∫ −a−bc+(1+c)x______________ 3∘___________ (−a+x)(−b+x)2 (−a2+b2d+2(a−bd)x+(−1+d)x2) dx

3.32.48 $\int \frac {-a-b c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2)} \, dx$

Optimal. Leaf size=1655 \[ \frac {(b-x)^{2/3} \sqrt [3]{x-a} \left (-\frac {\sqrt {3} b c \left (\sqrt {d}-1\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} a \left (\sqrt {d}-1\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} c \left (a-b \sqrt {d}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} \left (a-b \sqrt {d}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\sqrt {3} b c \left (\sqrt {d}+1\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\sqrt {3} a \left (\sqrt {d}+1\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} c \left (a+b \sqrt {d}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} \left (a+b \sqrt {d}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {b c \left (\sqrt {d}+1\right ) \log \left (\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{b-x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {a \left (\sqrt {d}+1\right ) \log \left (\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{b-x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-a-b \sqrt {d}\right ) \log \left (\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{b-x}\right )}{2 (a-b)^2 d^{2/3}}-\frac {c \left (a+b \sqrt {d}\right ) \log \left (\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{b-x}\right )}{2 (a-b)^2 d^{2/3}}-\frac {b c \left (\sqrt {d}-1\right ) \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{2/3}}-\frac {a \left (\sqrt {d}-1\right ) \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{2/3}}-\frac {c \left (a-b \sqrt {d}\right ) \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (b \sqrt {d}-a\right ) \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{2/3}}+\frac {b c \left (\sqrt {d}-1\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {a \left (\sqrt {d}-1\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {c \left (a-b \sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (a-b \sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}-\frac {b c \left (\sqrt {d}+1\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}-\frac {a \left (\sqrt {d}+1\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {c \left (a+b \sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (a+b \sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}\right )}{\sqrt [3]{(b-x)^2 (x-a)}} \]

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Rubi [A] time = 1.96, antiderivative size = 561, normalized size of antiderivative = 0.34, number of steps used = 5, number of rules used = 3, integrand size = 60, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.050, Rules used = {6719, 6728, 91} \begin {gather*} \frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (c+\sqrt {d}\right ) \log \left (2 \left (\sqrt {d}+1\right ) \left (a-b \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (c-\sqrt {d}\right ) \log \left (2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} \left (c-\sqrt {d}\right ) \log \left (-\frac {\sqrt [3]{x-a}}{\sqrt [6]{d}}-\sqrt [3]{x-b}\right )}{4 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} \left (c+\sqrt {d}\right ) \log \left (\frac {\sqrt [3]{x-a}}{\sqrt [6]{d}}-\sqrt [3]{x-b}\right )}{4 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} \left (c-\sqrt {d}\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-b}}\right )}{2 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} \left (c+\sqrt {d}\right ) \tan ^{-1}\left (\frac {2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{2 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(-a - b*c + (1 + c)*x)/(((-a + x)*(-b + x)^2)^(1/3)*(-a^2 + b^2*d + 2*(a - b*d)*x + (-1 + d)*x^2)),x]

[Out]

-1/2*(Sqrt[3]*(c - Sqrt[d])*(-a + x)^(1/3)*(-b + x)^(2/3)*ArcTan[1/Sqrt[3] - (2*(-a + x)^(1/3))/(Sqrt[3]*d^(1/

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

b + x)^(2/3)*ArcTan[1/Sqrt[3] + (2*(-a + x)^(1/3))/(Sqrt[3]*d^(1/6)*(-b + x)^(1/3))])/(2*(a - b)*d^(2/3)*(-((a

- x)*(b - x)^2))^(1/3)) + ((c + Sqrt[d])*(-a + x)^(1/3)*(-b + x)^(2/3)*Log[2*(1 + Sqrt[d])*(a - b*Sqrt[d]) -

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

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

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

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

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

Rule 91

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[

(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/

3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*

x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F

racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !

FreeQ[v, x] && !FreeQ[w, x]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +

b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [C] time = 0.36, size = 109, normalized size = 0.07 \begin {gather*} \frac {3 (x-b) \left (\left (\sqrt {d}-c\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\sqrt {d} (b-x)}{x-a}\right )+\left (c+\sqrt {d}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\sqrt {d} (x-b)}{x-a}\right )\right )}{2 \sqrt {d} (a-b) \sqrt [3]{(x-a) (b-x)^2}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-a - b*c + (1 + c)*x)/(((-a + x)*(-b + x)^2)^(1/3)*(-a^2 + b^2*d + 2*(a - b*d)*x + (-1 + d)*x^2)),x

]

[Out]

(3*(-b + x)*((-c + Sqrt[d])*Hypergeometric2F1[1/3, 1, 4/3, (Sqrt[d]*(b - x))/(-a + x)] + (c + Sqrt[d])*Hyperge

ometric2F1[1/3, 1, 4/3, (Sqrt[d]*(-b + x))/(-a + x)]))/(2*(a - b)*Sqrt[d]*((b - x)^2*(-a + x))^(1/3))

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IntegrateAlgebraic [F] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[(-a - b*c + (1 + c)*x)/(((-a + x)*(-b + x)^2)^(1/3)*(-a^2 + b^2*d + 2*(a - b*d)*x + (-1 + d

)*x^2)),x]

[Out]

$Aborted

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fricas [B] time = 1.46, size = 11598, normalized size = 7.01

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-a^2+b^2*d+2*(-b*d+a)*x+(-1+d)*x^2),x, algorithm="fricas")

[Out]

-sqrt(3)*(-((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20

*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) + c^3 + 3*c*d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3)*arcta

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

b^4 + b^5)*c*d^2)*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^

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

8*(a^2*b - 2*a*b^2 + b^3)*c^9*d^2 - 24*(a^2*b - 2*a*b^2 + b^3)*c^7*d^3 - 4*(a^2*b - 2*a*b^2 + b^3)*c^5*d^4 + 8

*(a^2*b - 2*a*b^2 + b^3)*c^3*d^5 + 2*(a^2*b - 2*a*b^2 + b^3)*c*d^6 - 2*(9*(a^2 - 2*a*b + b^2)*c^9*d^2 - 12*(a^

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

*c*d^6)*x - (3*(a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*c^8*d^3 - 2*(a^5*b - 5*a^4*b^2 +

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

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

10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*d^7 - (3*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^

8*d^3 - 2*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^6*d^4 - 4*(a^5 - 5*a^4*b + 10*a^3*b^2 -

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

^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d^7)*x)*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b +

15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)))*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)

^(1/3)*(-((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a

^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) + c^3 + 3*c*d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(2/3) + (9*c^

12 - 30*c^10*d + 31*c^8*d^2 - 4*c^6*d^3 - 9*c^4*d^4 + 2*c^2*d^5 + d^6)*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b

+ b^2)*x)^(2/3) - (9*(a*b^2 - b^3)*c^11*d - 21*(a*b^2 - b^3)*c^9*d^2 + 10*(a*b^2 - b^3)*c^7*d^3 + 6*(a*b^2 - b

^3)*c^5*d^4 - 3*(a*b^2 - b^3)*c^3*d^5 - (a*b^2 - b^3)*c*d^6 + (9*(a - b)*c^11*d - 21*(a - b)*c^9*d^2 + 10*(a -

b)*c^7*d^3 + 6*(a - b)*c^5*d^4 - 3*(a - b)*c^3*d^5 - (a - b)*c*d^6)*x^2 - 2*(9*(a*b - b^2)*c^11*d - 21*(a*b -

b^2)*c^9*d^2 + 10*(a*b - b^2)*c^7*d^3 + 6*(a*b - b^2)*c^5*d^4 - 3*(a*b - b^2)*c^3*d^5 - (a*b - b^2)*c*d^6)*x

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

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

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

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

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

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

4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d^7)*x)*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20

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

^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) + c^3 + 3*c*d)/((a^3 - 3*a^2

*b + 3*a*b^2 - b^3)*d^2))^(1/3))/(b^2 - 2*b*x + x^2))*(-((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c

^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) + c^3 + 3*c*d)/((a^3

- 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3) - 2*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(sqrt(3)*(3

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

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

((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) - sqrt(

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

^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4

- 6*a*b^5 + b^6)*d^3)) + c^3 + 3*c*d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3) - sqrt(3)*(9*b*c^10 - 21*b*

c^8*d + 10*b*c^6*d^2 + 6*b*c^4*d^3 - 3*b*c^2*d^4 - b*d^5 - (9*c^10 - 21*c^8*d + 10*c^6*d^2 + 6*c^4*d^3 - 3*c^2

*d^4 - d^5)*x))/(9*b*c^10 - 21*b*c^8*d + 10*b*c^6*d^2 + 6*b*c^4*d^3 - 3*b*c^2*d^4 - b*d^5 - (9*c^10 - 21*c^8*d

+ 10*c^6*d^2 + 6*c^4*d^3 - 3*c^2*d^4 - d^5)*x)) + sqrt(3)*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 +

6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) - c^3 - 3*c*d)/(

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

)*c*d^2*x - (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*c*d^2)*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*

b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) - sqrt(3)*(3*(a*b - b^2)*c^2*d + (a*b - b^2)*d

^2 - (3*(a - b)*c^2*d + (a - b)*d^2)*x))*sqrt(((18*(a^2*b - 2*a*b^2 + b^3)*c^9*d^2 - 24*(a^2*b - 2*a*b^2 + b^3

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

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

(a^2 - 2*a*b + b^2)*c^3*d^5 + (a^2 - 2*a*b + b^2)*c*d^6)*x + (3*(a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 +

5*a*b^5 - b^6)*c^8*d^3 - 2*(a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*c^6*d^4 - 4*(a^5*b -

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

+ 5*a*b^5 - b^6)*c^2*d^6 + (a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*d^7 - (3*(a^5 - 5*a^

4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^8*d^3 - 2*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4

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

^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^2*d^6 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d^7)*

x)*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)))

*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*

c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) - c^3 - 3*c*d)/((a^

3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(2/3) + (9*c^12 - 30*c^10*d + 31*c^8*d^2 - 4*c^6*d^3 - 9*c^4*d^4 + 2*c^2*d^

5 + d^6)*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3) - (9*(a*b^2 - b^3)*c^11*d - 21*(a*b^2 - b^3)*c

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

(9*(a - b)*c^11*d - 21*(a - b)*c^9*d^2 + 10*(a - b)*c^7*d^3 + 6*(a - b)*c^5*d^4 - 3*(a - b)*c^3*d^5 - (a - b)*

c*d^6)*x^2 - 2*(9*(a*b - b^2)*c^11*d - 21*(a*b - b^2)*c^9*d^2 + 10*(a*b - b^2)*c^7*d^3 + 6*(a*b - b^2)*c^5*d^4

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

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

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

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

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

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

+ 6*a^2*b^3 - 4*a*b^4 + b^5)*c^4*d^5 - (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d^7)*x)*sqrt((9*c^4 +

6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)))*(((a^3 - 3*a^2*b

+ 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*

a*b^5 + b^6)*d^3)) - c^3 - 3*c*d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3))/(b^2 - 2*b*x + x^2))*(((a^3 -

3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b

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

x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(sqrt(3)*(3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c^7*d^2 - 5*(a^4 -

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

a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*c*d^5)*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b

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

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

a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) - c^3 - 3*c*d)/((a^3 - 3*a^2*b + 3*a*b^2 -

b^3)*d^2))^(1/3) + sqrt(3)*(9*b*c^10 - 21*b*c^8*d + 10*b*c^6*d^2 + 6*b*c^4*d^3 - 3*b*c^2*d^4 - b*d^5 - (9*c^1

0 - 21*c^8*d + 10*c^6*d^2 + 6*c^4*d^3 - 3*c^2*d^4 - d^5)*x))/(9*b*c^10 - 21*b*c^8*d + 10*b*c^6*d^2 + 6*b*c^4*d

^3 - 3*b*c^2*d^4 - b*d^5 - (9*c^10 - 21*c^8*d + 10*c^6*d^2 + 6*c^4*d^3 - 3*c^2*d^4 - d^5)*x)) - 1/4*(-((a^3 -

3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b

^4 - 6*a*b^5 + b^6)*d^3)) + c^3 + 3*c*d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3)*log(((18*(a^2*b - 2*a*b^

2 + b^3)*c^9*d^2 - 24*(a^2*b - 2*a*b^2 + b^3)*c^7*d^3 - 4*(a^2*b - 2*a*b^2 + b^3)*c^5*d^4 + 8*(a^2*b - 2*a*b^2

+ b^3)*c^3*d^5 + 2*(a^2*b - 2*a*b^2 + b^3)*c*d^6 - 2*(9*(a^2 - 2*a*b + b^2)*c^9*d^2 - 12*(a^2 - 2*a*b + b^2)*

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

^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*c^8*d^3 - 2*(a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a

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

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

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

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

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

*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d^7)*x)*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*

a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)))*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(-((a^3 -

3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b

^4 - 6*a*b^5 + b^6)*d^3)) + c^3 + 3*c*d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(2/3) + (9*c^12 - 30*c^10*d +

31*c^8*d^2 - 4*c^6*d^3 - 9*c^4*d^4 + 2*c^2*d^5 + d^6)*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3) -

(9*(a*b^2 - b^3)*c^11*d - 21*(a*b^2 - b^3)*c^9*d^2 + 10*(a*b^2 - b^3)*c^7*d^3 + 6*(a*b^2 - b^3)*c^5*d^4 - 3*(

a*b^2 - b^3)*c^3*d^5 - (a*b^2 - b^3)*c*d^6 + (9*(a - b)*c^11*d - 21*(a - b)*c^9*d^2 + 10*(a - b)*c^7*d^3 + 6*(

a - b)*c^5*d^4 - 3*(a - b)*c^3*d^5 - (a - b)*c*d^6)*x^2 - 2*(9*(a*b - b^2)*c^11*d - 21*(a*b - b^2)*c^9*d^2 + 1

0*(a*b - b^2)*c^7*d^3 + 6*(a*b - b^2)*c^5*d^4 - 3*(a*b - b^2)*c^3*d^5 - (a*b - b^2)*c*d^6)*x - (3*(a^4*b^2 - 4

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

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

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

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

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

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

2*b^3 - 4*a*b^4 + b^5)*d^7)*x)*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2

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

*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) + c^3 + 3*c*d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^

3)*d^2))^(1/3))/(b^2 - 2*b*x + x^2)) - 1/4*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/

((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) - c^3 - 3*c*d)/((a^3 - 3*a^2*b +

3*a*b^2 - b^3)*d^2))^(1/3)*log(((18*(a^2*b - 2*a*b^2 + b^3)*c^9*d^2 - 24*(a^2*b - 2*a*b^2 + b^3)*c^7*d^3 - 4*

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

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

b^2)*c^3*d^5 + (a^2 - 2*a*b + b^2)*c*d^6)*x + (3*(a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6

)*c^8*d^3 - 2*(a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*c^6*d^4 - 4*(a^5*b - 5*a^4*b^2 + 1

0*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*c^4*d^5 + 2*(a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b

^6)*c^2*d^6 + (a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*d^7 - (3*(a^5 - 5*a^4*b + 10*a^3*b

^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^8*d^3 - 2*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^6*d^4

- 4*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^4*d^5 + 2*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^

2*b^3 + 5*a*b^4 - b^5)*c^2*d^6 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d^7)*x)*sqrt((9*c^4

+ 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)))*(-a*b^2 - (a

+ 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/(

(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) - c^3 - 3*c*d)/((a^3 - 3*a^2*b +

3*a*b^2 - b^3)*d^2))^(2/3) + (9*c^12 - 30*c^10*d + 31*c^8*d^2 - 4*c^6*d^3 - 9*c^4*d^4 + 2*c^2*d^5 + d^6)*(-a*b

^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3) - (9*(a*b^2 - b^3)*c^11*d - 21*(a*b^2 - b^3)*c^9*d^2 + 10*(a

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

1*d - 21*(a - b)*c^9*d^2 + 10*(a - b)*c^7*d^3 + 6*(a - b)*c^5*d^4 - 3*(a - b)*c^3*d^5 - (a - b)*c*d^6)*x^2 - 2

*(9*(a*b - b^2)*c^11*d - 21*(a*b - b^2)*c^9*d^2 + 10*(a*b - b^2)*c^7*d^3 + 6*(a*b - b^2)*c^5*d^4 - 3*(a*b - b^

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

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

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

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

4*d^5 - (a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d^7)*x^2 - 2*(3*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 +

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

4*a*b^4 + b^5)*c^4*d^5 - (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d^7)*x)*sqrt((9*c^4 + 6*c^2*d + d^2)

/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)))*(((a^3 - 3*a^2*b + 3*a*b^2 - b

^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d

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

+ 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*

a*b^5 + b^6)*d^3)) + c^3 + 3*c*d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3)*log(((6*(a^2*b - 2*a*b^2 + b^3)

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

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

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

a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d^4)*x)*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4

*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)))*(-((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*

c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) + c^3 + 3*c*d)/((a^

3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(2/3) - (3*c^6 - 5*c^4*d + c^2*d^2 + d^3)*(-a*b^2 - (a + 2*b)*x^2 + x^3 + (

2*a*b + b^2)*x)^(1/3))/(b - x)) + 1/2*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6

- 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)) - c^3 - 3*c*d)/((a^3 - 3*a^2*b + 3*a*

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

2*a*b + b^2)*c^3*d^2 + (a^2 - 2*a*b + b^2)*c*d^3)*x + ((a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5

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

*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^2*d^3 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*d^4)*x)

*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*d^3)))*(

((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*sqrt((9*c^4 + 6*c^2*d + d^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 +

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

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

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {b c - {\left (c + 1\right )} x + a}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-a^2+b^2*d+2*(-b*d+a)*x+(-1+d)*x^2),x, algorithm="giac")

[Out]

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

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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {-a -b c +\left (1+c \right ) x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (-a^{2}+b^{2} d +2 \left (-b d +a \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-a^2+b^2*d+2*(-b*d+a)*x+(-1+d)*x^2),x)

[Out]

int((-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-a^2+b^2*d+2*(-b*d+a)*x+(-1+d)*x^2),x)

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {b c - {\left (c + 1\right )} x + a}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a-b*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-a^2+b^2*d+2*(-b*d+a)*x+(-1+d)*x^2),x, algorithm="maxima")

[Out]

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

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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {a+b\,c-x\,\left (c+1\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (b^2\,d+2\,x\,\left (a-b\,d\right )-a^2+x^2\,\left (d-1\right )\right )} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(a + b*c - x*(c + 1))/((-(a - x)*(b - x)^2)^(1/3)*(b^2*d + 2*x*(a - b*d) - a^2 + x^2*(d - 1))),x)

[Out]

-int((a + b*c - x*(c + 1))/((-(a - x)*(b - x)^2)^(1/3)*(b^2*d + 2*x*(a - b*d) - a^2 + x^2*(d - 1))), x)

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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a-b*c+(1+c)*x)/((-a+x)*(-b+x)**2)**(1/3)/(-a**2+b**2*d+2*(-b*d+a)*x+(-1+d)*x**2),x)

[Out]

Timed out

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3.32.48 \(\int \frac {-a-b c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2)} \, dx\)