Integrand size = 29, antiderivative size = 325 \[ \int \frac {x^4 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\frac {x \sqrt {1-x^2}}{2 c}+\frac {(2 b+c) \arcsin (x)}{2 c^2}-\frac {\left (b^2-a c+b c-\frac {b^3-3 a b c+b^2 c-2 a c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {b+2 c-\sqrt {b^2-4 a c}} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{c^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {\left (b^2-a c+b c+\frac {b^3-3 a b c+b^2 c-2 a c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {b+2 c+\sqrt {b^2-4 a c}} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{c^2 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]
1/2*(2*b+c)*arcsin(x)/c^2+1/2*x*(-x^2+1)^(1/2)/c-arctan(x*(b+2*c-(-4*a*c+b ^2)^(1/2))^(1/2)/(-x^2+1)^(1/2)/(b-(-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^2/(b-(-4*a*c+b^2)^(1/2) )^(1/2)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2)-arctan(x*(b+2*c+(-4*a*c+b^2)^(1/2 ))^(1/2)/(-x^2+1)^(1/2)/(b+(-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^2/(b+(-4*a*c+b^2)^(1/2))^(1/2) /(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.64 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.81 \[ \int \frac {x^4 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\frac {2 c x \sqrt {1-x^2}+4 (2 b+c) \arctan \left (\frac {x}{-1+\sqrt {1-x^2}}\right )+\text {RootSum}\left [a+4 a \text {$\#$1}^2+4 b \text {$\#$1}^2+6 a \text {$\#$1}^4+8 b \text {$\#$1}^4+16 c \text {$\#$1}^4+4 a \text {$\#$1}^6+4 b \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {-a b \log (x)-a c \log (x)+a b \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right )+a c \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right )-3 a b \log (x) \text {$\#$1}^2-4 b^2 \log (x) \text {$\#$1}^2+a c \log (x) \text {$\#$1}^2-4 b c \log (x) \text {$\#$1}^2+3 a b \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 b^2 \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a c \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 b c \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-3 a b \log (x) \text {$\#$1}^4-4 b^2 \log (x) \text {$\#$1}^4+a c \log (x) \text {$\#$1}^4-4 b c \log (x) \text {$\#$1}^4+3 a b \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+4 b^2 \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4-a c \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+4 b c \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4-a b \log (x) \text {$\#$1}^6-a c \log (x) \text {$\#$1}^6+a b \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^6+a c \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^6}{a \text {$\#$1}+b \text {$\#$1}+3 a \text {$\#$1}^3+4 b \text {$\#$1}^3+8 c \text {$\#$1}^3+3 a \text {$\#$1}^5+3 b \text {$\#$1}^5+a \text {$\#$1}^7}\&\right ]}{4 c^2} \]
(2*c*x*Sqrt[1 - x^2] + 4*(2*b + c)*ArcTan[x/(-1 + Sqrt[1 - x^2])] + RootSu m[a + 4*a*#1^2 + 4*b*#1^2 + 6*a*#1^4 + 8*b*#1^4 + 16*c*#1^4 + 4*a*#1^6 + 4 *b*#1^6 + a*#1^8 & , (-(a*b*Log[x]) - a*c*Log[x] + a*b*Log[-1 + Sqrt[1 - x ^2] - x*#1] + a*c*Log[-1 + Sqrt[1 - x^2] - x*#1] - 3*a*b*Log[x]*#1^2 - 4*b ^2*Log[x]*#1^2 + a*c*Log[x]*#1^2 - 4*b*c*Log[x]*#1^2 + 3*a*b*Log[-1 + Sqrt [1 - x^2] - x*#1]*#1^2 + 4*b^2*Log[-1 + Sqrt[1 - x^2] - x*#1]*#1^2 - a*c*L og[-1 + Sqrt[1 - x^2] - x*#1]*#1^2 + 4*b*c*Log[-1 + Sqrt[1 - x^2] - x*#1]* #1^2 - 3*a*b*Log[x]*#1^4 - 4*b^2*Log[x]*#1^4 + a*c*Log[x]*#1^4 - 4*b*c*Log [x]*#1^4 + 3*a*b*Log[-1 + Sqrt[1 - x^2] - x*#1]*#1^4 + 4*b^2*Log[-1 + Sqrt [1 - x^2] - x*#1]*#1^4 - a*c*Log[-1 + Sqrt[1 - x^2] - x*#1]*#1^4 + 4*b*c*L og[-1 + Sqrt[1 - x^2] - x*#1]*#1^4 - a*b*Log[x]*#1^6 - a*c*Log[x]*#1^6 + a *b*Log[-1 + Sqrt[1 - x^2] - x*#1]*#1^6 + a*c*Log[-1 + Sqrt[1 - x^2] - x*#1 ]*#1^6)/(a*#1 + b*#1 + 3*a*#1^3 + 4*b*#1^3 + 8*c*#1^3 + 3*a*#1^5 + 3*b*#1^ 5 + a*#1^7) & ])/(4*c^2)
Time = 2.03 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1614, 299, 223, 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.
| \(\displaystyle \int \frac {x^4 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx\) |
| \(\Big \downarrow \) 1614 |
| \(\displaystyle \frac {\int \frac {-c x^2+b+c}{\sqrt {1-x^2}}dx}{c^2}-\frac {\int \frac {\left (b^2+c b-a c\right ) x^2+a (b+c)}{\sqrt {1-x^2} \left (c x^4+b x^2+a\right )}dx}{c^2}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {1}{2} (2 b+c) \int \frac {1}{\sqrt {1-x^2}}dx+\frac {1}{2} c \sqrt {1-x^2} x}{c^2}-\frac {\int \frac {\left (b^2+c b-a c\right ) x^2+a (b+c)}{\sqrt {1-x^2} \left (c x^4+b x^2+a\right )}dx}{c^2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {\frac {1}{2} \arcsin (x) (2 b+c)+\frac {1}{2} c \sqrt {1-x^2} x}{c^2}-\frac {\int \frac {\left (b^2+c b-a c\right ) x^2+a (b+c)}{\sqrt {1-x^2} \left (c x^4+b x^2+a\right )}dx}{c^2}\) |
| \(\Big \downarrow \) 2256 |
| \(\displaystyle \frac {\frac {1}{2} \arcsin (x) (2 b+c)+\frac {1}{2} c \sqrt {1-x^2} x}{c^2}-\frac {\int \left (\frac {b^2+c b-a c-\frac {-b^3-c b^2+3 a c b+2 a c^2}{\sqrt {b^2-4 a c}}}{\sqrt {1-x^2} \left (2 c x^2+b+\sqrt {b^2-4 a c}\right )}+\frac {b^2+c b-a c+\frac {-b^3-c b^2+3 a c b+2 a c^2}{\sqrt {b^2-4 a c}}}{\sqrt {1-x^2} \left (2 c x^2+b-\sqrt {b^2-4 a c}\right )}\right )dx}{c^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{2} \arcsin (x) (2 b+c)+\frac {1}{2} c \sqrt {1-x^2} x}{c^2}-\frac {\frac {\left (-\frac {-3 a b c-2 a c^2+b^3+b^2 c}{\sqrt {b^2-4 a c}}-a c+b^2+b c\right ) \arctan \left (\frac {x \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}{\sqrt {1-x^2} \sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}+\frac {\left (\frac {-3 a b c-2 a c^2+b^3+b^2 c}{\sqrt {b^2-4 a c}}-a c+b^2+b c\right ) \arctan \left (\frac {x \sqrt {\sqrt {b^2-4 a c}+b+2 c}}{\sqrt {1-x^2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}}{c^2}\) |
((c*x*Sqrt[1 - x^2])/2 + ((2*b + c)*ArcSin[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])*ArcTan[(Sqrt[b + 2 *c - Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[1 - x^2])])/( Sqrt[b - Sqrt[b^2 - 4*a*c]]*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)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sq rt[b + 2*c + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 - x ^2])])/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]))/c^ 2
3.4.81.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Simp[f^4/c^2 Int[(f*x)^(m - 4)*(c*d - b*e + c *e*x^2)*(d + e*x^2)^(q - 1), x], x] - Simp[f^4/c^2 Int[(f*x)^(m - 4)*(d + e*x^2)^(q - 1)*(Simp[a*(c*d - b*e) + (b*c*d - b^2*e + a*c*e)*x^2, x]/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 3]
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]
Time = 2.12 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.77
| method | result | size |
| risch | \(-\frac {x \left (x^{2}-1\right )}{2 c \sqrt {-x^{2}+1}}+\frac {\frac {\left (2 b +c \right ) \arcsin \left (x \right )}{c}+\frac {a \sqrt {2}\, \left (\frac {\left (b \sqrt {-4 a c +b^{2}}+\sqrt {-4 a c +b^{2}}\, c +2 a c -b^{2}-b c \right ) \arctan \left (\frac {a \sqrt {-x^{2}+1}\, \sqrt {2}}{x \sqrt {\left (2 a +b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (2 a +b +\sqrt {-4 a c +b^{2}}\right ) a}}-\frac {\left (b \sqrt {-4 a c +b^{2}}+\sqrt {-4 a c +b^{2}}\, c -2 a c +b^{2}+b c \right ) \operatorname {arctanh}\left (\frac {a \sqrt {-x^{2}+1}\, \sqrt {2}}{x \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}-2 a \right ) a}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}-2 a \right ) a}}\right )}{c \sqrt {-4 a c +b^{2}}}}{2 c}\) | \(250\) |
| pseudoelliptic | \(\frac {a \sqrt {2}\, \sqrt {\left (2 a +b +\sqrt {-4 a c +b^{2}}\right ) a}\, \left (\frac {\left (-b -c \right ) \sqrt {-4 a c +b^{2}}}{2}+a c -\frac {b \left (b +c \right )}{2}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {-x^{2}+1}\, \sqrt {2}}{x \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}-2 a \right ) a}}\right )+\left (a \sqrt {2}\, \left (\frac {\left (b +c \right ) \sqrt {-4 a c +b^{2}}}{2}+a c -\frac {b \left (b +c \right )}{2}\right ) \arctan \left (\frac {a \sqrt {-x^{2}+1}\, \sqrt {2}}{x \sqrt {\left (2 a +b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )-\left (\left (b +\frac {c}{2}\right ) \arctan \left (\frac {\sqrt {-x^{2}+1}}{x}\right )-\frac {\sqrt {-x^{2}+1}\, c x}{2}\right ) \sqrt {-4 a c +b^{2}}\, \sqrt {\left (2 a +b +\sqrt {-4 a c +b^{2}}\right ) a}\right ) \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}-2 a \right ) a}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}-2 a \right ) a}\, \sqrt {-4 a c +b^{2}}\, \sqrt {\left (2 a +b +\sqrt {-4 a c +b^{2}}\right ) a}\, c^{2}}\) | \(303\) |
| default | \(\frac {\frac {x \sqrt {-x^{2}+1}}{2}+\frac {\arcsin \left (x \right )}{2}}{c}+\frac {a \sqrt {2}\, \sqrt {\left (2 a +b +\sqrt {-4 a c +b^{2}}\right ) a}\, \left (\frac {\left (-b -c \right ) \sqrt {-4 a c +b^{2}}}{2}+a c -\frac {b \left (b +c \right )}{2}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {-x^{2}+1}\, \sqrt {2}}{x \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}-2 a \right ) a}}\right )+\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}-2 a \right ) a}\, \left (a \sqrt {2}\, \left (\frac {\left (b +c \right ) \sqrt {-4 a c +b^{2}}}{2}+a c -\frac {b \left (b +c \right )}{2}\right ) \arctan \left (\frac {a \sqrt {-x^{2}+1}\, \sqrt {2}}{x \sqrt {\left (2 a +b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )-\sqrt {\left (2 a +b +\sqrt {-4 a c +b^{2}}\right ) a}\, \sqrt {-4 a c +b^{2}}\, \arctan \left (\frac {\sqrt {-x^{2}+1}}{x}\right ) b \right )}{c^{2} \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}-2 a \right ) a}\, \sqrt {-4 a c +b^{2}}\, \sqrt {\left (2 a +b +\sqrt {-4 a c +b^{2}}\right ) a}}\) | \(306\) |
-1/2*x/c*(x^2-1)/(-x^2+1)^(1/2)+1/2/c*((2*b+c)/c*arcsin(x)+1/c*a*2^(1/2)/( -4*a*c+b^2)^(1/2)*((b*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^(1/2)*c+2*a*c-b^2-b* c)/((2*a+b+(-4*a*c+b^2)^(1/2))*a)^(1/2)*arctan(a/x*(-x^2+1)^(1/2)*2^(1/2)/ ((2*a+b+(-4*a*c+b^2)^(1/2))*a)^(1/2))-(b*(-4*a*c+b^2)^(1/2)+(-4*a*c+b^2)^( 1/2)*c-2*a*c+b^2+b*c)/((-b+(-4*a*c+b^2)^(1/2)-2*a)*a)^(1/2)*arctanh(a/x*(- x^2+1)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2)-2*a)*a)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 2860 vs. \(2 (279) = 558\).
Time = 1.30 (sec) , antiderivative size = 2860, normalized size of antiderivative = 8.80 \[ \int \frac {x^4 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
-1/2*(sqrt(1/2)*c^2*sqrt(-(b^4 + (2*a^2 - 3*a*b)*c^2 - (4*a*b^2 - b^3)*c + (b^2*c^4 - 4*a*c^5)*sqrt((b^6 + a^2*c^4 + 2*(2*a^2*b - a*b^2)*c^3 + (4*a^ 2*b^2 - 6*a*b^3 + b^4)*c^2 - 2*(2*a*b^4 - b^5)*c)/(b^2*c^8 - 4*a*c^9)))/(b ^2*c^4 - 4*a*c^5))*log(-(2*a^2*b^3 - 2*a^3*c^2 - 2*(a^2*b^3 - a^3*c^2 - (2 *a^3*b - a^2*b^2)*c)*x^2 - 2*(2*a^3*b - a^2*b^2)*c + sqrt(1/2)*((b^6 + 4*a ^2*b*c^3 + (8*a^2*b^2 - 5*a*b^3)*c^2 - (6*a*b^4 - b^5)*c)*sqrt(-x^2 + 1)*x - (b^6 + 4*a^2*b*c^3 + (8*a^2*b^2 - 5*a*b^3)*c^2 - (6*a*b^4 - b^5)*c)*x - ((b^4*c^4 - 6*a*b^2*c^5 + 8*a^2*c^6)*sqrt(-x^2 + 1)*x - (b^4*c^4 - 6*a*b^ 2*c^5 + 8*a^2*c^6)*x)*sqrt((b^6 + a^2*c^4 + 2*(2*a^2*b - a*b^2)*c^3 + (4*a ^2*b^2 - 6*a*b^3 + b^4)*c^2 - 2*(2*a*b^4 - b^5)*c)/(b^2*c^8 - 4*a*c^9)))*s qrt(-(b^4 + (2*a^2 - 3*a*b)*c^2 - (4*a*b^2 - b^3)*c + (b^2*c^4 - 4*a*c^5)* sqrt((b^6 + a^2*c^4 + 2*(2*a^2*b - a*b^2)*c^3 + (4*a^2*b^2 - 6*a*b^3 + b^4 )*c^2 - 2*(2*a*b^4 - b^5)*c)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4*a*c^5)) - 2*(a^2*b^3 - a^3*c^2 - (2*a^3*b - a^2*b^2)*c)*sqrt(-x^2 + 1))/x^2) - sqrt( 1/2)*c^2*sqrt(-(b^4 + (2*a^2 - 3*a*b)*c^2 - (4*a*b^2 - b^3)*c + (b^2*c^4 - 4*a*c^5)*sqrt((b^6 + a^2*c^4 + 2*(2*a^2*b - a*b^2)*c^3 + (4*a^2*b^2 - 6*a *b^3 + b^4)*c^2 - 2*(2*a*b^4 - b^5)*c)/(b^2*c^8 - 4*a*c^9)))/(b^2*c^4 - 4* a*c^5))*log(-(2*a^2*b^3 - 2*a^3*c^2 - 2*(a^2*b^3 - a^3*c^2 - (2*a^3*b - a^ 2*b^2)*c)*x^2 - 2*(2*a^3*b - a^2*b^2)*c - sqrt(1/2)*((b^6 + 4*a^2*b*c^3 + (8*a^2*b^2 - 5*a*b^3)*c^2 - (6*a*b^4 - b^5)*c)*sqrt(-x^2 + 1)*x - (b^6 ...
\[ \int \frac {x^4 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\int \frac {x^{4} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{a + b x^{2} + c x^{4}}\, dx \]
\[ \int \frac {x^4 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\int { \frac {\sqrt {-x^{2} + 1} x^{4}}{c x^{4} + b x^{2} + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1710 vs. \(2 (279) = 558\).
Time = 1.42 (sec) , antiderivative size = 1710, normalized size of antiderivative = 5.26 \[ \int \frac {x^4 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
1/4*(3*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^2*b^3 + 2*sqrt(2) *sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a*b^4 - 2*a^2*b^4 - sqrt(2)*sqrt( 2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*b^5 + 2*a*b^5 - 12*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^3*b*c - 8*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^ 2 - 4*a*c)*a)*a^2*b^2*c + 12*a^3*b^2*c + 8*sqrt(2)*sqrt(2*a^2 + a*b + sqrt (b^2 - 4*a*c)*a)*a*b^3*c - 16*a^2*b^3*c - 16*a^4*c^2 - 16*sqrt(2)*sqrt(2*a ^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^2*b*c^2 + 32*a^3*b*c^2 - 3*sqrt(2)*sqrt( 2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^2*b^2 - 2*sqrt(2)*s qrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a*b^3 + sqrt(2)*s qrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*b^4 + 6*sqrt(2)*s qrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^3*c + 4*sqrt(2) *sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^2*b*c - 6*sqr t(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a*b^2*c + 8 *sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^2*c^2 + 2*(b^2 - 4*a*c)*a^2*b^2 - 2*(b^2 - 4*a*c)*a*b^3 - 4*(b^2 - 4*a*c)*a^3*c + 8*(b^2 - 4*a*c)*a^2*b*c)*abs(a)*arctan(-1/2*sqrt(2)*(x/(sqrt(-x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)/sqrt((2*a*c^2 + b*c^2 + sqrt(-4*(a*c^2 + b* c^2 + c^3)*a*c^2 + (2*a*c^2 + b*c^2)^2))/(a*c^2)))/(3*a^4*b^2*c^2 + 2*a^3* b^3*c^2 - a^2*b^4*c^2 - 12*a^5*c^3 - 8*a^4*b*c^3 + 8*a^3*b^2*c^3 - 16*a^4* c^4) + 1/4*(3*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^2*b^3 +...
Time = 8.24 (sec) , antiderivative size = 1024, normalized size of antiderivative = 3.15 \[ \int \frac {x^4 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx=\mathrm {asin}\left (x\right )\,\left (\frac {\frac {b}{c}+1}{c}-\frac {1}{2\,c}\right )+\frac {x\,\sqrt {1-x^2}}{2\,c}-\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^2\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+a\,b\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+2\,a\,c\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+b\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{2\,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^2\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+a\,b\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+2\,a\,c\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+b\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{2\,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 (b^2\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+a\,b\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+2\,a\,c\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+b\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{2\,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^2\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+a\,b\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+2\,a\,c\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+b\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{2\,c\,\left (4\,a\,c-b^2\right )\,\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}} \]
asin(x)*((b/c + 1)/c - 1/(2*c)) + (x*(1 - x^2)^(1/2))/(2*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^2*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + a*b*(-(b - ( b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + 2*a*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c) )^(1/2) - 2*a*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + b*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2)))/(2*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^2*(-(b + (b^2 - 4*a*c )^(1/2))/(2*c))^(3/2) + a*b*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + 2*a *c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 2*a*c*(-(b + (b^2 - 4*a*c)^( 1/2))/(2*c))^(3/2) + b*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2)))/(2*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)))*(b^2*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + a*b*(-(b - (b^ 2 - 4*a*c)^(1/2))/(2*c))^(1/2) + 2*a*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^ (1/2) - 2*a*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + b*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2)))/(2*c*((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1...