3.4.82 \(\int \frac {x^2 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx\) [382]

Optimal. Leaf size=263 \[ -\frac {\sin ^{-1}(x)}{c}+\frac {\left (b+c-\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {b+2 c-\sqrt {b^2-4 a c}} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{c \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}+\frac {\left (b+c+\frac {b^2-2 a c+b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {b+2 c+\sqrt {b^2-4 a c}} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{c \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]

[Out]

-arcsin(x)/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+c+(2*a*
c-b^2-b*c)/(-4*a*c+b^2)^(1/2))/c/(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+c+(-2*a*c+b^2+b*c)/(-4*a*c+b^2)^(1/
2))/c/(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 = 1.30, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1307, 222, 1706, 385, 211} \begin {gather*} \frac {\left (-\frac {-2 a c+b^2+b c}{\sqrt {b^2-4 a c}}+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 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}+\frac {\left (\frac {-2 a c+b^2+b c}{\sqrt {b^2-4 a c}}+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 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}-\frac {\text {ArcSin}(x)}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcSin[x]/c) + ((b + c - (b^2 - 2*a*c + b*c)/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*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]
]) + ((b + c + (b^2 - 2*a*c + b*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[b + S
qrt[b^2 - 4*a*c]]*Sqrt[1 - x^2])])/(c*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 1307

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Dist[e
*(f^2/c), Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1), x], x] - Dist[f^2/c, Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1)*(S
imp[a*e - (c*d - b*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, 1] && LeQ[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^2 \sqrt {1-x^2}}{a+b x^2+c x^4} \, dx &=-\frac {\int \frac {1}{\sqrt {1-x^2}} \, dx}{c}-\frac {\int \frac {-a-(b+c) x^2}{\sqrt {1-x^2} \left (a+b x^2+c x^4\right )} \, dx}{c}\\ &=-\frac {\sin ^{-1}(x)}{c}-\frac {\int \left (\frac {-b-c+\frac {b^2-2 a c+b c}{\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-c-\frac {b^2-2 a c+b c}{\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}\\ &=-\frac {\sin ^{-1}(x)}{c}+\frac {\left (b+c-\frac {b^2-2 a c+b c}{\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}+\frac {\left (b+c+\frac {b^2-2 a c+b c}{\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}\\ &=-\frac {\sin ^{-1}(x)}{c}+\frac {\left (b+c-\frac {b^2-2 a c+b c}{\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}+\frac {\left (b+c+\frac {b^2-2 a c+b c}{\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}\\ &=-\frac {\sin ^{-1}(x)}{c}+\frac {\left (b+c-\frac {b^2-2 a c+b c}{\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 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}+\frac {\left (b+c+\frac {b^2-2 a c+b c}{\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 \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 = 0.43, size = 412, normalized size = 1.57 \begin {gather*} -\frac {8 \tan ^{-1}\left (\frac {x}{-1+\sqrt {1-x^2}}\right )+\text {RootSum}\left [a+4 a \text {$\#$1}^2+4 b \text {$\#$1}^2+6 a \text {$\#$1}^4+8 b \text {$\#$1}^4+16 c \text {$\#$1}^4+4 a \text {$\#$1}^6+4 b \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right )-3 a \log (x) \text {$\#$1}^2-4 b \log (x) \text {$\#$1}^2-4 c \log (x) \text {$\#$1}^2+3 a \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 b \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 c \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-3 a \log (x) \text {$\#$1}^4-4 b \log (x) \text {$\#$1}^4-4 c \log (x) \text {$\#$1}^4+3 a \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+4 b \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4+4 c \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4-a \log (x) \text {$\#$1}^6+a \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^6}{a \text {$\#$1}+b \text {$\#$1}+3 a \text {$\#$1}^3+4 b \text {$\#$1}^3+8 c \text {$\#$1}^3+3 a \text {$\#$1}^5+3 b \text {$\#$1}^5+a \text {$\#$1}^7}\&\right ]}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(8*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*Log[x]) + a*Log[-1 + Sqrt[1 - x^2] - x*#1] - 3*a*Log[x]*#1^2 - 4*b*Log[x]
*#1^2 - 4*c*Log[x]*#1^2 + 3*a*Log[-1 + Sqrt[1 - x^2] - x*#1]*#1^2 + 4*b*Log[-1 + Sqrt[1 - x^2] - x*#1]*#1^2 +
4*c*Log[-1 + Sqrt[1 - x^2] - x*#1]*#1^2 - 3*a*Log[x]*#1^4 - 4*b*Log[x]*#1^4 - 4*c*Log[x]*#1^4 + 3*a*Log[-1 + S
qrt[1 - x^2] - x*#1]*#1^4 + 4*b*Log[-1 + Sqrt[1 - x^2] - x*#1]*#1^4 + 4*c*Log[-1 + Sqrt[1 - x^2] - x*#1]*#1^4
- a*Log[x]*#1^6 + a*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) & ])/c

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.13, size = 175, normalized size = 0.67

method	result	size
default	\(\frac {2 \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )}{c}-\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}+\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (a \,\textit {\_R}^{6}+\left (4 c +3 a +4 b \right ) \textit {\_R}^{4}+\left (4 c +3 a +4 b \right ) \textit {\_R}^{2}+a \right ) \ln \left (\frac {\sqrt {-x^{2}+1}-1}{x}-\textit {\_R} \right )}{\textit {\_R}^{7} a +3 \textit {\_R}^{5} a +3 \textit {\_R}^{5} b +3 \textit {\_R}^{3} a +4 \textit {\_R}^{3} b +8 c \,\textit {\_R}^{3}+\textit {\_R} a +\textit {\_R} b}}{4 c}\)	\(175\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/c*arctan(((-x^2+1)^(1/2)-1)/x)-1/4/c*sum((a*_R^6+(4*c+3*a+4*b)*_R^4+(4*c+3*a+4*b)*_R^2+a)/(_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^2*(-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^2 + 1)*x^2/(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. 1430 vs. \(2 (223) = 446\).
time = 0.73, size = 1430, normalized size = 5.44 \begin {gather*} -\frac {\sqrt {\frac {1}{2}} c \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} \log \left (-\frac {2 \, {\left (a b + a c\right )} x^{2} - 2 \, a b - 2 \, a c + \sqrt {\frac {1}{2}} {\left ({\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} x - {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}\right )} \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} + 2 \, {\left (a b + a c\right )} \sqrt {-x^{2} + 1}}{x^{2}}\right ) - \sqrt {\frac {1}{2}} c \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} \log \left (-\frac {2 \, {\left (a b + a c\right )} x^{2} - 2 \, a b - 2 \, a c - \sqrt {\frac {1}{2}} {\left ({\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} x - {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}\right )} \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} + 2 \, {\left (a b + a c\right )} \sqrt {-x^{2} + 1}}{x^{2}}\right ) + \sqrt {\frac {1}{2}} c \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} \log \left (-\frac {2 \, {\left (a b + a c\right )} x^{2} - 2 \, a b - 2 \, a c + \sqrt {\frac {1}{2}} {\left ({\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} x + {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}\right )} \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} + 2 \, {\left (a b + a c\right )} \sqrt {-x^{2} + 1}}{x^{2}}\right ) - \sqrt {\frac {1}{2}} c \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} \log \left (-\frac {2 \, {\left (a b + a c\right )} x^{2} - 2 \, a b - 2 \, a c - \sqrt {\frac {1}{2}} {\left ({\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} - 4 \, a c^{2} - {\left (4 \, a b - b^{2}\right )} c\right )} x + {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} \sqrt {-x^{2} + 1} x - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}\right )} \sqrt {-\frac {b^{2} - {\left (2 \, a - b\right )} c - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {b^{2} + 2 \, b c + c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} + 2 \, {\left (a b + a c\right )} \sqrt {-x^{2} + 1}}{x^{2}}\right ) - 4 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(1/2)*c*sqrt(-(b^2 - (2*a - b)*c + (b^2*c^2 - 4*a*c^3)*sqrt((b^2 + 2*b*c + c^2)/(b^2*c^4 - 4*a*c^5))
)/(b^2*c^2 - 4*a*c^3))*log(-(2*(a*b + a*c)*x^2 - 2*a*b - 2*a*c + sqrt(1/2)*((b^3 - 4*a*c^2 - (4*a*b - b^2)*c)*
sqrt(-x^2 + 1)*x - (b^3 - 4*a*c^2 - (4*a*b - b^2)*c)*x - ((b^3*c^2 - 4*a*b*c^3)*sqrt(-x^2 + 1)*x - (b^3*c^2 -
4*a*b*c^3)*x)*sqrt((b^2 + 2*b*c + c^2)/(b^2*c^4 - 4*a*c^5)))*sqrt(-(b^2 - (2*a - b)*c + (b^2*c^2 - 4*a*c^3)*sq
rt((b^2 + 2*b*c + c^2)/(b^2*c^4 - 4*a*c^5)))/(b^2*c^2 - 4*a*c^3)) + 2*(a*b + a*c)*sqrt(-x^2 + 1))/x^2) - sqrt(
1/2)*c*sqrt(-(b^2 - (2*a - b)*c + (b^2*c^2 - 4*a*c^3)*sqrt((b^2 + 2*b*c + c^2)/(b^2*c^4 - 4*a*c^5)))/(b^2*c^2
- 4*a*c^3))*log(-(2*(a*b + a*c)*x^2 - 2*a*b - 2*a*c - sqrt(1/2)*((b^3 - 4*a*c^2 - (4*a*b - b^2)*c)*sqrt(-x^2 +
1)*x - (b^3 - 4*a*c^2 - (4*a*b - b^2)*c)*x - ((b^3*c^2 - 4*a*b*c^3)*sqrt(-x^2 + 1)*x - (b^3*c^2 - 4*a*b*c^3)*
x)*sqrt((b^2 + 2*b*c + c^2)/(b^2*c^4 - 4*a*c^5)))*sqrt(-(b^2 - (2*a - b)*c + (b^2*c^2 - 4*a*c^3)*sqrt((b^2 + 2
*b*c + c^2)/(b^2*c^4 - 4*a*c^5)))/(b^2*c^2 - 4*a*c^3)) + 2*(a*b + a*c)*sqrt(-x^2 + 1))/x^2) + sqrt(1/2)*c*sqrt
(-(b^2 - (2*a - b)*c - (b^2*c^2 - 4*a*c^3)*sqrt((b^2 + 2*b*c + c^2)/(b^2*c^4 - 4*a*c^5)))/(b^2*c^2 - 4*a*c^3))
*log(-(2*(a*b + a*c)*x^2 - 2*a*b - 2*a*c + sqrt(1/2)*((b^3 - 4*a*c^2 - (4*a*b - b^2)*c)*sqrt(-x^2 + 1)*x - (b^
3 - 4*a*c^2 - (4*a*b - b^2)*c)*x + ((b^3*c^2 - 4*a*b*c^3)*sqrt(-x^2 + 1)*x - (b^3*c^2 - 4*a*b*c^3)*x)*sqrt((b^
2 + 2*b*c + c^2)/(b^2*c^4 - 4*a*c^5)))*sqrt(-(b^2 - (2*a - b)*c - (b^2*c^2 - 4*a*c^3)*sqrt((b^2 + 2*b*c + c^2)
/(b^2*c^4 - 4*a*c^5)))/(b^2*c^2 - 4*a*c^3)) + 2*(a*b + a*c)*sqrt(-x^2 + 1))/x^2) - sqrt(1/2)*c*sqrt(-(b^2 - (2
*a - b)*c - (b^2*c^2 - 4*a*c^3)*sqrt((b^2 + 2*b*c + c^2)/(b^2*c^4 - 4*a*c^5)))/(b^2*c^2 - 4*a*c^3))*log(-(2*(a
*b + a*c)*x^2 - 2*a*b - 2*a*c - sqrt(1/2)*((b^3 - 4*a*c^2 - (4*a*b - b^2)*c)*sqrt(-x^2 + 1)*x - (b^3 - 4*a*c^2
- (4*a*b - b^2)*c)*x + ((b^3*c^2 - 4*a*b*c^3)*sqrt(-x^2 + 1)*x - (b^3*c^2 - 4*a*b*c^3)*x)*sqrt((b^2 + 2*b*c +
c^2)/(b^2*c^4 - 4*a*c^5)))*sqrt(-(b^2 - (2*a - b)*c - (b^2*c^2 - 4*a*c^3)*sqrt((b^2 + 2*b*c + c^2)/(b^2*c^4 -
4*a*c^5)))/(b^2*c^2 - 4*a*c^3)) + 2*(a*b + a*c)*sqrt(-x^2 + 1))/x^2) - 4*arctan((sqrt(-x^2 + 1) - 1)/x))/c

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

[Out]

Integral(x**2*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. 3580 vs. \(2 (223) = 446\).
time = 5.38, size = 3580, normalized size = 13.61 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(pi*sgn(x) + 2*arctan(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1)))/c - 1/8*((2*a^2*b^4
- 16*a^3*b^2*c + 32*a^4*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 - 12*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^3*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*b*c + 8*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 - 16*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 + 8*(b^2 - 4*a*c)*a^3*c)*c^2*abs(a) + 2*(3*sqrt(2)*sqrt(2*a^2 + a*b
+ sqrt(b^2 - 4*a*c)*a)*a^3*b^2*c + 5*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^2*b^3*c + sqrt(2)*sqrt(
2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a*b^4*c + 2*a^2*b^4*c - sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*b^5
*c + 2*a*b^5*c - 12*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^4*c^2 - 20*sqrt(2)*sqrt(2*a^2 + a*b + sq
rt(b^2 - 4*a*c)*a)*a^3*b*c^2 + 3*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^2*b^2*c^2 - 16*a^3*b^2*c^2
+ 10*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a*b^3*c^2 - 16*a^2*b^3*c^2 - sqrt(2)*sqrt(2*a^2 + a*b + s
qrt(b^2 - 4*a*c)*a)*b^4*c^2 + 2*a*b^4*c^2 - 28*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^3*c^3 + 32*a^
4*c^3 - 24*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^2*b*c^3 + 32*a^3*b*c^3 + 8*sqrt(2)*sqrt(2*a^2 + a
*b + sqrt(b^2 - 4*a*c)*a)*a*b^2*c^3 - 16*a^2*b^2*c^3 - 16*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^2*
c^4 + 32*a^3*c^4 - 2*(b^2 - 4*a*c)*a^2*b^2*c - 2*(b^2 - 4*a*c)*a*b^3*c + 8*(b^2 - 4*a*c)*a^3*c^2 + 8*(b^2 - 4*
a*c)*a^2*b*c^2 - 2*(b^2 - 4*a*c)*a*b^2*c^2 + 8*(b^2 - 4*a*c)*a^2*c^3)*abs(a)*abs(c) + (4*a^3*b^3*c^2 + 2*a^2*b
^4*c^2 - 16*a^4*b*c^3 + 4*a^2*b^3*c^3 - 32*a^4*c^4 - 16*a^3*b*c^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*b*c^2 + 7*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*c^2 - sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*b^4*c^2 + 12*sqrt(2)*sqrt(2*a^2 +
a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^3*c^3 + 22*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^3 + 4*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^3 - 2*sq
rt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*b^3*c^3 + 16*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b
^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^2*c^4 + 8*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)
*a*b*c^4 - 4*(b^2 - 4*a*c)*a^3*b*c^2 - 2*(b^2 - 4*a*c)*a^2*b^2*c^2 - 8*(b^2 - 4*a*c)*a^3*c^3 - 4*(b^2 - 4*a*c)
*a^2*b*c^3)*abs(a))*arctan(-1/2*sqrt(2)*(x/(sqrt(-x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)/sqrt((2*a*c + b*c +
sqrt(-4*(a*c + b*c + c^2)*a*c + (2*a*c + b*c)^2))/(a*c)))/((3*a^5*b^2*c^2 + 5*a^4*b^3*c^2 + a^3*b^4*c^2 - a^2*
b^5*c^2 - 12*a^6*c^3 - 20*a^5*b*c^3 + 3*a^4*b^2*c^3 + 10*a^3*b^3*c^3 - a^2*b^4*c^3 - 28*a^5*c^4 - 24*a^4*b*c^4
+ 8*a^3*b^2*c^4 - 16*a^4*c^5)*abs(c)) - 1/8*((2*a^2*b^4 - 16*a^3*b^2*c + 32*a^4*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 - 12*sqrt(2)*sqrt(2
*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^3*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*b*c + 8*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 - 16*s
qrt(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 + 8*(b^2 -
4*a*c)*a^3*c)*c^2*abs(a) + 2*(3*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^3*b^2*c + 5*sqrt(2)*sqrt(2*a
^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^2*b^3*c + sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a*b^4*c - 2*a^2*b^
4*c - sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*b^5*c - 2*a*b^5*c - 12*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b
^2 - 4*a*c)*a)*a^4*c^2 - 20*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^3*b*c^2 + 3*sqrt(2)*sqrt(2*a^2 +
a*b - sqrt(b^2 - 4*a*c)*a)*a^2*b^2*c^2 + 16*a^3*b^2*c^2 + 10*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*
a*b^3*c^2 + 16*a^2*b^3*c^2 - sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*b^4*c^2 - 2*a*b^4*c^2 - 28*sqrt(2
)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^3*c^3 - 32*a^4*c^3 - 24*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*
c)*a)*a^2*b*c^3 - 32*a^3*b*c^3 + 8*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a*b^2*c^3 + 16*a^2*b^2*c^3
- 16*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^2*c^4 - 32*a^3*c^4 + 2*(b^2 - 4*a*c)*a^2*b^2*c + 2*(b^2
- 4*a*c)*a*b^3*c - 8*(b^2 - 4*a*c)*a^3*c^2 - 8*(b^2 - 4*a*c)*a^2*b*c^2 + 2*(b^2 - 4*a*c)*a*b^2*c^2 - 8*(b^2 -
4*a*c)*a^2*c^3)*abs(a)*abs(c) + (4*a^3*b^3*c^2 + 2*a^2*b^4*c^2 - 16*a^4*b*c^3 + 4*a^2*b^3*c^3 - 32*a^4*c^4 -
16*a^3*b*c^4 + 6*sqrt(2)*sqrt(2*a^2 + a*b - sqr...

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Mupad [B]
time = 1.27, size = 870, normalized size = 3.31 \begin {gather*} -\frac {\mathrm {asin}\left (x\right )}{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 (2\,a\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+2\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (8\,a\,c-2\,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 (2\,a\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+2\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{\left (8\,a\,c-2\,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 (2\,a\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+2\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (8\,a\,c-2\,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 (2\,a\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+b\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+2\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{\left (8\,a\,c-2\,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^2*(1 - x^2)^(1/2))/(a + b*x^2 + c*x^4),x)

[Out]

(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)))*(2*a*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)
+ b*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + b*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + 2*c*(-(b - (b^2 -
4*a*c)^(1/2))/(2*c))^(3/2)))/(((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2)*(8*a*c - 2*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)))*(2*a*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + b*(-(b - (b^2 -
4*a*c)^(1/2))/(2*c))^(1/2) + b*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + 2*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*
c))^(3/2)))/(((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2)*(8*a*c - 2*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)))*(2*a*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + b*(-(b + (b^2 - 4*a*c)^(1/2))/(2
*c))^(1/2) + b*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + 2*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2)))/((8*a
*c - 2*b^2)*((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2)) - asin(x)/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)))*(2*a*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + b*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^
(1/2) + b*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + 2*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2)))/((8*a*c -
2*b^2)*((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2))

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