3.4.81 \(\int \frac {x^4 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx\) [381]

Optimal. Leaf size=325 \[ \frac {x \sqrt {1-x^2}}{2 c}+\frac {(2 b+c) \sin ^{-1}(x)}{2 c^2}-\frac {\left (b^2-a c+b c-\frac {b^3-3 a b c+b^2 c-2 a c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {b+2 c-\sqrt {b^2-4 a c}} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{c^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {\left (b^2-a c+b c+\frac {b^3-3 a b c+b^2 c-2 a c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {b+2 c+\sqrt {b^2-4 a c}} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{c^2 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]

[Out]

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)

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Rubi [A]
time = 3.68, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1305, 396, 222, 1706, 385, 211} \begin {gather*} -\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 {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 )}{c^2 \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 ) \text {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 )}{c^2 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}+\frac {\text {ArcSin}(x) (2 b+c)}{2 c^2}+\frac {\sqrt {1-x^2} x}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*Sqrt[1 - x^2])/(a + b*x^2 + c*x^4),x]

[Out]

(x*Sqrt[1 - x^2])/(2*c) + ((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])]
)/(c^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[(Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*S
qrt[1 - x^2])])/(c^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 222

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]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 1305

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Dist[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] - Dist[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] /; Free
Q[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 3]

Rule 1706

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[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^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {x^4 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx &=\frac {\int \frac {b+c-c x^2}{\sqrt {1-x^2}} \, dx}{c^2}-\frac {\int \frac {a (b+c)+\left (b^2-a c+b c\right ) x^2}{\sqrt {1-x^2} \left (a+b x^2+c x^4\right )} \, dx}{c^2}\\ &=\frac {x \sqrt {1-x^2}}{2 c}-\frac {\int \left (\frac {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}}}{\sqrt {1-x^2} \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}+\frac {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}}}{\sqrt {1-x^2} \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}\right ) \, dx}{c^2}+\frac {(2 b+c) \int \frac {1}{\sqrt {1-x^2}} \, dx}{2 c^2}\\ &=\frac {x \sqrt {1-x^2}}{2 c}+\frac {(2 b+c) \sin ^{-1}(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 ) \int \frac {1}{\sqrt {1-x^2} \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )} \, dx}{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 ) \int \frac {1}{\sqrt {1-x^2} \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )} \, dx}{c^2}\\ &=\frac {x \sqrt {1-x^2}}{2 c}+\frac {(2 b+c) \sin ^{-1}(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 ) \text {Subst}\left (\int \frac {1}{b-\sqrt {b^2-4 a c}-\left (-b-2 c+\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{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 ) \text {Subst}\left (\int \frac {1}{b+\sqrt {b^2-4 a c}-\left (-b-2 c-\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{c^2}\\ &=\frac {x \sqrt {1-x^2}}{2 c}+\frac {(2 b+c) \sin ^{-1}(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 ) \tan ^{-1}\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 ) \tan ^{-1}\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}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.64, size = 588, normalized size = 1.81 \begin {gather*} \frac {2 c x \sqrt {1-x^2}+4 (2 b+c) \tan ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^4*Sqrt[1 - x^2])/(a + b*x^2 + c*x^4),x]

[Out]

(2*c*x*Sqrt[1 - x^2] + 4*(2*b + c)*ArcTan[x/(-1 + Sqrt[1 - x^2])] + RootSum[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*Log[-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*Log[-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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.15, size = 226, normalized size = 0.70

method	result	size
risch	\(-\frac {x \left (x^{2}-1\right )}{2 c \sqrt {-x^{2}+1}}+\frac {\arcsin \left (x \right ) b}{c^{2}}+\frac {\arcsin \left (x \right )}{2 c}+\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}+\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (a \left (b +c \right ) \textit {\_R}^{6}+\left (3 a b -a c +4 b^{2}+4 b c \right ) \textit {\_R}^{4}+\left (3 a b -a c +4 b^{2}+4 b c \right ) \textit {\_R}^{2}+a b +a c \right ) \ln \left (\frac {\sqrt {-x^{2}+1}-1}{x}-\textit {\_R} \right )}{\textit {\_R}^{7} a +3 \textit {\_R}^{5} a +3 \textit {\_R}^{5} b +3 \textit {\_R}^{3} a +4 \textit {\_R}^{3} b +8 c \,\textit {\_R}^{3}+\textit {\_R} a +\textit {\_R} b}}{4 c^{2}}\)	\(212\)
default	\(\frac {\frac {x \sqrt {-x^{2}+1}}{2}+\frac {\arcsin \left (x \right )}{2}}{c}+\frac {-\frac {2 b \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )}{c}+\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}+\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (a \left (b +c \right ) \textit {\_R}^{6}+\left (3 a b -a c +4 b^{2}+4 b c \right ) \textit {\_R}^{4}+\left (3 a b -a c +4 b^{2}+4 b c \right ) \textit {\_R}^{2}+a b +a c \right ) \ln \left (\frac {\sqrt {-x^{2}+1}-1}{x}-\textit {\_R} \right )}{\textit {\_R}^{7} a +3 \textit {\_R}^{5} a +3 \textit {\_R}^{5} b +3 \textit {\_R}^{3} a +4 \textit {\_R}^{3} b +8 c \,\textit {\_R}^{3}+\textit {\_R} a +\textit {\_R} b}}{4 c}}{c}\)	\(226\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

1/c*(1/2*x*(-x^2+1)^(1/2)+1/2*arcsin(x))+1/c*(-2*b/c*arctan(((-x^2+1)^(1/2)-1)/x)+1/4/c*sum((a*(b+c)*_R^6+(3*a
*b-a*c+4*b^2+4*b*c)*_R^4+(3*a*b-a*c+4*b^2+4*b*c)*_R^2+a*b+a*c)/(_R^7*a+3*_R^5*a+3*_R^5*b+3*_R^3*a+4*_R^3*b+8*_
R^3*c+_R*a+_R*b)*ln(((-x^2+1)^(1/2)-1)/x-_R),_R=RootOf(a*_Z^8+(4*a+4*b)*_Z^6+(6*a+8*b+16*c)*_Z^4+(4*a+4*b)*_Z^
2+a)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^2 + 1)*x^4/(c*x^4 + b*x^2 + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2860 vs. \(2 (279) = 558\).
time = 1.34, size = 2860, normalized size = 8.80 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

-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)))*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)) - 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 + 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
)))*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)) - 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 + 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)))*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)) - 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 + 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)))*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)) - 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(-x^2 + 1)*c*x + 2*(2*b + c)*arctan((sqrt(-x^2 + 1) - 1)/x))/c^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{a + b x^{2} + c x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-x**2+1)**(1/2)/(c*x**4+b*x**2+a),x)

[Out]

Integral(x**4*sqrt(-(x - 1)*(x + 1))/(a + b*x**2 + c*x**4), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1709 vs. \(2 (279) = 558\).
time = 7.54, size = 1709, normalized size = 5.26 \begin {gather*} \frac {{\left (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} \sqrt {2 \, a^{2} + a b + \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} a b^{3} + \sqrt {2} \sqrt {2 \, a^{2} + a b + \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} b^{4} + 6 \, \sqrt {2} \sqrt {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 \, \sqrt {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 \, {\left (b^{2} - 4 \, a c\right )} a^{2} b^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} a b^{3} - 4 \, {\left (b^{2} - 4 \, a c\right )} a^{3} c + 8 \, {\left (b^{2} - 4 \, a c\right )} a^{2} b c\right )} {\left | a \right |} \arctan \left (-\frac {\sqrt {2} {\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}}{2 \, \sqrt {\frac {2 \, a c^{2} + b c^{2} + \sqrt {-4 \, {\left (a c^{2} + b c^{2} + c^{3}\right )} a c^{2} + {\left (2 \, a c^{2} + b c^{2}\right )}^{2}}}{a c^{2}}}}\right )}{4 \, {\left (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}\right )}} + \frac {{\left (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} \sqrt {2 \, a^{2} + a b - \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} a b^{3} + \sqrt {2} \sqrt {2 \, a^{2} + a b - \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} b^{4} + 6 \, \sqrt {2} \sqrt {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 \, \sqrt {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 \, {\left (b^{2} - 4 \, a c\right )} a^{2} b^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} a b^{3} - 4 \, {\left (b^{2} - 4 \, a c\right )} a^{3} c - 8 \, {\left (b^{2} - 4 \, a c\right )} a^{2} b c\right )} {\left | a \right |} \arctan \left (-\frac {\sqrt {2} {\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}}{2 \, \sqrt {\frac {2 \, a c^{2} + b c^{2} - \sqrt {-4 \, {\left (a c^{2} + b c^{2} + c^{3}\right )} a c^{2} + {\left (2 \, a c^{2} + b c^{2}\right )}^{2}}}{a c^{2}}}}\right )}{4 \, {\left (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}\right )}} + \frac {\sqrt {-x^{2} + 1} x}{2 \, c} + \frac {{\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (-\frac {x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} {\left (2 \, b + c\right )}}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

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)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a*b^3 + sqr
t(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*b^4 + 6*sqrt(2)*sqrt(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*sqrt(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)*sq
rt(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 - sq
rt(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)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a*b^3 + sqrt(2)*sqrt(2*a^2 +
a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*b^4 + 6*sqrt(2)*sqrt(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*sqrt(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) + 1/2*sqrt(-x^2 + 1)*x/c + 1/4*(pi*s
gn(x) + 2*arctan(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1)))*(2*b + c)/c^2

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Mupad [B]
time = 1.30, size = 1024, normalized size = 3.15 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(1 - x^2)^(1/2))/(a + b*x^2 + c*x^4),x)

[Out]

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)^(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))

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