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

Optimal. Leaf size=272 \[ \frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (\sqrt [3]{e} \left (2 b c d-a^2 e\right )-\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right )}{6 d^{2/3} e^{5/3}}-\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (\sqrt [3]{e} \left (2 b c d-a^2 e\right )-\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right )}{3 d^{2/3} e^{5/3}}+\frac{\left (2 a c+b^2\right ) \log \left (d+e x^3\right )}{3 e}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (-a e \left (a \sqrt [3]{e}+2 b \sqrt [3]{d}\right )+2 b c d \sqrt [3]{e}+c^2 d^{4/3}\right )}{\sqrt{3} d^{2/3} e^{5/3}}+\frac{2 b c x}{e}+\frac{c^2 x^2}{2 e} \]

[Out]

(2*b*c*x)/e + (c^2*x^2)/(2*e) + ((c^2*d^(4/3) + 2*b*c*d*e^(1/3) - a*(2*b*d^(1/3) + a*e^(1/3))*e)*ArcTan[(d^(1/
3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(2/3)*e^(5/3)) - ((e^(1/3)*(2*b*c*d - a^2*e) - d^(1/3)*(c^2*d
- 2*a*b*e))*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(5/3)) + ((e^(1/3)*(2*b*c*d - a^2*e) - d^(1/3)*(c^2*d - 2*
a*b*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(5/3)) + ((b^2 + 2*a*c)*Log[d + e*x^3])/(
3*e)

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Rubi [A]  time = 0.490332, antiderivative size = 270, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a^2 (-e)-\frac{\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}+2 b c d\right )}{6 d^{2/3} e^{4/3}}-\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (\sqrt [3]{e} \left (2 b c d-a^2 e\right )-\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right )}{3 d^{2/3} e^{5/3}}+\frac{\left (2 a c+b^2\right ) \log \left (d+e x^3\right )}{3 e}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (-a e \left (a \sqrt [3]{e}+2 b \sqrt [3]{d}\right )+2 b c d \sqrt [3]{e}+c^2 d^{4/3}\right )}{\sqrt{3} d^{2/3} e^{5/3}}+\frac{2 b c x}{e}+\frac{c^2 x^2}{2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*c*x)/e + (c^2*x^2)/(2*e) + ((c^2*d^(4/3) + 2*b*c*d*e^(1/3) - a*(2*b*d^(1/3) + a*e^(1/3))*e)*ArcTan[(d^(1/
3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(2/3)*e^(5/3)) - ((e^(1/3)*(2*b*c*d - a^2*e) - d^(1/3)*(c^2*d
- 2*a*b*e))*Log[d^(1/3) + e^(1/3)*x])/(3*d^(2/3)*e^(5/3)) + ((2*b*c*d - a^2*e - (d^(1/3)*(c^2*d - 2*a*b*e))/e
^(1/3))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(4/3)) + ((b^2 + 2*a*c)*Log[d + e*x^3])/(
3*e)

Rule 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 31

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

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{d+e x^3} \, dx &=\int \left (\frac{2 b c}{e}+\frac{c^2 x}{e}-\frac{2 b c d-a^2 e+\left (c^2 d-2 a b e\right ) x-\left (b^2+2 a c\right ) e x^2}{e \left (d+e x^3\right )}\right ) \, dx\\ &=\frac{2 b c x}{e}+\frac{c^2 x^2}{2 e}-\frac{\int \frac{2 b c d-a^2 e+\left (c^2 d-2 a b e\right ) x-\left (b^2+2 a c\right ) e x^2}{d+e x^3} \, dx}{e}\\ &=\frac{2 b c x}{e}+\frac{c^2 x^2}{2 e}-\left (-b^2-2 a c\right ) \int \frac{x^2}{d+e x^3} \, dx-\frac{\int \frac{2 b c d-a^2 e+\left (c^2 d-2 a b e\right ) x}{d+e x^3} \, dx}{e}\\ &=\frac{2 b c x}{e}+\frac{c^2 x^2}{2 e}+\frac{\left (b^2+2 a c\right ) \log \left (d+e x^3\right )}{3 e}-\frac{\int \frac{\sqrt [3]{d} \left (2 \sqrt [3]{e} \left (2 b c d-a^2 e\right )+\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right )+\sqrt [3]{e} \left (-\sqrt [3]{e} \left (2 b c d-a^2 e\right )+\sqrt [3]{d} \left (c^2 d-2 a b e\right )\right ) x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{3 d^{2/3} e^{4/3}}-\frac{\left (2 b c d-a^2 e-\frac{\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \int \frac{1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{3 d^{2/3} e}\\ &=\frac{2 b c x}{e}+\frac{c^2 x^2}{2 e}-\frac{\left (2 b c d-a^2 e-\frac{\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{4/3}}+\frac{\left (b^2+2 a c\right ) \log \left (d+e x^3\right )}{3 e}-\frac{\left (c^2 d^{4/3}+2 b c d \sqrt [3]{e}-a \left (2 b \sqrt [3]{d}+a \sqrt [3]{e}\right ) e\right ) \int \frac{1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 \sqrt [3]{d} e^{4/3}}+\frac{\left (2 b c d-a^2 e-\frac{\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \int \frac{-\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{6 d^{2/3} e^{4/3}}\\ &=\frac{2 b c x}{e}+\frac{c^2 x^2}{2 e}-\frac{\left (2 b c d-a^2 e-\frac{\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{4/3}}+\frac{\left (2 b c d-a^2 e-\frac{\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{4/3}}+\frac{\left (b^2+2 a c\right ) \log \left (d+e x^3\right )}{3 e}-\frac{\left (c^2 d^{4/3}+2 b c d \sqrt [3]{e}-a \left (2 b \sqrt [3]{d}+a \sqrt [3]{e}\right ) e\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{d^{2/3} e^{5/3}}\\ &=\frac{2 b c x}{e}+\frac{c^2 x^2}{2 e}+\frac{\left (c^2 d^{4/3}+2 b c d \sqrt [3]{e}-a \left (2 b \sqrt [3]{d}+a \sqrt [3]{e}\right ) e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} e^{5/3}}-\frac{\left (2 b c d-a^2 e-\frac{\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{4/3}}+\frac{\left (2 b c d-a^2 e-\frac{\sqrt [3]{d} \left (c^2 d-2 a b e\right )}{\sqrt [3]{e}}\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{4/3}}+\frac{\left (b^2+2 a c\right ) \log \left (d+e x^3\right )}{3 e}\\ \end{align*}

Mathematica [A]  time = 0.284016, size = 269, normalized size = 0.99 \[ \frac{2 e^{2/3} \left (2 a c+b^2\right ) \log \left (d+e x^3\right )-\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a e \left (a \sqrt [3]{e}-2 b \sqrt [3]{d}\right )-2 b c d \sqrt [3]{e}+c^2 d^{4/3}\right )}{d^{2/3}}+\frac{2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (a e \left (a \sqrt [3]{e}-2 b \sqrt [3]{d}\right )-2 b c d \sqrt [3]{e}+c^2 d^{4/3}\right )}{d^{2/3}}+\frac{2 \sqrt{3} \left (c d^{2/3}-a e^{2/3}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right ) \left (a e^{2/3}+2 b \sqrt [3]{d} \sqrt [3]{e}+c d^{2/3}\right )}{d^{2/3}}+12 b c e^{2/3} x+3 c^2 e^{2/3} x^2}{6 e^{5/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*b*c*e^(2/3)*x + 3*c^2*e^(2/3)*x^2 + (2*Sqrt[3]*(c*d^(2/3) - a*e^(2/3))*(c*d^(2/3) + 2*b*d^(1/3)*e^(1/3) +
a*e^(2/3))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(2/3) + (2*(c^2*d^(4/3) - 2*b*c*d*e^(1/3) + a*(-2*b*
d^(1/3) + a*e^(1/3))*e)*Log[d^(1/3) + e^(1/3)*x])/d^(2/3) - ((c^2*d^(4/3) - 2*b*c*d*e^(1/3) + a*(-2*b*d^(1/3)
+ a*e^(1/3))*e)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(2/3) + 2*(b^2 + 2*a*c)*e^(2/3)*Log[d + e*x^
3])/(6*e^(5/3))

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Maple [B]  time = 0.004, size = 444, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 15.671, size = 26617, normalized size = 97.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*c^2*x^2 + 24*b*c*x - 2*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^3*d^2 + b^4*d*e +
3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a
^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c
*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 -
a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) + (1/2)^(1/3)*(
I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/
(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^
3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3
- 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)*e*log(-4*b*c^5*d^4
- (5*b^4*c^2 - 4*a*b^2*c^3 + 2*a^2*c^4)*d^3*e + 2*(a*b^5 - 2*a^2*b^3*c + 4*a^3*b*c^2)*d^2*e^2 + (7*a^4*b^2 -
2*a^5*c)*d*e^3 - 1/4*(c^2*d^3*e^3 - 2*a*b*d^2*e^4)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b
*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e +
3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2
*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d
^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(
1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b
*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e
^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6
*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e
)^2 - 1/2*(a^4*d*e^4 + 2*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 4*(a*b^3 + 3*a^2*b*c)*d^2*e^3)*(2*(1/2)^(2/3)*(-I*sqr
t(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*
c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^
3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 +
6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3
*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^
2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 +
9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d
^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)
/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 -
3*a^4*b*c)*d*e^3)*x) + (6*b^2 + 12*a*c + (2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^3*d^2
+ b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^
2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2
+ 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) -
(c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) + (1
/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^
2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*
e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*
(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)*e + 3*s
qrt(1/3)*e*sqrt(-(32*b*c^3*d^2 + 32*a^3*b*e^2 + (2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^
3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*
a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^
2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*
e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3
) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^
2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3
+ a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^
4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)^2
*d*e^3 + 4*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a
^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a
*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2
*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*
c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b
^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 -
2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^
3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e -
2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)*(b^2 + 2*a*c)*d*e^2 + 4*(b^4 - 12*a*b^
2*c)*d*e)/(d*e^3)))*log(4*b*c^5*d^4 + (5*b^4*c^2 - 4*a*b^2*c^3 + 2*a^2*c^4)*d^3*e - 2*(a*b^5 - 2*a^2*b^3*c + 4
*a^3*b*c^2)*d^2*e^2 - (7*a^4*b^2 - 2*a^5*c)*d*e^3 + 1/4*(c^2*d^3*e^3 - 2*a*b*d^2*e^4)*(2*(1/2)^(2/3)*(-I*sqrt(
3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)
^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*
d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*
(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b
^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2
+ b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*
a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2
*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(
d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)^2 + 1/2*(a^4*d*e^4 + 2*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 4*(a*b^3 + 3*a^2*b
*c)*d^2*e^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2
*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2
*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 +
2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^
3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*
(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4
- 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*
e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e
- 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e) - 2*(c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^
3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)*x + 3/4*sqrt(1/3)*(4*a*b^3*d^2*e^3 + 2*a^4*d*e^4 + 2*(
3*b^2*c^2 - 2*a*c^3)*d^3*e^2 - (c^2*d^3*e^3 - 2*a*b*d^2*e^4)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/
e^2 - (2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 +
b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a
^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*
e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d
^2*e^5))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e
+ 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a
^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6
*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2
+ 2*a*c)/e))*sqrt(-(32*b*c^3*d^2 + 32*a^3*b*e^2 + (2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*
c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e +
3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*
d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^
2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1
/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*
e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^
3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*
e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)
^2*d*e^3 + 4*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2
*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2
*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 +
2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^
3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*
(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4
- 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*
e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e
- 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)*(b^2 + 2*a*c)*d*e^2 + 4*(b^4 - 12*a*
b^2*c)*d*e)/(d*e^3))) + (6*b^2 + 12*a*c + (2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^3*d^2
+ b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^
2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2
+ 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) -
(c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) + (1
/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^
2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*
e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*
(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)*e - 3*s
qrt(1/3)*e*sqrt(-(32*b*c^3*d^2 + 32*a^3*b*e^2 + (2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^
3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*
a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^
2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*
e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3
) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^
2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3
+ a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^
4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)^2
*d*e^3 + 4*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a
^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a
*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2
*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*
c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b
^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 -
2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^
3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e -
2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)*(b^2 + 2*a*c)*d*e^2 + 4*(b^4 - 12*a*b^
2*c)*d*e)/(d*e^3)))*log(4*b*c^5*d^4 + (5*b^4*c^2 - 4*a*b^2*c^3 + 2*a^2*c^4)*d^3*e - 2*(a*b^5 - 2*a^2*b^3*c + 4
*a^3*b*c^2)*d^2*e^2 - (7*a^4*b^2 - 2*a^5*c)*d*e^3 + 1/4*(c^2*d^3*e^3 - 2*a*b*d^2*e^4)*(2*(1/2)^(2/3)*(-I*sqrt(
3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)
^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*
d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*
(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b
^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2
+ b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*
a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2
*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(
d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)^2 + 1/2*(a^4*d*e^4 + 2*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 4*(a*b^3 + 3*a^2*b
*c)*d^2*e^3)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2
*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2
*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 +
2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^
3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*
(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4
- 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*
e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e
- 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e) - 2*(c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^
3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)*x - 3/4*sqrt(1/3)*(4*a*b^3*d^2*e^3 + 2*a^4*d*e^4 + 2*(
3*b^2*c^2 - 2*a*c^3)*d^3*e^2 - (c^2*d^3*e^3 - 2*a*b*d^2*e^4)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/
e^2 - (2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 +
b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a
^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*
e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d
^2*e^5))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e
+ 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a
^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6
*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2
+ 2*a*c)/e))*sqrt(-(32*b*c^3*d^2 + 32*a^3*b*e^2 + (2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*
c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e +
3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*
d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^
2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1
/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*
e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^
3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*
e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)
^2*d*e^3 + 4*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((b^2 + 2*a*c)^2/e^2 - (2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2
*a^3*b*e^2)/(d*e^3))/(2*(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2
*a*c)/(d*e^4) + (c^6*d^4 - 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 +
2*(c^3*d^2*e^2 - b^3*d*e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^
3*c^3 - 3*a*b*c^4)*d^3*e - 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*
(b^2 + 2*a*c)^3/e^3 - 3*(2*b*c^3*d^2 + b^4*d*e + 3*a^2*c^2*d*e + 2*a^3*b*e^2)*(b^2 + 2*a*c)/(d*e^4) + (c^6*d^4
- 2*b^3*c^3*d^3*e + b^6*d^2*e^2 + 9*a^2*b^2*c^2*d^2*e^2 + 6*a^4*b*c*d*e^3 + a^6*e^4 + 2*(c^3*d^2*e^2 - b^3*d*
e^3)*a^3 + 6*(b*c^4*d^3*e - b^4*c*d^2*e^2)*a)/(d^2*e^5) - (c^6*d^4 - a^6*e^4 + 2*(4*b^3*c^3 - 3*a*b*c^4)*d^3*e
- 2*(4*a^3*b^3 - 3*a^4*b*c)*d*e^3)/(d^2*e^5))^(1/3) - 2*(b^2 + 2*a*c)/e)*(b^2 + 2*a*c)*d*e^2 + 4*(b^4 - 12*a*
b^2*c)*d*e)/(d*e^3))))/e

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Sympy [B]  time = 8.7991, size = 546, normalized size = 2.01 \begin{align*} \frac{2 b c x}{e} + \frac{c^{2} x^{2}}{2 e} + \operatorname{RootSum}{\left (27 t^{3} d^{2} e^{5} + t^{2} \left (- 54 a c d^{2} e^{4} - 27 b^{2} d^{2} e^{4}\right ) + t \left (18 a^{3} b d e^{4} + 27 a^{2} c^{2} d^{2} e^{3} + 9 b^{4} d^{2} e^{3} + 18 b c^{3} d^{3} e^{2}\right ) - a^{6} e^{4} - 6 a^{4} b c d e^{3} + 2 a^{3} b^{3} d e^{3} - 2 a^{3} c^{3} d^{2} e^{2} - 9 a^{2} b^{2} c^{2} d^{2} e^{2} + 6 a b^{4} c d^{2} e^{2} - 6 a b c^{4} d^{3} e - b^{6} d^{2} e^{2} + 2 b^{3} c^{3} d^{3} e - c^{6} d^{4}, \left ( t \mapsto t \log{\left (x + \frac{18 t^{2} a b d^{2} e^{4} - 9 t^{2} c^{2} d^{3} e^{3} + 3 t a^{4} d e^{4} - 36 t a^{2} b c d^{2} e^{3} - 12 t a b^{3} d^{2} e^{3} + 12 t a c^{3} d^{3} e^{2} + 18 t b^{2} c^{2} d^{3} e^{2} - 2 a^{5} c d e^{3} + 7 a^{4} b^{2} d e^{3} + 8 a^{3} b c^{2} d^{2} e^{2} - 4 a^{2} b^{3} c d^{2} e^{2} - 2 a^{2} c^{4} d^{3} e + 2 a b^{5} d^{2} e^{2} + 4 a b^{2} c^{3} d^{3} e - 5 b^{4} c^{2} d^{3} e - 4 b c^{5} d^{4}}{a^{6} e^{4} - 6 a^{4} b c d e^{3} + 8 a^{3} b^{3} d e^{3} + 6 a b c^{4} d^{3} e - 8 b^{3} c^{3} d^{3} e - c^{6} d^{4}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(e*x**3+d),x)

[Out]

2*b*c*x/e + c**2*x**2/(2*e) + RootSum(27*_t**3*d**2*e**5 + _t**2*(-54*a*c*d**2*e**4 - 27*b**2*d**2*e**4) + _t*
(18*a**3*b*d*e**4 + 27*a**2*c**2*d**2*e**3 + 9*b**4*d**2*e**3 + 18*b*c**3*d**3*e**2) - a**6*e**4 - 6*a**4*b*c*
d*e**3 + 2*a**3*b**3*d*e**3 - 2*a**3*c**3*d**2*e**2 - 9*a**2*b**2*c**2*d**2*e**2 + 6*a*b**4*c*d**2*e**2 - 6*a*
b*c**4*d**3*e - b**6*d**2*e**2 + 2*b**3*c**3*d**3*e - c**6*d**4, Lambda(_t, _t*log(x + (18*_t**2*a*b*d**2*e**4
- 9*_t**2*c**2*d**3*e**3 + 3*_t*a**4*d*e**4 - 36*_t*a**2*b*c*d**2*e**3 - 12*_t*a*b**3*d**2*e**3 + 12*_t*a*c**
3*d**3*e**2 + 18*_t*b**2*c**2*d**3*e**2 - 2*a**5*c*d*e**3 + 7*a**4*b**2*d*e**3 + 8*a**3*b*c**2*d**2*e**2 - 4*a
**2*b**3*c*d**2*e**2 - 2*a**2*c**4*d**3*e + 2*a*b**5*d**2*e**2 + 4*a*b**2*c**3*d**3*e - 5*b**4*c**2*d**3*e - 4
*b*c**5*d**4)/(a**6*e**4 - 6*a**4*b*c*d*e**3 + 8*a**3*b**3*d*e**3 + 6*a*b*c**4*d**3*e - 8*b**3*c**3*d**3*e - c
**6*d**4))))

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Giac [A]  time = 1.07009, size = 383, normalized size = 1.41 \begin{align*} \frac{1}{3} \,{\left (b^{2} + 2 \, a c\right )} e^{\left (-1\right )} \log \left ({\left | x^{3} e + d \right |}\right ) - \frac{\sqrt{3}{\left (2 \, \left (-d e^{2}\right )^{\frac{1}{3}} b c d e - \left (-d e^{2}\right )^{\frac{2}{3}} c^{2} d + 2 \, \left (-d e^{2}\right )^{\frac{2}{3}} a b e - \left (-d e^{2}\right )^{\frac{1}{3}} a^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}}\right ) e^{\left (-3\right )}}{3 \, d} + \frac{{\left (\left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} c^{2} d e^{4} + 2 \, b c d e^{4} - 2 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} a b e^{5} - a^{2} e^{5}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} e^{\left (-5\right )} \log \left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} \right |}\right )}{3 \, d} + \frac{1}{2} \,{\left (c^{2} x^{2} e + 4 \, b c x e\right )} e^{\left (-2\right )} - \frac{{\left (2 \, \left (-d e^{2}\right )^{\frac{1}{3}} b c d e + \left (-d e^{2}\right )^{\frac{2}{3}} c^{2} d - 2 \, \left (-d e^{2}\right )^{\frac{2}{3}} a b e - \left (-d e^{2}\right )^{\frac{1}{3}} a^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac{2}{3}}\right )}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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