∫ −b−ac+(1+c)x______________ 3∘___________ (−a+x)(−b+x)2 (−b2+a2d+2(b−ad)x+(−1+d)x2) dx

3.32.49 $\int \frac {-b-a c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2)} \, dx$

Optimal. Leaf size=1707 \[ \frac {(b-x)^{2/3} \sqrt [3]{x-a} \left (-\frac {\sqrt {3} b \left (\sqrt {d}-1\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt [6]{d} \sqrt [3]{x-a}-2 \sqrt [3]{b-x}}\right )}{2 (a-b)^2 d^{5/6}}-\frac {\sqrt {3} a c \left (\sqrt {d}-1\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt [6]{d} \sqrt [3]{x-a}-2 \sqrt [3]{b-x}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} c \left (a \sqrt {d}-b\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt [6]{d} \sqrt [3]{x-a}-2 \sqrt [3]{b-x}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} \left (a \sqrt {d}-b\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt [6]{d} \sqrt [3]{x-a}-2 \sqrt [3]{b-x}}\right )}{2 (a-b)^2 d^{5/6}}-\frac {\sqrt {3} b \left (\sqrt {d}+1\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{x-a}}\right )}{2 (a-b)^2 d^{5/6}}-\frac {\sqrt {3} a c \left (\sqrt {d}+1\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{x-a}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} c \left (\sqrt {d} a+b\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{x-a}}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\sqrt {3} \left (\sqrt {d} a+b\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{x-a}}\right )}{2 (a-b)^2 d^{5/6}}-\frac {b \left (\sqrt {d}+1\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{5/6}}-\frac {a c \left (\sqrt {d}+1\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{5/6}}+\frac {c \left (\sqrt {d} a+b\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\left (\sqrt {d} a+b\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{5/6}}-\frac {b \left (\sqrt {d}-1\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{5/6}}-\frac {a c \left (\sqrt {d}-1\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{5/6}}+\frac {c \left (a \sqrt {d}-b\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{5/6}}+\frac {\left (a \sqrt {d}-b\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{x-a}\right )}{2 (a-b)^2 d^{5/6}}+\frac {b \left (\sqrt {d}-1\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+\sqrt [3]{d} (x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {a c \left (\sqrt {d}-1\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+\sqrt [3]{d} (x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {\left (b-a \sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+\sqrt [3]{d} (x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}-\frac {c \left (a \sqrt {d}-b\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+\sqrt [3]{d} (x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {b \left (\sqrt {d}+1\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+\sqrt [3]{d} (x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {a c \left (\sqrt {d}+1\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+\sqrt [3]{d} (x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}+\frac {\left (-\sqrt {d} a-b\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+\sqrt [3]{d} (x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}-\frac {c \left (\sqrt {d} a+b\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+\sqrt [3]{d} (x-a)^{2/3}\right )}{4 (a-b)^2 d^{5/6}}\right )}{\sqrt [3]{(b-x)^2 (x-a)}} \]

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Rubi [A] time = 1.87, antiderivative size = 561, normalized size of antiderivative = 0.33, 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 (b-a \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{5/6} (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 \sqrt {d}+b\right )-2 (1-d) x\right )}{4 d^{5/6} (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 (-\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x-b}\right )}{4 d^{5/6} (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 (\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x-b}\right )}{4 d^{5/6} (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 [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x-b}}\right )}{2 d^{5/6} (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 [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{2 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

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

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

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

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

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

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

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

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Mathematica [C] time = 0.36, size = 107, normalized size = 0.06 \begin {gather*} -\frac {3 (x-b) \left (\left (c-\sqrt {d}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {b-x}{\sqrt {d} (x-a)}\right )+\left (c+\sqrt {d}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {x-b}{\sqrt {d} (x-a)}\right )\right )}{2 d (a-b) \sqrt [3]{(x-a) (b-x)^2}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

]

[Out]

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

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

)*x^2)),x]

[Out]

$Aborted

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fricas [B] time = 1.40, size = 11788, normalized size = 6.91

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b-a*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-b^2+a^2*d+2*(-a*d+b)*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((c^6 + 6*c^4*d + 9*c^2*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^5)) + 3*c^2 + d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3)*arct

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

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

*sqrt(((2*(a^2*b - 2*a*b^2 + b^3)*c^11*d^2 + 8*(a^2*b - 2*a*b^2 + b^3)*c^9*d^3 - 4*(a^2*b - 2*a*b^2 + b^3)*c^7

*d^4 - 24*(a^2*b - 2*a*b^2 + b^3)*c^5*d^5 + 18*(a^2*b - 2*a*b^2 + b^3)*c^3*d^6 - 2*((a^2 - 2*a*b + b^2)*c^11*d

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

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

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

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

5)*c^3*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*d^8)*x)*sqrt((c^6 + 6*c^4*d + 9*c^2

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

b^3)*d^2))^(2/3) + (c^14 + 2*c^12*d - 9*c^10*d^2 - 4*c^8*d^3 + 31*c^6*d^4 - 30*c^4*d^5 + 9*c^2*d^6)*(-a*b^2 -

(a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3) - ((a*b^2 - b^3)*c^12*d + 3*(a*b^2 - b^3)*c^10*d^2 - 6*(a*b^2 - b

^3)*c^8*d^3 - 10*(a*b^2 - b^3)*c^6*d^4 + 21*(a*b^2 - b^3)*c^4*d^5 - 9*(a*b^2 - b^3)*c^2*d^6 + ((a - b)*c^12*d

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

2*((a*b - b^2)*c^12*d + 3*(a*b - b^2)*c^10*d^2 - 6*(a*b - b^2)*c^8*d^3 - 10*(a*b - b^2)*c^6*d^4 + 21*(a*b - b^

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

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

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

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

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

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

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

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

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

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

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

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

*d + 9*c^2*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^5)) + 3*c^2 + d)/((a

^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3) - sqrt(3)*(b*c^12 + 3*b*c^10*d - 6*b*c^8*d^2 - 10*b*c^6*d^3 + 21*b*c

^4*d^4 - 9*b*c^2*d^5 - (c^12 + 3*c^10*d - 6*c^8*d^2 - 10*c^6*d^3 + 21*c^4*d^4 - 9*c^2*d^5)*x))/(b*c^12 + 3*b*c

^10*d - 6*b*c^8*d^2 - 10*b*c^6*d^3 + 21*b*c^4*d^4 - 9*b*c^2*d^5 - (c^12 + 3*c^10*d - 6*c^8*d^2 - 10*c^6*d^3 +

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

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

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

c^4*d + 3*(a - b)*c^2*d^2)*x))*sqrt(((2*(a^2*b - 2*a*b^2 + b^3)*c^11*d^2 + 8*(a^2*b - 2*a*b^2 + b^3)*c^9*d^3 -

4*(a^2*b - 2*a*b^2 + b^3)*c^7*d^4 - 24*(a^2*b - 2*a*b^2 + b^3)*c^5*d^5 + 18*(a^2*b - 2*a*b^2 + b^3)*c^3*d^6 -

2*((a^2 - 2*a*b + b^2)*c^11*d^2 + 4*(a^2 - 2*a*b + b^2)*c^9*d^3 - 2*(a^2 - 2*a*b + b^2)*c^7*d^4 - 12*(a^2 - 2

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

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

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

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

x)*sqrt((c^6 + 6*c^4*d + 9*c^2*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^

5)))*(-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((c^6 +

6*c^4*d + 9*c^2*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^5)) - 3*c^2 -

d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(2/3) + (c^14 + 2*c^12*d - 9*c^10*d^2 - 4*c^8*d^3 + 31*c^6*d^4 - 30*

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

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

b^3)*c^2*d^6 + ((a - b)*c^12*d + 3*(a - b)*c^10*d^2 - 6*(a - b)*c^8*d^3 - 10*(a - b)*c^6*d^4 + 21*(a - b)*c^4*

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

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

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

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

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

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

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

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

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

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

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

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

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

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

*b*c^8*d^2 - 10*b*c^6*d^3 + 21*b*c^4*d^4 - 9*b*c^2*d^5 - (c^12 + 3*c^10*d - 6*c^8*d^2 - 10*c^6*d^3 + 21*c^4*d^

4 - 9*c^2*d^5)*x))/(b*c^12 + 3*b*c^10*d - 6*b*c^8*d^2 - 10*b*c^6*d^3 + 21*b*c^4*d^4 - 9*b*c^2*d^5 - (c^12 + 3*

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

((c^6 + 6*c^4*d + 9*c^2*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^5)) + 3

*c^2 + d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3)*log(((2*(a^2*b - 2*a*b^2 + b^3)*c^11*d^2 + 8*(a^2*b - 2

*a*b^2 + b^3)*c^9*d^3 - 4*(a^2*b - 2*a*b^2 + b^3)*c^7*d^4 - 24*(a^2*b - 2*a*b^2 + b^3)*c^5*d^5 + 18*(a^2*b - 2

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

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

*d^5 - 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^5*d^6 - 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^3*d^7 + 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

*d^8 - ((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^9*d^4 + 2*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10

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

^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^3*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*d^8)*x)*sqrt((c^6 + 6*c^4*d + 9*c^2*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^5)))*(-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((c^6 + 6*c^4*d + 9*c^2*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^5)) + 3*c^2 + d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(2/3) + (c^14 + 2*c^12*d - 9*c^10*d^2 - 4*c^8*

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

b^3)*c^12*d + 3*(a*b^2 - b^3)*c^10*d^2 - 6*(a*b^2 - b^3)*c^8*d^3 - 10*(a*b^2 - b^3)*c^6*d^4 + 21*(a*b^2 - b^3)

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

*d^4 + 21*(a - b)*c^4*d^5 - 9*(a - b)*c^2*d^6)*x^2 - 2*((a*b - b^2)*c^12*d + 3*(a*b - b^2)*c^10*d^2 - 6*(a*b -

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

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

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

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

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

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

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

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

2*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^5)) + 3*c^2 + 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((c^6

+ 6*c^4*d + 9*c^2*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^5)) - 3*c^2 -

d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(1/3)*log(((2*(a^2*b - 2*a*b^2 + b^3)*c^11*d^2 + 8*(a^2*b - 2*a*b^2

+ b^3)*c^9*d^3 - 4*(a^2*b - 2*a*b^2 + b^3)*c^7*d^4 - 24*(a^2*b - 2*a*b^2 + b^3)*c^5*d^5 + 18*(a^2*b - 2*a*b^2

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

d^4 - 12*(a^2 - 2*a*b + b^2)*c^5*d^5 + 9*(a^2 - 2*a*b + b^2)*c^3*d^6)*x + ((a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 1

0*a^2*b^4 + 5*a*b^5 - b^6)*c^9*d^4 + 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^7*d^5 -

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^5*d^6 - 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^3*d^7 + 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*d^8 -

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

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

*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*c^3*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*d^8)*x)*sqrt((c^6 + 6*c^4*d + 9*c^2*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^5)))*(-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((c^6 + 6*c^4*d + 9*c^2*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^5)) - 3*c^2 - d)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2))^(2/3) + (c^14 + 2*c^12*d - 9*c^10*d^2 - 4*c^8*d^3 +

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

c^12*d + 3*(a*b^2 - b^3)*c^10*d^2 - 6*(a*b^2 - b^3)*c^8*d^3 - 10*(a*b^2 - b^3)*c^6*d^4 + 21*(a*b^2 - b^3)*c^4*

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

+ 21*(a - b)*c^4*d^5 - 9*(a - b)*c^2*d^6)*x^2 - 2*((a*b - b^2)*c^12*d + 3*(a*b - b^2)*c^10*d^2 - 6*(a*b - b^2)

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

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

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

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

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

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

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

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

^4*d + 9*c^2*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^5)) + 3*c^2 + d)/(

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

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

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

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

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

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

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

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

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

+ 3*c*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 {a c - {\left (c + 1\right )} x + b}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b^{2} - 2 \, {\left (a d - b\right )} x\right )}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {-b -a c +\left (1+c \right ) x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (-b^{2}+a^{2} d +2 \left (-a d +b \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((-b-a*c+(1+c)*x)/((-a+x)*(-b+x)^2)^(1/3)/(-b^2+a^2*d+2*(-a*d+b)*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 {a c - {\left (c + 1\right )} x + b}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b^{2} - 2 \, {\left (a d - b\right )} x\right )}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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