b)*c-1/3*b^3)*arctanh(c*(-x^2+1)^(1/2)*2^(1/2)/((b+2*c+ (-4*a*c+b^2)^(1/2))*c)^(1/2))+2/3*(b+1/3*c-1/3*c*x^2)*(-4*a*c+b^2)^(1/2)*( (b+2*c+(-4*a*c+b^2)^(1/2))*c)^(1/2)*(-x^2+1)^(1/2)))/c^2
Leaf count of result is larger than twice the leaf count of optimal. 3615 vs. \(2 (237) = 474\).
Time = 3.73 (sec) , antiderivative size = 3615, normalized size of antiderivative = 12.86 \[ \int \frac {x^5 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
-1/6*(3*sqrt(1/2)*c^2*sqrt((b^5 + 2*a^2*c^3 + (5*a^2*b - 4*a*b^2)*c^2 - (5 *a*b^3 - b^4)*c - (b^2*c^5 - 4*a*c^6)*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b ^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a* c^6))*log(-(2*a^3*b^4 + (a^2*b^2*c^5 - 4*a^3*c^6)*x^2*sqrt((b^8 + (a^4 - 4 *a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^ 2*b^4 - 10*a*b^5 + b^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)) + 2*(a^5 - 2*a^4*b)*c^2 + (a^2*b^5 + (a^4*b - 2*a^3*b^2)*c^2 - (3*a^3*b^3 - a^2*b^4)*c)*x^2 - 2*(3*a^4*b^2 - a^3*b^3)*c + sqrt(1/2)*((b^5*c^5 - 7*a* b^3*c^6 + 12*a^2*b*c^7)*x^2*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)*c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b^6)*c^ 2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)) + (b^8 + 4*(a^4 - 2*a^3*b) *c^4 - (17*a^3*b^2 - 14*a^2*b^3)*c^3 + (20*a^2*b^4 - 7*a*b^5)*c^2 - (8*a*b ^6 - b^7)*c)*x^2)*sqrt((b^5 + 2*a^2*c^3 + (5*a^2*b - 4*a*b^2)*c^2 - (5*a*b ^3 - b^4)*c - (b^2*c^5 - 4*a*c^6)*sqrt((b^8 + (a^4 - 4*a^3*b + 4*a^2*b^2)* c^4 - 2*(3*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4)*c^3 + (11*a^2*b^4 - 10*a*b^5 + b ^6)*c^2 - 2*(3*a*b^6 - b^7)*c)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6) ) - 2*(a^3*b^4 + (a^5 - 2*a^4*b)*c^2 - (3*a^4*b^2 - a^3*b^3)*c)*sqrt(-x^2 + 1))/x^2) - 3*sqrt(1/2)*c^2*sqrt((b^5 + 2*a^2*c^3 + (5*a^2*b - 4*a*b^2)*c ^2 - (5*a*b^3 - b^4)*c - (b^2*c^5 - 4*a*c^6)*sqrt((b^8 + (a^4 - 4*a^3*b...
\[ \int \frac {x^5 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\int \frac {x^{5} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{a + b x^{2} + c x^{4}}\, dx \]
\[ \int \frac {x^5 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\int { \frac {\sqrt {-x^{2} + 1} x^{5}}{c x^{4} + b x^{2} + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 4637 vs. \(2 (237) = 474\).
Time = 1.15 (sec) , antiderivative size = 4637, normalized size of antiderivative = 16.50 \[ \int \frac {x^5 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
1/8*(2*b^6*c^4 - 14*a*b^4*c^5 + 6*b^5*c^5 + 24*a^2*b^2*c^6 - 40*a*b^3*c^6 + 4*b^4*c^6 + 64*a^2*b*c^7 - 24*a*b^2*c^7 + 32*a^2*c^8 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^6*c^2 + 7*sqrt(2)*sqrt (b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^4*c^3 - 5*sqrt( 2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^5*c^3 - 12 *sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*b^ 2*c^4 + 26*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c) *c)*a*b^3*c^4 - 13*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^4*c^4 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqr t(b^2 - 4*a*c)*c)*a^2*b*c^5 + 43*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c ^2 - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^5 - 19*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b *c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^3*c^5 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*s qrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*c^6 + 48*sqrt(2)*sqrt(b^2 - 4* a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b*c^6 - 10*sqrt(2)*sqrt(b^ 2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^2*c^6 + 20*sqrt(2)*s qrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*c^7 - 2*(b^2 - 4*a*c)*b^4*c^4 + 6*(b^2 - 4*a*c)*a*b^2*c^5 - 6*(b^2 - 4*a*c)*b^3*c^5 + 16 *(b^2 - 4*a*c)*a*b*c^6 - 4*(b^2 - 4*a*c)*b^2*c^6 + 8*(b^2 - 4*a*c)*a*c^7 - (2*b^6*c^2 - 18*a*b^4*c^3 + 2*b^5*c^3 + 48*a^2*b^2*c^4 - 16*a*b^3*c^4 - 3 2*a^3*c^5 + 32*a^2*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 ...
Time = 8.34 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.26 \[ \int \frac {x^5 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\sqrt {1-x^2}\,\left (\frac {2}{3\,c}-\frac {\frac {b}{c}+1}{c}+\frac {x^2}{3\,c}\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^3\,c+b^4-b^3\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2+2\,a\,c^2\,\sqrt {b^2-4\,a\,c}-b^2\,c\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c^2-5\,a\,b^2\,c+3\,a\,b\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,c^3\,\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^3\,c+b^4+b^3\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2-2\,a\,c^2\,\sqrt {b^2-4\,a\,c}+b^2\,c\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c^2-5\,a\,b^2\,c-3\,a\,b\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,c^3\,\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 (b^3\,c+b^4+b^3\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2-2\,a\,c^2\,\sqrt {b^2-4\,a\,c}+b^2\,c\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c^2-5\,a\,b^2\,c-3\,a\,b\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,c^3\,\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 (b^3\,c+b^4-b^3\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2+2\,a\,c^2\,\sqrt {b^2-4\,a\,c}-b^2\,c\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c^2-5\,a\,b^2\,c+3\,a\,b\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,c^3\,\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (4\,a\,c-b^2\right )} \]
(1 - x^2)^(1/2)*(2/(3*c) - (b/c + 1)/c + x^2/(3*c)) - (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^3*c + b^4 - b^3*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2 + 2*a*c^2*(b^2 - 4*a*c)^(1/2) - b^2*c*(b^2 - 4*a*c)^(1/2) - 4*a*b*c^2 - 5*a*b^2*c + 3*a*b*c *(b^2 - 4*a*c)^(1/2)))/(4*c^3*((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)*1 i)/((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^3*c + b^4 + b^3*(b^2 - 4*a*c )^(1/2) + 4*a^2*c^2 - 2*a*c^2*(b^2 - 4*a*c)^(1/2) + b^2*c*(b^2 - 4*a*c)^(1 /2) - 4*a*b*c^2 - 5*a*b^2*c - 3*a*b*c*(b^2 - 4*a*c)^(1/2)))/(4*c^3*(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)))*(b^3*c + b^4 + b^3*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2 - 2*a*c^2*(b^2 - 4*a*c)^(1/2) + b^2*c*(b^2 - 4*a*c)^(1/2) - 4*a*b*c^2 - 5*a*b^2*c - 3*a*b*c *(b^2 - 4*a*c)^(1/2)))/(4*c^3*(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)*1 i)/((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^3*c + b^4 - b^3*(b^2 - 4*...