∫ (d+ex2)3 ____________________

3.2.93 \(\int \frac {(d+e x^2)^3}{(a+b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=563 \[ \frac {x \left (c \left (-\frac {a b e \left (a e^2+3 c d^2\right )}{c}-2 a d \left (c d^2-3 a e^2\right )+b^2 d^3\right )-x^2 \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (a b^3 e^3-b^2 \left (a e^3 \sqrt {b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (e \sqrt {b^2-4 a c}+2 c d\right )-b c \left (c d^2 \left (d \sqrt {b^2-4 a c}+12 a e\right )+a e^2 \left (3 d \sqrt {b^2-4 a c}+8 a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (a b^3 e^3-b^2 \left (-a e^3 \sqrt {b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (2 c d-e \sqrt {b^2-4 a c}\right )+b c \left (c d^2 \left (d \sqrt {b^2-4 a c}-12 a e\right )+a e^2 \left (3 d \sqrt {b^2-4 a c}-8 a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}} \]

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Rubi [A] time = 3.52, antiderivative size = 563, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1205, 1166, 205} \begin {gather*} -\frac {\left (-b^2 \left (a e^3 \sqrt {b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (e \sqrt {b^2-4 a c}+2 c d\right )-b c \left (c d^2 \left (d \sqrt {b^2-4 a c}+12 a e\right )+a e^2 \left (3 d \sqrt {b^2-4 a c}+8 a e\right )\right )+a b^3 e^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (-b^2 \left (-a e^3 \sqrt {b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (2 c d-e \sqrt {b^2-4 a c}\right )+b c \left (c d^2 \left (d \sqrt {b^2-4 a c}-12 a e\right )+a e^2 \left (3 d \sqrt {b^2-4 a c}-8 a e\right )\right )+a b^3 e^3\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {x \left (c \left (-\frac {a b e \left (a e^2+3 c d^2\right )}{c}-2 a d \left (c d^2-3 a e^2\right )+b^2 d^3\right )-x^2 \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x^2)^3/(a + b*x^2 + c*x^4)^2,x]

[Out]

(x*(c*(b^2*d^3 - 2*a*d*(c*d^2 - 3*a*e^2) - (a*b*e*(3*c*d^2 + a*e^2))/c) - (a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^

2) - b*c*d*(c*d^2 + 3*a*e^2))*x^2))/(2*a*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((a*b^3*e^3 + 6*a*c*(2*c*d + S

qrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) - b^2*(c^2*d^3 - 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) - b*c*(a*e^2*(3*Sq

rt[b^2 - 4*a*c]*d + 8*a*e) + c*d^2*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b

^2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((a*b^3*e^3 + 6*a*c*(2*

c*d - Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) - b^2*(c^2*d^3 - 3*a*c*d*e^2 - a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(c*d^

2*(Sqrt[b^2 - 4*a*c]*d - 12*a*e) + a*e^2*(3*Sqrt[b^2 - 4*a*c]*d - 8*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b +

Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

Rule 205

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

&& PosQ[a/b]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di

st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +

q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^

2 - 4*a*c]

Rule 1205

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom

ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x

^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2))/(

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

um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c

*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*

a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac {x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac {a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {-b^2 d^3+6 a d \left (c d^2+a e^2\right )-\frac {a b e \left (3 c d^2+a e^2\right )}{c}+\left (-\frac {a b^2 e^3}{c}+6 a e \left (c d^2+a e^2\right )-b \left (c d^3+3 a d e^2\right )\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac {x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac {a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (a b^3 e^3+6 a c \left (2 c d-\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2-a \sqrt {b^2-4 a c} e^3\right )+b c \left (c d^2 \left (\sqrt {b^2-4 a c} d-12 a e\right )+a e^2 \left (3 \sqrt {b^2-4 a c} d-8 a e\right )\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{4 a c \left (b^2-4 a c\right )^{3/2}}-\frac {\left (a b^3 e^3+6 a c \left (2 c d+\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2+a \sqrt {b^2-4 a c} e^3\right )-b c \left (a e^2 \left (3 \sqrt {b^2-4 a c} d+8 a e\right )+c d^2 \left (\sqrt {b^2-4 a c} d+12 a e\right )\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{4 a c \left (b^2-4 a c\right )^{3/2}}\\ &=\frac {x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac {a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (a b^3 e^3+6 a c \left (2 c d+\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2+a \sqrt {b^2-4 a c} e^3\right )-b c \left (a e^2 \left (3 \sqrt {b^2-4 a c} d+8 a e\right )+c d^2 \left (\sqrt {b^2-4 a c} d+12 a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (a b^3 e^3+6 a c \left (2 c d-\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2-a \sqrt {b^2-4 a c} e^3\right )+b c \left (c d^2 \left (\sqrt {b^2-4 a c} d-12 a e\right )+a e^2 \left (3 \sqrt {b^2-4 a c} d-8 a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A] time = 1.63, size = 540, normalized size = 0.96 \begin {gather*} \frac {\frac {2 \sqrt {c} x \left (b \left (-a^2 e^3-3 a c d e \left (d-e x^2\right )+c^2 d^3 x^2\right )+b^2 \left (c d^3-a e^3 x^2\right )+2 a c \left (a e^2 \left (3 d+e x^2\right )-c d^2 \left (d+3 e x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \left (-a b^3 e^3+b^2 \left (a e^3 \sqrt {b^2-4 a c}-3 a c d e^2+c^2 d^3\right )-6 a c \left (a e^2+c d^2\right ) \left (e \sqrt {b^2-4 a c}+2 c d\right )+b c \left (c d^2 \left (d \sqrt {b^2-4 a c}+12 a e\right )+a e^2 \left (3 d \sqrt {b^2-4 a c}+8 a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (a b^3 e^3+b^2 \left (a e^3 \sqrt {b^2-4 a c}+3 a c d e^2-c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (2 c d-e \sqrt {b^2-4 a c}\right )+b c \left (c d^2 \left (d \sqrt {b^2-4 a c}-12 a e\right )+a e^2 \left (3 d \sqrt {b^2-4 a c}-8 a e\right )\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}}{4 a c^{3/2}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x^2)^3/(a + b*x^2 + c*x^4)^2,x]

[Out]

((2*Sqrt[c]*x*(b^2*(c*d^3 - a*e^3*x^2) + b*(-(a^2*e^3) + c^2*d^3*x^2 - 3*a*c*d*e*(d - e*x^2)) + 2*a*c*(a*e^2*(

3*d + e*x^2) - c*d^2*(d + 3*e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(a*b^3*e^3) - 6*a*c*(2*

c*d + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) + b^2*(c^2*d^3 - 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(a*e^

2*(3*Sqrt[b^2 - 4*a*c]*d + 8*a*e) + c*d^2*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b -

Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(a*b^3*e^3 + 6*a*c*(2*c*d -

Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) + b^2*(-(c^2*d^3) + 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(c*d^2*

(Sqrt[b^2 - 4*a*c]*d - 12*a*e) + a*e^2*(3*Sqrt[b^2 - 4*a*c]*d - 8*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + S

qrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a*c^(3/2))

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x^2)^3/(a + b*x^2 + c*x^4)^2,x]

[Out]

IntegrateAlgebraic[(d + e*x^2)^3/(a + b*x^2 + c*x^4)^2, x]

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fricas [B] time = 111.89, size = 12117, normalized size = 21.52

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/4*(2*(b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - (a*b^2 - 2*a^2*c)*e^3)*x^3 - sqrt(1/2)*(a^2*b^2*c - 4*a^3*

c^2 + (a*b^2*c^2 - 4*a^2*c^3)*x^4 + (a*b^3*c - 4*a^2*b*c^2)*x^2)*sqrt(-((b^5*c^3 - 15*a*b^3*c^4 + 60*a^2*b*c^5

)*d^6 + 6*(a*b^4*c^3 - 6*a^2*b^2*c^4 - 24*a^3*c^5)*d^5*e - 3*(3*a^2*b^3*c^3 - 92*a^3*b*c^4)*d^4*e^2 - 8*(11*a^

3*b^2*c^3 + 36*a^4*c^4)*d^3*e^3 - 3*(3*a^3*b^3*c^2 - 92*a^4*b*c^3)*d^2*e^4 + 6*(a^3*b^4*c - 6*a^4*b^2*c^2 - 24

*a^5*c^3)*d*e^5 + (a^3*b^5 - 15*a^4*b^3*c + 60*a^5*b*c^2)*e^6 + (a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5

- 64*a^6*c^6)*sqrt(-(108*a^3*b*c^6*d^9*e^3 + 108*a^6*b*c^3*d^3*e^9 - (b^4*c^6 - 18*a*b^2*c^7 + 81*a^2*c^8)*d^

12 - 12*(a*b^3*c^6 - 9*a^2*b*c^7)*d^11*e - 18*(a^2*b^2*c^6 + 9*a^3*c^7)*d^10*e^2 - 9*(2*a^3*b^2*c^5 - 9*a^4*c^

6)*d^8*e^4 + 12*(a^3*b^3*c^4 - 18*a^4*b*c^5)*d^7*e^5 + 2*(a^3*b^4*c^3 + 18*a^4*b^2*c^4 + 162*a^5*c^5)*d^6*e^6

+ 12*(a^4*b^3*c^3 - 18*a^5*b*c^4)*d^5*e^7 - 9*(2*a^5*b^2*c^3 - 9*a^6*c^4)*d^4*e^8 - 18*(a^6*b^2*c^2 + 9*a^7*c^

3)*d^2*e^10 - 12*(a^6*b^3*c - 9*a^7*b*c^2)*d*e^11 - (a^6*b^4 - 18*a^7*b^2*c + 81*a^8*c^2)*e^12)/(a^6*b^6*c^6 -

12*a^7*b^4*c^7 + 48*a^8*b^2*c^8 - 64*a^9*c^9)))/(a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 - 64*a^6*c^6))

*log(-((5*b^4*c^6 - 81*a*b^2*c^7 + 324*a^2*c^8)*d^12 - 3*(3*b^5*c^5 - 65*a*b^3*c^6 + 324*a^2*b*c^7)*d^11*e + 3

*(b^6*c^4 - 42*a*b^4*c^5 + 252*a^2*b^2*c^6 + 432*a^3*c^7)*d^10*e^2 + (b^7*c^3 + 3*a*b^5*c^4 + 33*a^2*b^3*c^5 -

2916*a^3*b*c^6)*d^9*e^3 + 9*(a*b^6*c^3 - 15*a^2*b^4*c^4 + 195*a^3*b^2*c^5 + 180*a^4*c^6)*d^8*e^4 - 162*(a^3*b

^3*c^4 + 12*a^4*b*c^5)*d^7*e^5 + 162*(a^4*b^3*c^3 + 12*a^5*b*c^4)*d^5*e^7 - 9*(a^3*b^6*c - 15*a^4*b^4*c^2 + 19

5*a^5*b^2*c^3 + 180*a^6*c^4)*d^4*e^8 - (a^3*b^7 + 3*a^4*b^5*c + 33*a^5*b^3*c^2 - 2916*a^6*b*c^3)*d^3*e^9 - 3*(

a^4*b^6 - 42*a^5*b^4*c + 252*a^6*b^2*c^2 + 432*a^7*c^3)*d^2*e^10 + 3*(3*a^5*b^5 - 65*a^6*b^3*c + 324*a^7*b*c^2

)*d*e^11 - (5*a^6*b^4 - 81*a^7*b^2*c + 324*a^8*c^2)*e^12)*x + 1/2*sqrt(1/2)*((b^8*c^4 - 23*a*b^6*c^5 + 190*a^2

*b^4*c^6 - 672*a^3*b^2*c^7 + 864*a^4*c^8)*d^9 + 9*(a*b^7*c^4 - 15*a^2*b^5*c^5 + 72*a^3*b^3*c^6 - 112*a^4*b*c^7

)*d^8*e + 3*(a^2*b^6*c^4 + 28*a^3*b^4*c^5 - 272*a^4*b^2*c^6 + 576*a^5*c^7)*d^7*e^2 + (a^2*b^7*c^3 - 80*a^3*b^5

*c^4 + 592*a^4*b^3*c^5 - 1152*a^5*b*c^6)*d^6*e^3 + 15*(a^3*b^6*c^3 - 8*a^4*b^4*c^4 + 16*a^5*b^2*c^5)*d^5*e^4 -

6*(a^3*b^7*c^2 - 17*a^4*b^5*c^3 + 88*a^5*b^3*c^4 - 144*a^6*b*c^5)*d^4*e^5 - (a^3*b^8*c - 5*a^4*b^6*c^2 + 100*

a^5*b^4*c^3 - 816*a^6*b^2*c^4 + 1728*a^7*c^5)*d^3*e^6 - 3*(a^4*b^7*c - 32*a^5*b^5*c^2 + 208*a^6*b^3*c^3 - 384*

a^7*b*c^4)*d^2*e^7 - 54*(a^6*b^4*c^2 - 8*a^7*b^2*c^3 + 16*a^8*c^4)*d*e^8 - (a^5*b^7 - 17*a^6*b^5*c + 88*a^7*b^

3*c^2 - 144*a^8*b*c^3)*e^9 - ((a^3*b^9*c^4 - 20*a^4*b^7*c^5 + 144*a^5*b^5*c^6 - 448*a^6*b^3*c^7 + 512*a^7*b*c^

8)*d^3 + 3*(a^4*b^8*c^4 - 8*a^5*b^6*c^5 + 128*a^7*b^2*c^7 - 256*a^8*c^8)*d^2*e - 12*(a^5*b^7*c^4 - 12*a^6*b^5*

c^5 + 48*a^7*b^3*c^6 - 64*a^8*b*c^7)*d*e^2 - (a^5*b^8*c^3 - 24*a^6*b^6*c^4 + 192*a^7*b^4*c^5 - 640*a^8*b^2*c^6

+ 768*a^9*c^7)*e^3)*sqrt(-(108*a^3*b*c^6*d^9*e^3 + 108*a^6*b*c^3*d^3*e^9 - (b^4*c^6 - 18*a*b^2*c^7 + 81*a^2*c

^8)*d^12 - 12*(a*b^3*c^6 - 9*a^2*b*c^7)*d^11*e - 18*(a^2*b^2*c^6 + 9*a^3*c^7)*d^10*e^2 - 9*(2*a^3*b^2*c^5 - 9*

a^4*c^6)*d^8*e^4 + 12*(a^3*b^3*c^4 - 18*a^4*b*c^5)*d^7*e^5 + 2*(a^3*b^4*c^3 + 18*a^4*b^2*c^4 + 162*a^5*c^5)*d^

6*e^6 + 12*(a^4*b^3*c^3 - 18*a^5*b*c^4)*d^5*e^7 - 9*(2*a^5*b^2*c^3 - 9*a^6*c^4)*d^4*e^8 - 18*(a^6*b^2*c^2 + 9*

a^7*c^3)*d^2*e^10 - 12*(a^6*b^3*c - 9*a^7*b*c^2)*d*e^11 - (a^6*b^4 - 18*a^7*b^2*c + 81*a^8*c^2)*e^12)/(a^6*b^6

*c^6 - 12*a^7*b^4*c^7 + 48*a^8*b^2*c^8 - 64*a^9*c^9)))*sqrt(-((b^5*c^3 - 15*a*b^3*c^4 + 60*a^2*b*c^5)*d^6 + 6*

(a*b^4*c^3 - 6*a^2*b^2*c^4 - 24*a^3*c^5)*d^5*e - 3*(3*a^2*b^3*c^3 - 92*a^3*b*c^4)*d^4*e^2 - 8*(11*a^3*b^2*c^3

+ 36*a^4*c^4)*d^3*e^3 - 3*(3*a^3*b^3*c^2 - 92*a^4*b*c^3)*d^2*e^4 + 6*(a^3*b^4*c - 6*a^4*b^2*c^2 - 24*a^5*c^3)*

d*e^5 + (a^3*b^5 - 15*a^4*b^3*c + 60*a^5*b*c^2)*e^6 + (a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 - 64*a^6*

c^6)*sqrt(-(108*a^3*b*c^6*d^9*e^3 + 108*a^6*b*c^3*d^3*e^9 - (b^4*c^6 - 18*a*b^2*c^7 + 81*a^2*c^8)*d^12 - 12*(a

*b^3*c^6 - 9*a^2*b*c^7)*d^11*e - 18*(a^2*b^2*c^6 + 9*a^3*c^7)*d^10*e^2 - 9*(2*a^3*b^2*c^5 - 9*a^4*c^6)*d^8*e^4

+ 12*(a^3*b^3*c^4 - 18*a^4*b*c^5)*d^7*e^5 + 2*(a^3*b^4*c^3 + 18*a^4*b^2*c^4 + 162*a^5*c^5)*d^6*e^6 + 12*(a^4*

b^3*c^3 - 18*a^5*b*c^4)*d^5*e^7 - 9*(2*a^5*b^2*c^3 - 9*a^6*c^4)*d^4*e^8 - 18*(a^6*b^2*c^2 + 9*a^7*c^3)*d^2*e^1

0 - 12*(a^6*b^3*c - 9*a^7*b*c^2)*d*e^11 - (a^6*b^4 - 18*a^7*b^2*c + 81*a^8*c^2)*e^12)/(a^6*b^6*c^6 - 12*a^7*b^

4*c^7 + 48*a^8*b^2*c^8 - 64*a^9*c^9)))/(a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 - 64*a^6*c^6))) + sqrt(1

/2)*(a^2*b^2*c - 4*a^3*c^2 + (a*b^2*c^2 - 4*a^2*c^3)*x^4 + (a*b^3*c - 4*a^2*b*c^2)*x^2)*sqrt(-((b^5*c^3 - 15*a

*b^3*c^4 + 60*a^2*b*c^5)*d^6 + 6*(a*b^4*c^3 - 6*a^2*b^2*c^4 - 24*a^3*c^5)*d^5*e - 3*(3*a^2*b^3*c^3 - 92*a^3*b*

c^4)*d^4*e^2 - 8*(11*a^3*b^2*c^3 + 36*a^4*c^4)*d^3*e^3 - 3*(3*a^3*b^3*c^2 - 92*a^4*b*c^3)*d^2*e^4 + 6*(a^3*b^4

*c - 6*a^4*b^2*c^2 - 24*a^5*c^3)*d*e^5 + (a^3*b^5 - 15*a^4*b^3*c + 60*a^5*b*c^2)*e^6 + (a^3*b^6*c^3 - 12*a^4*b

^4*c^4 + 48*a^5*b^2*c^5 - 64*a^6*c^6)*sqrt(-(108*a^3*b*c^6*d^9*e^3 + 108*a^6*b*c^3*d^3*e^9 - (b^4*c^6 - 18*a*b

^2*c^7 + 81*a^2*c^8)*d^12 - 12*(a*b^3*c^6 - 9*a^2*b*c^7)*d^11*e - 18*(a^2*b^2*c^6 + 9*a^3*c^7)*d^10*e^2 - 9*(2

*a^3*b^2*c^5 - 9*a^4*c^6)*d^8*e^4 + 12*(a^3*b^3*c^4 - 18*a^4*b*c^5)*d^7*e^5 + 2*(a^3*b^4*c^3 + 18*a^4*b^2*c^4

+ 162*a^5*c^5)*d^6*e^6 + 12*(a^4*b^3*c^3 - 18*a^5*b*c^4)*d^5*e^7 - 9*(2*a^5*b^2*c^3 - 9*a^6*c^4)*d^4*e^8 - 18*

(a^6*b^2*c^2 + 9*a^7*c^3)*d^2*e^10 - 12*(a^6*b^3*c - 9*a^7*b*c^2)*d*e^11 - (a^6*b^4 - 18*a^7*b^2*c + 81*a^8*c^

2)*e^12)/(a^6*b^6*c^6 - 12*a^7*b^4*c^7 + 48*a^8*b^2*c^8 - 64*a^9*c^9)))/(a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5

*b^2*c^5 - 64*a^6*c^6))*log(-((5*b^4*c^6 - 81*a*b^2*c^7 + 324*a^2*c^8)*d^12 - 3*(3*b^5*c^5 - 65*a*b^3*c^6 + 32

4*a^2*b*c^7)*d^11*e + 3*(b^6*c^4 - 42*a*b^4*c^5 + 252*a^2*b^2*c^6 + 432*a^3*c^7)*d^10*e^2 + (b^7*c^3 + 3*a*b^5

*c^4 + 33*a^2*b^3*c^5 - 2916*a^3*b*c^6)*d^9*e^3 + 9*(a*b^6*c^3 - 15*a^2*b^4*c^4 + 195*a^3*b^2*c^5 + 180*a^4*c^

6)*d^8*e^4 - 162*(a^3*b^3*c^4 + 12*a^4*b*c^5)*d^7*e^5 + 162*(a^4*b^3*c^3 + 12*a^5*b*c^4)*d^5*e^7 - 9*(a^3*b^6*

c - 15*a^4*b^4*c^2 + 195*a^5*b^2*c^3 + 180*a^6*c^4)*d^4*e^8 - (a^3*b^7 + 3*a^4*b^5*c + 33*a^5*b^3*c^2 - 2916*a

^6*b*c^3)*d^3*e^9 - 3*(a^4*b^6 - 42*a^5*b^4*c + 252*a^6*b^2*c^2 + 432*a^7*c^3)*d^2*e^10 + 3*(3*a^5*b^5 - 65*a^

6*b^3*c + 324*a^7*b*c^2)*d*e^11 - (5*a^6*b^4 - 81*a^7*b^2*c + 324*a^8*c^2)*e^12)*x - 1/2*sqrt(1/2)*((b^8*c^4 -

23*a*b^6*c^5 + 190*a^2*b^4*c^6 - 672*a^3*b^2*c^7 + 864*a^4*c^8)*d^9 + 9*(a*b^7*c^4 - 15*a^2*b^5*c^5 + 72*a^3*

b^3*c^6 - 112*a^4*b*c^7)*d^8*e + 3*(a^2*b^6*c^4 + 28*a^3*b^4*c^5 - 272*a^4*b^2*c^6 + 576*a^5*c^7)*d^7*e^2 + (a

^2*b^7*c^3 - 80*a^3*b^5*c^4 + 592*a^4*b^3*c^5 - 1152*a^5*b*c^6)*d^6*e^3 + 15*(a^3*b^6*c^3 - 8*a^4*b^4*c^4 + 16

*a^5*b^2*c^5)*d^5*e^4 - 6*(a^3*b^7*c^2 - 17*a^4*b^5*c^3 + 88*a^5*b^3*c^4 - 144*a^6*b*c^5)*d^4*e^5 - (a^3*b^8*c

- 5*a^4*b^6*c^2 + 100*a^5*b^4*c^3 - 816*a^6*b^2*c^4 + 1728*a^7*c^5)*d^3*e^6 - 3*(a^4*b^7*c - 32*a^5*b^5*c^2 +

208*a^6*b^3*c^3 - 384*a^7*b*c^4)*d^2*e^7 - 54*(a^6*b^4*c^2 - 8*a^7*b^2*c^3 + 16*a^8*c^4)*d*e^8 - (a^5*b^7 - 1

7*a^6*b^5*c + 88*a^7*b^3*c^2 - 144*a^8*b*c^3)*e^9 - ((a^3*b^9*c^4 - 20*a^4*b^7*c^5 + 144*a^5*b^5*c^6 - 448*a^6

*b^3*c^7 + 512*a^7*b*c^8)*d^3 + 3*(a^4*b^8*c^4 - 8*a^5*b^6*c^5 + 128*a^7*b^2*c^7 - 256*a^8*c^8)*d^2*e - 12*(a^

5*b^7*c^4 - 12*a^6*b^5*c^5 + 48*a^7*b^3*c^6 - 64*a^8*b*c^7)*d*e^2 - (a^5*b^8*c^3 - 24*a^6*b^6*c^4 + 192*a^7*b^

4*c^5 - 640*a^8*b^2*c^6 + 768*a^9*c^7)*e^3)*sqrt(-(108*a^3*b*c^6*d^9*e^3 + 108*a^6*b*c^3*d^3*e^9 - (b^4*c^6 -

18*a*b^2*c^7 + 81*a^2*c^8)*d^12 - 12*(a*b^3*c^6 - 9*a^2*b*c^7)*d^11*e - 18*(a^2*b^2*c^6 + 9*a^3*c^7)*d^10*e^2

- 9*(2*a^3*b^2*c^5 - 9*a^4*c^6)*d^8*e^4 + 12*(a^3*b^3*c^4 - 18*a^4*b*c^5)*d^7*e^5 + 2*(a^3*b^4*c^3 + 18*a^4*b^

2*c^4 + 162*a^5*c^5)*d^6*e^6 + 12*(a^4*b^3*c^3 - 18*a^5*b*c^4)*d^5*e^7 - 9*(2*a^5*b^2*c^3 - 9*a^6*c^4)*d^4*e^8

- 18*(a^6*b^2*c^2 + 9*a^7*c^3)*d^2*e^10 - 12*(a^6*b^3*c - 9*a^7*b*c^2)*d*e^11 - (a^6*b^4 - 18*a^7*b^2*c + 81*

a^8*c^2)*e^12)/(a^6*b^6*c^6 - 12*a^7*b^4*c^7 + 48*a^8*b^2*c^8 - 64*a^9*c^9)))*sqrt(-((b^5*c^3 - 15*a*b^3*c^4 +

60*a^2*b*c^5)*d^6 + 6*(a*b^4*c^3 - 6*a^2*b^2*c^4 - 24*a^3*c^5)*d^5*e - 3*(3*a^2*b^3*c^3 - 92*a^3*b*c^4)*d^4*e

^2 - 8*(11*a^3*b^2*c^3 + 36*a^4*c^4)*d^3*e^3 - 3*(3*a^3*b^3*c^2 - 92*a^4*b*c^3)*d^2*e^4 + 6*(a^3*b^4*c - 6*a^4

*b^2*c^2 - 24*a^5*c^3)*d*e^5 + (a^3*b^5 - 15*a^4*b^3*c + 60*a^5*b*c^2)*e^6 + (a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 4

8*a^5*b^2*c^5 - 64*a^6*c^6)*sqrt(-(108*a^3*b*c^6*d^9*e^3 + 108*a^6*b*c^3*d^3*e^9 - (b^4*c^6 - 18*a*b^2*c^7 + 8

1*a^2*c^8)*d^12 - 12*(a*b^3*c^6 - 9*a^2*b*c^7)*d^11*e - 18*(a^2*b^2*c^6 + 9*a^3*c^7)*d^10*e^2 - 9*(2*a^3*b^2*c

^5 - 9*a^4*c^6)*d^8*e^4 + 12*(a^3*b^3*c^4 - 18*a^4*b*c^5)*d^7*e^5 + 2*(a^3*b^4*c^3 + 18*a^4*b^2*c^4 + 162*a^5*

c^5)*d^6*e^6 + 12*(a^4*b^3*c^3 - 18*a^5*b*c^4)*d^5*e^7 - 9*(2*a^5*b^2*c^3 - 9*a^6*c^4)*d^4*e^8 - 18*(a^6*b^2*c

^2 + 9*a^7*c^3)*d^2*e^10 - 12*(a^6*b^3*c - 9*a^7*b*c^2)*d*e^11 - (a^6*b^4 - 18*a^7*b^2*c + 81*a^8*c^2)*e^12)/(

a^6*b^6*c^6 - 12*a^7*b^4*c^7 + 48*a^8*b^2*c^8 - 64*a^9*c^9)))/(a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 -

64*a^6*c^6))) - sqrt(1/2)*(a^2*b^2*c - 4*a^3*c^2 + (a*b^2*c^2 - 4*a^2*c^3)*x^4 + (a*b^3*c - 4*a^2*b*c^2)*x^2)

*sqrt(-((b^5*c^3 - 15*a*b^3*c^4 + 60*a^2*b*c^5)*d^6 + 6*(a*b^4*c^3 - 6*a^2*b^2*c^4 - 24*a^3*c^5)*d^5*e - 3*(3*

a^2*b^3*c^3 - 92*a^3*b*c^4)*d^4*e^2 - 8*(11*a^3*b^2*c^3 + 36*a^4*c^4)*d^3*e^3 - 3*(3*a^3*b^3*c^2 - 92*a^4*b*c^

3)*d^2*e^4 + 6*(a^3*b^4*c - 6*a^4*b^2*c^2 - 24*a^5*c^3)*d*e^5 + (a^3*b^5 - 15*a^4*b^3*c + 60*a^5*b*c^2)*e^6 -

(a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 - 64*a^6*c^6)*sqrt(-(108*a^3*b*c^6*d^9*e^3 + 108*a^6*b*c^3*d^3*

e^9 - (b^4*c^6 - 18*a*b^2*c^7 + 81*a^2*c^8)*d^12 - 12*(a*b^3*c^6 - 9*a^2*b*c^7)*d^11*e - 18*(a^2*b^2*c^6 + 9*a

^3*c^7)*d^10*e^2 - 9*(2*a^3*b^2*c^5 - 9*a^4*c^6)*d^8*e^4 + 12*(a^3*b^3*c^4 - 18*a^4*b*c^5)*d^7*e^5 + 2*(a^3*b^

4*c^3 + 18*a^4*b^2*c^4 + 162*a^5*c^5)*d^6*e^6 + 12*(a^4*b^3*c^3 - 18*a^5*b*c^4)*d^5*e^7 - 9*(2*a^5*b^2*c^3 - 9

*a^6*c^4)*d^4*e^8 - 18*(a^6*b^2*c^2 + 9*a^7*c^3)*d^2*e^10 - 12*(a^6*b^3*c - 9*a^7*b*c^2)*d*e^11 - (a^6*b^4 - 1

8*a^7*b^2*c + 81*a^8*c^2)*e^12)/(a^6*b^6*c^6 - 12*a^7*b^4*c^7 + 48*a^8*b^2*c^8 - 64*a^9*c^9)))/(a^3*b^6*c^3 -

12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 - 64*a^6*c^6))*log(-((5*b^4*c^6 - 81*a*b^2*c^7 + 324*a^2*c^8)*d^12 - 3*(3*b^5*

c^5 - 65*a*b^3*c^6 + 324*a^2*b*c^7)*d^11*e + 3*(b^6*c^4 - 42*a*b^4*c^5 + 252*a^2*b^2*c^6 + 432*a^3*c^7)*d^10*e

^2 + (b^7*c^3 + 3*a*b^5*c^4 + 33*a^2*b^3*c^5 - 2916*a^3*b*c^6)*d^9*e^3 + 9*(a*b^6*c^3 - 15*a^2*b^4*c^4 + 195*a

^3*b^2*c^5 + 180*a^4*c^6)*d^8*e^4 - 162*(a^3*b^3*c^4 + 12*a^4*b*c^5)*d^7*e^5 + 162*(a^4*b^3*c^3 + 12*a^5*b*c^4

)*d^5*e^7 - 9*(a^3*b^6*c - 15*a^4*b^4*c^2 + 195*a^5*b^2*c^3 + 180*a^6*c^4)*d^4*e^8 - (a^3*b^7 + 3*a^4*b^5*c +

33*a^5*b^3*c^2 - 2916*a^6*b*c^3)*d^3*e^9 - 3*(a^4*b^6 - 42*a^5*b^4*c + 252*a^6*b^2*c^2 + 432*a^7*c^3)*d^2*e^10

+ 3*(3*a^5*b^5 - 65*a^6*b^3*c + 324*a^7*b*c^2)*d*e^11 - (5*a^6*b^4 - 81*a^7*b^2*c + 324*a^8*c^2)*e^12)*x + 1/

2*sqrt(1/2)*((b^8*c^4 - 23*a*b^6*c^5 + 190*a^2*b^4*c^6 - 672*a^3*b^2*c^7 + 864*a^4*c^8)*d^9 + 9*(a*b^7*c^4 - 1

5*a^2*b^5*c^5 + 72*a^3*b^3*c^6 - 112*a^4*b*c^7)*d^8*e + 3*(a^2*b^6*c^4 + 28*a^3*b^4*c^5 - 272*a^4*b^2*c^6 + 57

6*a^5*c^7)*d^7*e^2 + (a^2*b^7*c^3 - 80*a^3*b^5*c^4 + 592*a^4*b^3*c^5 - 1152*a^5*b*c^6)*d^6*e^3 + 15*(a^3*b^6*c

^3 - 8*a^4*b^4*c^4 + 16*a^5*b^2*c^5)*d^5*e^4 - 6*(a^3*b^7*c^2 - 17*a^4*b^5*c^3 + 88*a^5*b^3*c^4 - 144*a^6*b*c^

5)*d^4*e^5 - (a^3*b^8*c - 5*a^4*b^6*c^2 + 100*a^5*b^4*c^3 - 816*a^6*b^2*c^4 + 1728*a^7*c^5)*d^3*e^6 - 3*(a^4*b

^7*c - 32*a^5*b^5*c^2 + 208*a^6*b^3*c^3 - 384*a^7*b*c^4)*d^2*e^7 - 54*(a^6*b^4*c^2 - 8*a^7*b^2*c^3 + 16*a^8*c^

4)*d*e^8 - (a^5*b^7 - 17*a^6*b^5*c + 88*a^7*b^3*c^2 - 144*a^8*b*c^3)*e^9 + ((a^3*b^9*c^4 - 20*a^4*b^7*c^5 + 14

4*a^5*b^5*c^6 - 448*a^6*b^3*c^7 + 512*a^7*b*c^8)*d^3 + 3*(a^4*b^8*c^4 - 8*a^5*b^6*c^5 + 128*a^7*b^2*c^7 - 256*

a^8*c^8)*d^2*e - 12*(a^5*b^7*c^4 - 12*a^6*b^5*c^5 + 48*a^7*b^3*c^6 - 64*a^8*b*c^7)*d*e^2 - (a^5*b^8*c^3 - 24*a

^6*b^6*c^4 + 192*a^7*b^4*c^5 - 640*a^8*b^2*c^6 + 768*a^9*c^7)*e^3)*sqrt(-(108*a^3*b*c^6*d^9*e^3 + 108*a^6*b*c^

3*d^3*e^9 - (b^4*c^6 - 18*a*b^2*c^7 + 81*a^2*c^8)*d^12 - 12*(a*b^3*c^6 - 9*a^2*b*c^7)*d^11*e - 18*(a^2*b^2*c^6

+ 9*a^3*c^7)*d^10*e^2 - 9*(2*a^3*b^2*c^5 - 9*a^4*c^6)*d^8*e^4 + 12*(a^3*b^3*c^4 - 18*a^4*b*c^5)*d^7*e^5 + 2*(

a^3*b^4*c^3 + 18*a^4*b^2*c^4 + 162*a^5*c^5)*d^6*e^6 + 12*(a^4*b^3*c^3 - 18*a^5*b*c^4)*d^5*e^7 - 9*(2*a^5*b^2*c

^3 - 9*a^6*c^4)*d^4*e^8 - 18*(a^6*b^2*c^2 + 9*a^7*c^3)*d^2*e^10 - 12*(a^6*b^3*c - 9*a^7*b*c^2)*d*e^11 - (a^6*b

^4 - 18*a^7*b^2*c + 81*a^8*c^2)*e^12)/(a^6*b^6*c^6 - 12*a^7*b^4*c^7 + 48*a^8*b^2*c^8 - 64*a^9*c^9)))*sqrt(-((b

^5*c^3 - 15*a*b^3*c^4 + 60*a^2*b*c^5)*d^6 + 6*(a*b^4*c^3 - 6*a^2*b^2*c^4 - 24*a^3*c^5)*d^5*e - 3*(3*a^2*b^3*c^

3 - 92*a^3*b*c^4)*d^4*e^2 - 8*(11*a^3*b^2*c^3 + 36*a^4*c^4)*d^3*e^3 - 3*(3*a^3*b^3*c^2 - 92*a^4*b*c^3)*d^2*e^4

+ 6*(a^3*b^4*c - 6*a^4*b^2*c^2 - 24*a^5*c^3)*d*e^5 + (a^3*b^5 - 15*a^4*b^3*c + 60*a^5*b*c^2)*e^6 - (a^3*b^6*c

^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 - 64*a^6*c^6)*sqrt(-(108*a^3*b*c^6*d^9*e^3 + 108*a^6*b*c^3*d^3*e^9 - (b^4

*c^6 - 18*a*b^2*c^7 + 81*a^2*c^8)*d^12 - 12*(a*b^3*c^6 - 9*a^2*b*c^7)*d^11*e - 18*(a^2*b^2*c^6 + 9*a^3*c^7)*d^

10*e^2 - 9*(2*a^3*b^2*c^5 - 9*a^4*c^6)*d^8*e^4 + 12*(a^3*b^3*c^4 - 18*a^4*b*c^5)*d^7*e^5 + 2*(a^3*b^4*c^3 + 18

*a^4*b^2*c^4 + 162*a^5*c^5)*d^6*e^6 + 12*(a^4*b^3*c^3 - 18*a^5*b*c^4)*d^5*e^7 - 9*(2*a^5*b^2*c^3 - 9*a^6*c^4)*

d^4*e^8 - 18*(a^6*b^2*c^2 + 9*a^7*c^3)*d^2*e^10 - 12*(a^6*b^3*c - 9*a^7*b*c^2)*d*e^11 - (a^6*b^4 - 18*a^7*b^2*

c + 81*a^8*c^2)*e^12)/(a^6*b^6*c^6 - 12*a^7*b^4*c^7 + 48*a^8*b^2*c^8 - 64*a^9*c^9)))/(a^3*b^6*c^3 - 12*a^4*b^4

*c^4 + 48*a^5*b^2*c^5 - 64*a^6*c^6))) + sqrt(1/2)*(a^2*b^2*c - 4*a^3*c^2 + (a*b^2*c^2 - 4*a^2*c^3)*x^4 + (a*b^

3*c - 4*a^2*b*c^2)*x^2)*sqrt(-((b^5*c^3 - 15*a*b^3*c^4 + 60*a^2*b*c^5)*d^6 + 6*(a*b^4*c^3 - 6*a^2*b^2*c^4 - 24

*a^3*c^5)*d^5*e - 3*(3*a^2*b^3*c^3 - 92*a^3*b*c^4)*d^4*e^2 - 8*(11*a^3*b^2*c^3 + 36*a^4*c^4)*d^3*e^3 - 3*(3*a^

3*b^3*c^2 - 92*a^4*b*c^3)*d^2*e^4 + 6*(a^3*b^4*c - 6*a^4*b^2*c^2 - 24*a^5*c^3)*d*e^5 + (a^3*b^5 - 15*a^4*b^3*c

+ 60*a^5*b*c^2)*e^6 - (a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 - 64*a^6*c^6)*sqrt(-(108*a^3*b*c^6*d^9*e

^3 + 108*a^6*b*c^3*d^3*e^9 - (b^4*c^6 - 18*a*b^2*c^7 + 81*a^2*c^8)*d^12 - 12*(a*b^3*c^6 - 9*a^2*b*c^7)*d^11*e

- 18*(a^2*b^2*c^6 + 9*a^3*c^7)*d^10*e^2 - 9*(2*a^3*b^2*c^5 - 9*a^4*c^6)*d^8*e^4 + 12*(a^3*b^3*c^4 - 18*a^4*b*c

^5)*d^7*e^5 + 2*(a^3*b^4*c^3 + 18*a^4*b^2*c^4 + 162*a^5*c^5)*d^6*e^6 + 12*(a^4*b^3*c^3 - 18*a^5*b*c^4)*d^5*e^7

- 9*(2*a^5*b^2*c^3 - 9*a^6*c^4)*d^4*e^8 - 18*(a^6*b^2*c^2 + 9*a^7*c^3)*d^2*e^10 - 12*(a^6*b^3*c - 9*a^7*b*c^2

)*d*e^11 - (a^6*b^4 - 18*a^7*b^2*c + 81*a^8*c^2)*e^12)/(a^6*b^6*c^6 - 12*a^7*b^4*c^7 + 48*a^8*b^2*c^8 - 64*a^9

*c^9)))/(a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 - 64*a^6*c^6))*log(-((5*b^4*c^6 - 81*a*b^2*c^7 + 324*a^

2*c^8)*d^12 - 3*(3*b^5*c^5 - 65*a*b^3*c^6 + 324*a^2*b*c^7)*d^11*e + 3*(b^6*c^4 - 42*a*b^4*c^5 + 252*a^2*b^2*c^

6 + 432*a^3*c^7)*d^10*e^2 + (b^7*c^3 + 3*a*b^5*c^4 + 33*a^2*b^3*c^5 - 2916*a^3*b*c^6)*d^9*e^3 + 9*(a*b^6*c^3 -

15*a^2*b^4*c^4 + 195*a^3*b^2*c^5 + 180*a^4*c^6)*d^8*e^4 - 162*(a^3*b^3*c^4 + 12*a^4*b*c^5)*d^7*e^5 + 162*(a^4

*b^3*c^3 + 12*a^5*b*c^4)*d^5*e^7 - 9*(a^3*b^6*c - 15*a^4*b^4*c^2 + 195*a^5*b^2*c^3 + 180*a^6*c^4)*d^4*e^8 - (a

^3*b^7 + 3*a^4*b^5*c + 33*a^5*b^3*c^2 - 2916*a^6*b*c^3)*d^3*e^9 - 3*(a^4*b^6 - 42*a^5*b^4*c + 252*a^6*b^2*c^2

+ 432*a^7*c^3)*d^2*e^10 + 3*(3*a^5*b^5 - 65*a^6*b^3*c + 324*a^7*b*c^2)*d*e^11 - (5*a^6*b^4 - 81*a^7*b^2*c + 32

4*a^8*c^2)*e^12)*x - 1/2*sqrt(1/2)*((b^8*c^4 - 23*a*b^6*c^5 + 190*a^2*b^4*c^6 - 672*a^3*b^2*c^7 + 864*a^4*c^8)

*d^9 + 9*(a*b^7*c^4 - 15*a^2*b^5*c^5 + 72*a^3*b^3*c^6 - 112*a^4*b*c^7)*d^8*e + 3*(a^2*b^6*c^4 + 28*a^3*b^4*c^5

- 272*a^4*b^2*c^6 + 576*a^5*c^7)*d^7*e^2 + (a^2*b^7*c^3 - 80*a^3*b^5*c^4 + 592*a^4*b^3*c^5 - 1152*a^5*b*c^6)*

d^6*e^3 + 15*(a^3*b^6*c^3 - 8*a^4*b^4*c^4 + 16*a^5*b^2*c^5)*d^5*e^4 - 6*(a^3*b^7*c^2 - 17*a^4*b^5*c^3 + 88*a^5

*b^3*c^4 - 144*a^6*b*c^5)*d^4*e^5 - (a^3*b^8*c - 5*a^4*b^6*c^2 + 100*a^5*b^4*c^3 - 816*a^6*b^2*c^4 + 1728*a^7*

c^5)*d^3*e^6 - 3*(a^4*b^7*c - 32*a^5*b^5*c^2 + 208*a^6*b^3*c^3 - 384*a^7*b*c^4)*d^2*e^7 - 54*(a^6*b^4*c^2 - 8*

a^7*b^2*c^3 + 16*a^8*c^4)*d*e^8 - (a^5*b^7 - 17*a^6*b^5*c + 88*a^7*b^3*c^2 - 144*a^8*b*c^3)*e^9 + ((a^3*b^9*c^

4 - 20*a^4*b^7*c^5 + 144*a^5*b^5*c^6 - 448*a^6*b^3*c^7 + 512*a^7*b*c^8)*d^3 + 3*(a^4*b^8*c^4 - 8*a^5*b^6*c^5 +

128*a^7*b^2*c^7 - 256*a^8*c^8)*d^2*e - 12*(a^5*b^7*c^4 - 12*a^6*b^5*c^5 + 48*a^7*b^3*c^6 - 64*a^8*b*c^7)*d*e^

2 - (a^5*b^8*c^3 - 24*a^6*b^6*c^4 + 192*a^7*b^4*c^5 - 640*a^8*b^2*c^6 + 768*a^9*c^7)*e^3)*sqrt(-(108*a^3*b*c^6

*d^9*e^3 + 108*a^6*b*c^3*d^3*e^9 - (b^4*c^6 - 18*a*b^2*c^7 + 81*a^2*c^8)*d^12 - 12*(a*b^3*c^6 - 9*a^2*b*c^7)*d

^11*e - 18*(a^2*b^2*c^6 + 9*a^3*c^7)*d^10*e^2 - 9*(2*a^3*b^2*c^5 - 9*a^4*c^6)*d^8*e^4 + 12*(a^3*b^3*c^4 - 18*a

^4*b*c^5)*d^7*e^5 + 2*(a^3*b^4*c^3 + 18*a^4*b^2*c^4 + 162*a^5*c^5)*d^6*e^6 + 12*(a^4*b^3*c^3 - 18*a^5*b*c^4)*d

^5*e^7 - 9*(2*a^5*b^2*c^3 - 9*a^6*c^4)*d^4*e^8 - 18*(a^6*b^2*c^2 + 9*a^7*c^3)*d^2*e^10 - 12*(a^6*b^3*c - 9*a^7

*b*c^2)*d*e^11 - (a^6*b^4 - 18*a^7*b^2*c + 81*a^8*c^2)*e^12)/(a^6*b^6*c^6 - 12*a^7*b^4*c^7 + 48*a^8*b^2*c^8 -

64*a^9*c^9)))*sqrt(-((b^5*c^3 - 15*a*b^3*c^4 + 60*a^2*b*c^5)*d^6 + 6*(a*b^4*c^3 - 6*a^2*b^2*c^4 - 24*a^3*c^5)*

d^5*e - 3*(3*a^2*b^3*c^3 - 92*a^3*b*c^4)*d^4*e^2 - 8*(11*a^3*b^2*c^3 + 36*a^4*c^4)*d^3*e^3 - 3*(3*a^3*b^3*c^2

- 92*a^4*b*c^3)*d^2*e^4 + 6*(a^3*b^4*c - 6*a^4*b^2*c^2 - 24*a^5*c^3)*d*e^5 + (a^3*b^5 - 15*a^4*b^3*c + 60*a^5*

b*c^2)*e^6 - (a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 - 64*a^6*c^6)*sqrt(-(108*a^3*b*c^6*d^9*e^3 + 108*a

^6*b*c^3*d^3*e^9 - (b^4*c^6 - 18*a*b^2*c^7 + 81*a^2*c^8)*d^12 - 12*(a*b^3*c^6 - 9*a^2*b*c^7)*d^11*e - 18*(a^2*

b^2*c^6 + 9*a^3*c^7)*d^10*e^2 - 9*(2*a^3*b^2*c^5 - 9*a^4*c^6)*d^8*e^4 + 12*(a^3*b^3*c^4 - 18*a^4*b*c^5)*d^7*e^

5 + 2*(a^3*b^4*c^3 + 18*a^4*b^2*c^4 + 162*a^5*c^5)*d^6*e^6 + 12*(a^4*b^3*c^3 - 18*a^5*b*c^4)*d^5*e^7 - 9*(2*a^

5*b^2*c^3 - 9*a^6*c^4)*d^4*e^8 - 18*(a^6*b^2*c^2 + 9*a^7*c^3)*d^2*e^10 - 12*(a^6*b^3*c - 9*a^7*b*c^2)*d*e^11 -

(a^6*b^4 - 18*a^7*b^2*c + 81*a^8*c^2)*e^12)/(a^6*b^6*c^6 - 12*a^7*b^4*c^7 + 48*a^8*b^2*c^8 - 64*a^9*c^9)))/(a

^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 - 64*a^6*c^6))) - 2*(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 -

(b^2*c - 2*a*c^2)*d^3)*x)/(a^2*b^2*c - 4*a^3*c^2 + (a*b^2*c^2 - 4*a^2*c^3)*x^4 + (a*b^3*c - 4*a^2*b*c^2)*x^2)

________________________________________________________________________________________

giac [B] time = 2.46, size = 8983, normalized size = 15.96

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

1/2*(b*c^2*d^3*x^3 - 6*a*c^2*d^2*x^3*e + 3*a*b*c*d*x^3*e^2 + b^2*c*d^3*x - 2*a*c^2*d^3*x - a*b^2*x^3*e^3 + 2*a

^2*c*x^3*e^3 - 3*a*b*c*d^2*x*e + 6*a^2*c*d*x*e^2 - a^2*b*x*e^3)/((c*x^4 + b*x^2 + a)*(a*b^2*c - 4*a^2*c^2)) +

1/16*((2*b^3*c^4 - 8*a*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*s

qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4

*a*c)*c)*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^4 - 2*(b^2 - 4*a*c)*b*c^4)*(a

*b^2*c - 4*a^2*c^2)^2*d^3 - 6*(2*a*b^2*c^4 - 8*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c

)*c)*a*b^2*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*

a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^4

- 2*(b^2 - 4*a*c)*a*c^4)*(a*b^2*c - 4*a^2*c^2)^2*d^2*e + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c^3

- 14*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^4 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^

4 - 2*a*b^6*c^4 + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^5 + 20*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*

a*c)*c)*a^2*b^3*c^5 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^5 + 28*a^2*b^4*c^5 - 96*sqrt(2)*sqrt(b*c

+ sqrt(b^2 - 4*a*c)*c)*a^4*c^6 - 48*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^6 - 10*sqrt(2)*sqrt(b*c +

sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^6 - 128*a^3*b^2*c^6 + 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^7 + 192

*a^4*c^7 + 2*(b^2 - 4*a*c)*a*b^4*c^4 - 20*(b^2 - 4*a*c)*a^2*b^2*c^5 + 48*(b^2 - 4*a*c)*a^3*c^6)*d^3*abs(a*b^2*

c - 4*a^2*c^2) + 3*(2*a*b^3*c^3 - 8*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^

3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt

(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*

(b^2 - 4*a*c)*a*b*c^3)*(a*b^2*c - 4*a^2*c^2)^2*d*e^2 + 6*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^3

- 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^4 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^

4 - 2*a^2*b^5*c^4 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^5 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a

*c)*c)*a^3*b^2*c^5 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^5 + 16*a^3*b^3*c^5 - 4*sqrt(2)*sqrt(b*c

+ sqrt(b^2 - 4*a*c)*c)*a^3*b*c^6 - 32*a^4*b*c^6 + 2*(b^2 - 4*a*c)*a^2*b^3*c^4 - 8*(b^2 - 4*a*c)*a^3*b*c^5)*d^

2*abs(a*b^2*c - 4*a^2*c^2)*e + (2*a^2*b^7*c^6 - 40*a^3*b^5*c^7 + 224*a^4*b^3*c^8 - 384*a^5*b*c^9 - sqrt(2)*sqr

t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^7*c^4 + 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2

- 4*a*c)*c)*a^3*b^5*c^5 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^6*c^5 - 112*sqrt(2

)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^6 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt

(b^2 - 4*a*c)*c)*a^3*b^4*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^6 + 192*sqr

t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b*c^7 + 96*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr

t(b^2 - 4*a*c)*c)*a^4*b^2*c^7 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^7 - 48*

sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^8 - 2*(b^2 - 4*a*c)*a^2*b^5*c^6 + 32*(b^2 -

4*a*c)*a^3*b^3*c^7 - 96*(b^2 - 4*a*c)*a^4*b*c^8)*d^3 + (2*a*b^4*c^2 - 20*a^2*b^2*c^3 + 48*a^3*c^4 - sqrt(2)*sq

rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a

*c)*c)*a^2*b^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c - 24*sqrt(2)*sqrt(b^2 -

4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^2 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)

*a^2*b*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)

*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 - 2*(b^2 - 4*a*c)*a*b^2*c^2 + 12*(b^2 - 4*a*c)*a^2*c^3)*(a*b^2*c - 4*

a^2*c^2)^2*e^3 - 12*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^3 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a

*c)*c)*a^4*b^2*c^4 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^4 - 2*a^3*b^4*c^4 + 16*sqrt(2)*sqrt(b

*c + sqrt(b^2 - 4*a*c)*c)*a^5*c^5 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^5 + sqrt(2)*sqrt(b*c + s

qrt(b^2 - 4*a*c)*c)*a^3*b^2*c^5 + 16*a^4*b^2*c^5 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*c^6 - 32*a^5*

c^6 + 2*(b^2 - 4*a*c)*a^3*b^2*c^4 - 8*(b^2 - 4*a*c)*a^4*c^5)*d*abs(a*b^2*c - 4*a^2*c^2)*e^2 + 12*(2*a^3*b^6*c^

6 - 16*a^4*b^4*c^7 + 32*a^5*b^2*c^8 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c^4 +

8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^4*c^5 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c

+ sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^5 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^6

- 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*

c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^6 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^7

- 2*(b^2 - 4*a*c)*a^3*b^4*c^6 + 8*(b^2 - 4*a*c)*a^4*b^2*c^7)*d^2*e + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*

c)*a^3*b^5*c^2 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^3 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c

)*c)*a^3*b^4*c^3 - 2*a^3*b^5*c^3 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b*c^4 + 8*sqrt(2)*sqrt(b*c +

sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^4 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^4 + 16*a^4*b^3*c^4 - 4*s

qrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^5 - 32*a^5*b*c^5 + 2*(b^2 - 4*a*c)*a^3*b^3*c^3 - 8*(b^2 - 4*a*c

)*a^4*b*c^4)*abs(a*b^2*c - 4*a^2*c^2)*e^3 - 3*(2*a^3*b^7*c^5 - 8*a^4*b^5*c^6 - 32*a^5*b^3*c^7 + 128*a^6*b*c^8

- sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^7*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c

+ sqrt(b^2 - 4*a*c)*c)*a^4*b^5*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c^4

+ 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*

c + sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^5 - 64*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b*c^6

- 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^6 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt

(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b*c^7 - 2*(b^2 - 4*a*c)*a^3*b^5*c^5 + 32*(b^2 - 4*a*c)*a^5*b*c^7)*d*e^2 - (2*a

^3*b^8*c^4 - 32*a^4*b^6*c^5 + 160*a^5*b^4*c^6 - 256*a^6*b^2*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^

2 - 4*a*c)*c)*a^3*b^8*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c^3 + 2*sqrt(

2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^7*c^3 - 80*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr

t(b^2 - 4*a*c)*c)*a^5*b^4*c^4 - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^5*c^4 - sqr

t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c^4 + 128*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c +

sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^5 + 64*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^3*c^5 +

12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^4*c^5 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b

*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^6 - 2*(b^2 - 4*a*c)*a^3*b^6*c^4 + 24*(b^2 - 4*a*c)*a^4*b^4*c^5 - 64*(b^2 -

4*a*c)*a^5*b^2*c^6)*e^3)*arctan(2*sqrt(1/2)*x/sqrt((a*b^3*c - 4*a^2*b*c^2 + sqrt((a*b^3*c - 4*a^2*b*c^2)^2 -

4*(a^2*b^2*c - 4*a^3*c^2)*(a*b^2*c^2 - 4*a^2*c^3)))/(a*b^2*c^2 - 4*a^2*c^3)))/((a^3*b^6*c^3 - 12*a^4*b^4*c^4 -

2*a^3*b^5*c^4 + 48*a^5*b^2*c^5 + 16*a^4*b^3*c^5 + a^3*b^4*c^5 - 64*a^6*c^6 - 32*a^5*b*c^6 - 8*a^4*b^2*c^6 + 1

6*a^5*c^7)*abs(a*b^2*c - 4*a^2*c^2)*abs(c)) - 1/16*((2*b^3*c^4 - 8*a*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*

c - sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + 2*sqr

t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2

- 4*a*c)*c)*b*c^4 - 2*(b^2 - 4*a*c)*b*c^4)*(a*b^2*c - 4*a^2*c^2)^2*d^3 - 6*(2*a*b^2*c^4 - 8*a^2*c^5 - sqrt(2)

*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2

- 4*a*c)*c)*a^2*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2

- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^4 - 2*(b^2 - 4*a*c)*a*c^4)*(a*b^2*c - 4*a^2*c^2)^2*d^2*e - 2*(sqr

t(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^6*c^3 - 14*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^4 - 2*sq

rt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^5*c^4 + 2*a*b^6*c^4 + 64*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3

*b^2*c^5 + 20*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^5 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*

b^4*c^5 - 28*a^2*b^4*c^5 - 96*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*c^6 - 48*sqrt(2)*sqrt(b*c - sqrt(b^2

- 4*a*c)*c)*a^3*b*c^6 - 10*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^6 + 128*a^3*b^2*c^6 + 24*sqrt(2)

*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*c^7 - 192*a^4*c^7 - 2*(b^2 - 4*a*c)*a*b^4*c^4 + 20*(b^2 - 4*a*c)*a^2*b^2*

c^5 - 48*(b^2 - 4*a*c)*a^3*c^6)*d^3*abs(a*b^2*c - 4*a^2*c^2) + 3*(2*a*b^3*c^3 - 8*a^2*b*c^4 - sqrt(2)*sqrt(b^2

- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c

)*a^2*b*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c

)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*a*b*c^3)*(a*b^2*c - 4*a^2*c^2)^2*d*e^2 - 6*(sqrt(2

)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^3 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^4 - 2*sqrt

(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^4 + 2*a^2*b^5*c^4 + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a

^4*b*c^5 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^5 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2

*b^3*c^5 - 16*a^3*b^3*c^5 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^6 + 32*a^4*b*c^6 - 2*(b^2 - 4*a*

c)*a^2*b^3*c^4 + 8*(b^2 - 4*a*c)*a^3*b*c^5)*d^2*abs(a*b^2*c - 4*a^2*c^2)*e + (2*a^2*b^7*c^6 - 40*a^3*b^5*c^7 +

224*a^4*b^3*c^8 - 384*a^5*b*c^9 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^7*c^4 + 20*

sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^5 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c -

sqrt(b^2 - 4*a*c)*c)*a^2*b^6*c^5 - 112*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^6

- 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*

c - sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^6 + 192*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b*c^7

+ 96*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^7 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr

t(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^7 - 48*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b*

c^8 - 2*(b^2 - 4*a*c)*a^2*b^5*c^6 + 32*(b^2 - 4*a*c)*a^3*b^3*c^7 - 96*(b^2 - 4*a*c)*a^4*b*c^8)*d^3 + (2*a*b^4*

c^2 - 20*a^2*b^2*c^3 + 48*a^3*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4 + 10*sqrt(

2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b

^2 - 4*a*c)*c)*a*b^3*c - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*c^2 - 12*sqrt(2)*sqr

t(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a

*c)*c)*a*b^2*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*c^3 - 2*(b^2 - 4*a*c)*a*b^2

*c^2 + 12*(b^2 - 4*a*c)*a^2*c^3)*(a*b^2*c - 4*a^2*c^2)^2*e^3 + 12*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3

*b^4*c^3 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^4 - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a

^3*b^3*c^4 + 2*a^3*b^4*c^4 + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*c^5 + 8*sqrt(2)*sqrt(b*c - sqrt(b^

2 - 4*a*c)*c)*a^4*b*c^5 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^5 - 16*a^4*b^2*c^5 - 4*sqrt(2)*sqr

t(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*c^6 + 32*a^5*c^6 - 2*(b^2 - 4*a*c)*a^3*b^2*c^4 + 8*(b^2 - 4*a*c)*a^4*c^5)*d*a

bs(a*b^2*c - 4*a^2*c^2)*e^2 + 12*(2*a^3*b^6*c^6 - 16*a^4*b^4*c^7 + 32*a^5*b^2*c^8 - sqrt(2)*sqrt(b^2 - 4*a*c)*

sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*

b^4*c^5 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^5 - 16*sqrt(2)*sqrt(b^2 - 4*a*

c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^6 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a

^4*b^3*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^6 + 4*sqrt(2)*sqrt(b^2 - 4*a*

c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^7 - 2*(b^2 - 4*a*c)*a^3*b^4*c^6 + 8*(b^2 - 4*a*c)*a^4*b^2*c^7)*d^

2*e - 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^2 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b

^3*c^3 - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^3 + 2*a^3*b^5*c^3 + 16*sqrt(2)*sqrt(b*c - sqrt(b^

2 - 4*a*c)*c)*a^5*b*c^4 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^4 + sqrt(2)*sqrt(b*c - sqrt(b^2

- 4*a*c)*c)*a^3*b^3*c^4 - 16*a^4*b^3*c^4 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b*c^5 + 32*a^5*b*c^5

- 2*(b^2 - 4*a*c)*a^3*b^3*c^3 + 8*(b^2 - 4*a*c)*a^4*b*c^4)*abs(a*b^2*c - 4*a^2*c^2)*e^3 - 3*(2*a^3*b^7*c^5 - 8

*a^4*b^5*c^6 - 32*a^5*b^3*c^7 + 128*a^6*b*c^8 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*

b^7*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^5*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c

)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c^4 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a

^5*b^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c^5 - 64*sqrt(2)*sqrt(b^2 - 4*a

*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b*c^6 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a

^5*b^2*c^6 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b*c^7 - 2*(b^2 - 4*a*c)*a^3*b^5*

c^5 + 32*(b^2 - 4*a*c)*a^5*b*c^7)*d*e^2 - (2*a^3*b^8*c^4 - 32*a^4*b^6*c^5 + 160*a^5*b^4*c^6 - 256*a^6*b^2*c^7

- sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^8*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*

c - sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^7*c^3

- 80*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^4 - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr

t(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^5*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^6*c

^4 + 128*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^5 + 64*sqrt(2)*sqrt(b^2 - 4*a*c)*

sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^3*c^5 + 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4

*b^4*c^5 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^6 - 2*(b^2 - 4*a*c)*a^3*b^6*

c^4 + 24*(b^2 - 4*a*c)*a^4*b^4*c^5 - 64*(b^2 - 4*a*c)*a^5*b^2*c^6)*e^3)*arctan(2*sqrt(1/2)*x/sqrt((a*b^3*c - 4

*a^2*b*c^2 - sqrt((a*b^3*c - 4*a^2*b*c^2)^2 - 4*(a^2*b^2*c - 4*a^3*c^2)*(a*b^2*c^2 - 4*a^2*c^3)))/(a*b^2*c^2 -

4*a^2*c^3)))/((a^3*b^6*c^3 - 12*a^4*b^4*c^4 - 2*a^3*b^5*c^4 + 48*a^5*b^2*c^5 + 16*a^4*b^3*c^5 + a^3*b^4*c^5 -

64*a^6*c^6 - 32*a^5*b*c^6 - 8*a^4*b^2*c^6 + 16*a^5*c^7)*abs(a*b^2*c - 4*a^2*c^2)*abs(c))

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maple [B] time = 0.05, size = 1846, normalized size = 3.28

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^3/(c*x^4+b*x^2+a)^2,x)

[Out]

-1/4/a/(4*a*c-b^2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*

c*x)*b*d^3+2*a/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*

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

/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b^3*e^3-3/4/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((

b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b^2*d*e^2+1/4/a/(4*a*c-b^2

)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b*d^3+2*a

/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1

/2))*c)^(1/2)*c*x)*b*e^3-1/4/(4*a*c-b^2)/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan

h(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b^3*e^3-3/4/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a

*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b^2*d*e^2-3*a/(4*a*c-b^2)*c/(-4

*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*

x)*d*e^2+3/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4

*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b*d^2*e+1/4/a/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))

*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b^2*d^3-3*a/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*

2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*d*e^2+3/(4*a*c-b

^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1

/2)*c*x)*b*d^2*e+1/4/a/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2

)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b^2*d^3+1/4/(4*a*c-b^2)/c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*ar

ctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b^2*e^3+3/4/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*

c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*b*d*e^2-3/2/(4*a*c-b^2)*c*2^(1/2)/((-b+(-4*a*c

+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*d^2*e-3/(4*a*c-b^2)*c^2/(-4*a*c+b

^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*d^3

-1/4/(4*a*c-b^2)/c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*

x)*b^2*e^3-3/4/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^

(1/2)*c*x)*b*d*e^2+3/2/(4*a*c-b^2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^

(1/2))*c)^(1/2)*c*x)*d^2*e-3/(4*a*c-b^2)*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arcta

n(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*d^3-3/2*a/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2

)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*e^3+3/2*a/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))

*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x)*e^3+(-1/2*(2*a^2*c*e^3-a*b^2*e^3+3*a*b*c*d*e^2-

6*a*c^2*d^2*e+b*c^2*d^3)/a/c/(4*a*c-b^2)*x^3+1/2/c*(a^2*b*e^3-6*a^2*c*d*e^2+3*a*b*c*d^2*e+2*a*c^2*d^3-b^2*c*d^

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

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {{\left (b c^{2} d^{3} - 6 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} - {\left (a b^{2} - 2 \, a^{2} c\right )} e^{3}\right )} x^{3} - {\left (3 \, a b c d^{2} e - 6 \, a^{2} c d e^{2} + a^{2} b e^{3} - {\left (b^{2} c - 2 \, a c^{2}\right )} d^{3}\right )} x}{2 \, {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x^{4} + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} x^{2}\right )}} - \frac {-\int \frac {3 \, a b c d^{2} e - 6 \, a^{2} c d e^{2} + a^{2} b e^{3} + {\left (b^{2} c - 6 \, a c^{2}\right )} d^{3} + {\left (b c^{2} d^{3} - 6 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} + {\left (a b^{2} - 6 \, a^{2} c\right )} e^{3}\right )} x^{2}}{c x^{4} + b x^{2} + a}\,{d x}}{2 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/2*((b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - (a*b^2 - 2*a^2*c)*e^3)*x^3 - (3*a*b*c*d^2*e - 6*a^2*c*d*e^2

+ a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3)*x)/(a^2*b^2*c - 4*a^3*c^2 + (a*b^2*c^2 - 4*a^2*c^3)*x^4 + (a*b^3*c - 4*a^

2*b*c^2)*x^2) - 1/2*integrate(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 + (b^2*c - 6*a*c^2)*d^3 + (b*c^2*d^3

- 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 + (a*b^2 - 6*a^2*c)*e^3)*x^2)/(c*x^4 + b*x^2 + a), x)/(a*b^2*c - 4*a^2*c^2)

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mupad [B] time = 8.79, size = 29030, normalized size = 51.56

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2)^3/(a + b*x^2 + c*x^4)^2,x)

[Out]

- ((x^3*(b*c^2*d^3 - a*b^2*e^3 + 2*a^2*c*e^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2))/(2*a*c*(4*a*c - b^2)) - (x*(2*a

*c^2*d^3 + a^2*b*e^3 - b^2*c*d^3 - 6*a^2*c*d*e^2 + 3*a*b*c*d^2*e))/(2*a*c*(4*a*c - b^2)))/(a + b*x^2 + c*x^4)

- atan(((((6144*a^5*c^7*d^3 + 16*a*b^8*c^3*d^3 - 1024*a^6*b*c^5*e^3 + 6144*a^6*c^6*d*e^2 - 288*a^2*b^6*c^4*d^3

+ 1920*a^3*b^4*c^5*d^3 - 5632*a^4*b^2*c^6*d^3 + 16*a^3*b^7*c^2*e^3 - 192*a^4*b^5*c^3*e^3 + 768*a^5*b^3*c^4*e^

3 - 3072*a^5*b*c^6*d^2*e + 48*a^2*b^7*c^3*d^2*e - 576*a^3*b^5*c^4*d^2*e - 96*a^3*b^6*c^3*d*e^2 + 2304*a^4*b^3*

c^5*d^2*e + 1152*a^4*b^4*c^4*d*e^2 - 4608*a^5*b^2*c^5*d*e^2)/(8*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*

a^4*b^2*c^3)) - (x*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*

c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^

8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6

*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-

(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*

c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2

*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e

^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2

)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4

*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*

e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^

2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 2

40*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*(1024*a^5*b*c^6 - 16*a^2*b^7*

c^3 + 192*a^3*b^5*c^4 - 768*a^4*b^3*c^5))/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b

^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 38

40*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^

7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c

^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^

3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 374

4*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^

5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^

3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b

^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 +

4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b

*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c -

b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*

a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2) - (x*(72*a^5*c^3*e^6 - 72*a^2*c^6*d^6 - a^2*b^6*e^6 - b^4*c^4*d^6 + 14

*a*b^2*c^5*d^6 + 16*a^3*b^4*c*e^6 - 74*a^4*b^2*c^2*e^6 - 72*a^3*c^5*d^4*e^2 + 72*a^4*c^4*d^2*e^4 - 102*a^2*b^2

*c^4*d^4*e^2 + 44*a^2*b^3*c^3*d^3*e^3 + 9*a^2*b^4*c^2*d^2*e^4 - 174*a^3*b^2*c^3*d^2*e^4 - 6*a*b^3*c^4*d^5*e +

120*a^2*b*c^5*d^5*e - 6*a^2*b^5*c*d*e^5 + 24*a^4*b*c^3*d*e^5 + 144*a^3*b*c^4*d^3*e^3 + 42*a^3*b^3*c^2*d*e^5))/

(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^

8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c -

b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^

4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b

^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^

7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^

3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e

^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3

*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*

a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d

*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*

a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^1

2*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*1i

- (((6144*a^5*c^7*d^3 + 16*a*b^8*c^3*d^3 - 1024*a^6*b*c^5*e^3 + 6144*a^6*c^6*d*e^2 - 288*a^2*b^6*c^4*d^3 + 19

20*a^3*b^4*c^5*d^3 - 5632*a^4*b^2*c^6*d^3 + 16*a^3*b^7*c^2*e^3 - 192*a^4*b^5*c^3*e^3 + 768*a^5*b^3*c^4*e^3 - 3

072*a^5*b*c^6*d^2*e + 48*a^2*b^7*c^3*d^2*e - 576*a^3*b^5*c^4*d^2*e - 96*a^3*b^6*c^3*d*e^2 + 2304*a^4*b^3*c^5*d

^2*e + 1152*a^4*b^4*c^4*d*e^2 - 4608*a^5*b^2*c^5*d*e^2)/(8*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b

^2*c^3)) + (x*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b

^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5

*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4

*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*

c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d

^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4

- 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 -

13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^

(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*

c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 +

384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)

^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^

5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*(1024*a^5*b*c^6 - 16*a^2*b^7*c^3 +

192*a^3*b^5*c^4 - 768*a^4*b^3*c^5))/(2*(16*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c

^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^

8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5

*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^

6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3

+ 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4

*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4

*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4

*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c

*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*

a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5 + 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*

d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^

9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3 - 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b

^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2) + (x*(72*a^5*c^3*e^6 - 72*a^2*c^6*d^6 - a^2*b^6*e^6 - b^4*c^4*d^6 + 14*a*b^

2*c^5*d^6 + 16*a^3*b^4*c*e^6 - 74*a^4*b^2*c^2*e^6 - 72*a^3*c^5*d^4*e^2 + 72*a^4*c^4*d^2*e^4 - 102*a^2*b^2*c^4*

d^4*e^2 + 44*a^2*b^3*c^3*d^3*e^3 + 9*a^2*b^4*c^2*d^2*e^4 - 174*a^3*b^2*c^3*d^2*e^4 - 6*a*b^3*c^4*d^5*e + 120*a

^2*b*c^5*d^5*e - 6*a^2*b^5*c*d*e^5 + 24*a^4*b*c^3*d*e^5 + 144*a^3*b*c^4*d^3*e^3 + 42*a^3*b^3*c^2*d*e^5))/(2*(1

6*a^4*c^3 + a^2*b^4*c - 8*a^3*b^2*c^2)))*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6

- 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840*a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^

9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a^4*b^3

*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^

4*e^6 + b^2*c^3*d^6*(-(4*a*c - b^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7*c^4

*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3

- 384*a^4*b^7*c^3*d^2*e^4 - 13824*a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 +

8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*e^2*(-(4*a*c - b^2)^9)^(1/2) + 9*a^3*c^2*

d^2*e^4*(-(4*a*c - b^2)^9)^(1/2) - 6*a*b^10*c^3*d^5*e - 6*a^3*b^10*c*d*e^5 + 108*a^2*b^8*c^4*d^5*e - 576*a^3*b

^6*c^5*d^5*e + 384*a^4*b^4*c^6*d^5*e + 108*a^4*b^8*c^2*d*e^5 + 4608*a^5*b^2*c^7*d^5*e - 576*a^5*b^6*c^3*d*e^5

+ 17664*a^6*b*c^7*d^4*e^2 + 384*a^6*b^4*c^4*d*e^5 + 17664*a^7*b*c^6*d^2*e^4 + 4608*a^7*b^2*c^5*d*e^5 + 6*a*b*c

^3*d^5*e*(-(4*a*c - b^2)^9)^(1/2) - 6*a^3*b*c*d*e^5*(-(4*a*c - b^2)^9)^(1/2))/(32*(4096*a^9*c^9 + a^3*b^12*c^3

- 24*a^4*b^10*c^4 + 240*a^5*b^8*c^5 - 1280*a^6*b^6*c^6 + 3840*a^7*b^4*c^7 - 6144*a^8*b^2*c^8)))^(1/2)*1i)/((5

*a^4*b^4*e^9 + 216*a^6*c^2*e^9 + 5*b^3*c^5*d^9 - 66*a^5*b^2*c*e^9 + a*b^7*d^3*e^6 - 9*a^3*b^5*d*e^8 + 216*a^2*

c^6*d^8*e - 9*b^4*c^4*d^8*e + 3*a^2*b^6*d^2*e^7 + 864*a^3*c^5*d^6*e^3 + 1296*a^4*c^4*d^4*e^5 + 864*a^5*c^3*d^2

*e^7 + 3*b^5*c^3*d^7*e^2 + b^6*c^2*d^6*e^3 - 36*a*b*c^6*d^9 + 624*a^2*b^2*c^4*d^6*e^3 - 6*a^2*b^3*c^3*d^5*e^4

- 108*a^2*b^4*c^2*d^4*e^5 + 1020*a^3*b^2*c^3*d^4*e^5 + 128*a^3*b^3*c^2*d^3*e^6 + 384*a^4*b^2*c^2*d^2*e^7 + 54*

a*b^2*c^5*d^8*e + 6*a*b^6*c*d^4*e^5 + 153*a^4*b^3*c*d*e^8 - 612*a^5*b*c^2*d*e^8 + 24*a*b^3*c^4*d^7*e^2 - 46*a*

b^4*c^3*d^6*e^3 - 3*a*b^5*c^2*d^5*e^4 - 720*a^2*b*c^5*d^7*e^2 - 3*a^2*b^5*c*d^3*e^6 - 1944*a^3*b*c^4*d^5*e^4 -

90*a^3*b^4*c*d^2*e^7 - 1872*a^4*b*c^3*d^3*e^6)/(4*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b^2*c^3))

+ (((6144*a^5*c^7*d^3 + 16*a*b^8*c^3*d^3 - 1024*a^6*b*c^5*e^3 + 6144*a^6*c^6*d*e^2 - 288*a^2*b^6*c^4*d^3 + 19

20*a^3*b^4*c^5*d^3 - 5632*a^4*b^2*c^6*d^3 + 16*a^3*b^7*c^2*e^3 - 192*a^4*b^5*c^3*e^3 + 768*a^5*b^3*c^4*e^3 - 3

072*a^5*b*c^6*d^2*e + 48*a^2*b^7*c^3*d^2*e - 576*a^3*b^5*c^4*d^2*e - 96*a^3*b^6*c^3*d*e^2 + 2304*a^4*b^3*c^5*d

^2*e + 1152*a^4*b^4*c^4*d*e^2 - 4608*a^5*b^2*c^5*d*e^2)/(8*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b

^2*c^3)) - (x*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b*c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b