Integrand size = 57, antiderivative size = 88 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-2 a^{2/3} C-(-b)^{2/3} B x-(-b)^{2/3} C x^2}{a+b x^3} \, dx=\frac {2 \left (b B+\sqrt [3]{a} (-b)^{2/3} C\right ) \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b}+\frac {C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}} \]
C*ln(a^(1/3)-(-b)^(1/3)*x)/(-b)^(1/3)+2/3*(b*B+a^(1/3)*(-b)^(2/3)*C)*arcta n(1/3*(a^(1/3)+2*(-b)^(1/3)*x)/a^(1/3)*3^(1/2))/a^(1/3)/b*3^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(238\) vs. \(2(88)=176\).
Time = 0.57 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.70 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-2 a^{2/3} C-(-b)^{2/3} B x-(-b)^{2/3} C x^2}{a+b x^3} \, dx=\frac {2 \sqrt {3} \sqrt [3]{b} \left (\left ((-b)^{2/3}-\sqrt [3]{-b^2}\right ) B+2 \sqrt [3]{a} \sqrt [3]{b} C\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+\frac {-2 b \left (\left (-(-b)^{2/3}+b^{2/3}\right ) B+2 \sqrt [3]{a} \sqrt [3]{-b} C\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\left ((-b)^{5/3} B+b^{5/3} B+2 \sqrt [3]{a} \sqrt [3]{-b} b C\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt [3]{a} (-b)^{2/3} \sqrt [3]{-b^2} C \log \left (a+b x^3\right )}{\sqrt [3]{-b^2}}}{6 \sqrt [3]{a} b} \]
Integrate[(a^(1/3)*(-b)^(1/3)*B - 2*a^(2/3)*C - (-b)^(2/3)*B*x - (-b)^(2/3 )*C*x^2)/(a + b*x^3),x]
(2*Sqrt[3]*b^(1/3)*(((-b)^(2/3) - (-b^2)^(1/3))*B + 2*a^(1/3)*b^(1/3)*C)*A rcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + (-2*b*((-(-b)^(2/3) + b^(2/3) )*B + 2*a^(1/3)*(-b)^(1/3)*C)*Log[a^(1/3) + b^(1/3)*x] + ((-b)^(5/3)*B + b ^(5/3)*B + 2*a^(1/3)*(-b)^(1/3)*b*C)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^( 2/3)*x^2] - 2*a^(1/3)*(-b)^(2/3)*(-b^2)^(1/3)*C*Log[a + b*x^3])/(-b^2)^(1/ 3))/(6*a^(1/3)*b)
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2405, 16, 1082, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-2 a^{2/3} C+\sqrt [3]{a} \sqrt [3]{-b} B-(-b)^{2/3} B x-(-b)^{2/3} C x^2}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 2405 |
| \(\displaystyle \frac {\left (\sqrt [3]{-b} B-\sqrt [3]{a} C\right ) \int \frac {1}{x^2+\frac {\sqrt [3]{a} x}{\sqrt [3]{-b}}+\frac {a^{2/3}}{(-b)^{2/3}}}dx}{(-b)^{2/3}}-\frac {C \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{-b}}-x}dx}{\sqrt [3]{-b}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\left (\sqrt [3]{-b} B-\sqrt [3]{a} C\right ) \int \frac {1}{x^2+\frac {\sqrt [3]{a} x}{\sqrt [3]{-b}}+\frac {a^{2/3}}{(-b)^{2/3}}}dx}{(-b)^{2/3}}+\frac {C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}-\frac {2 \left (\sqrt [3]{-b} B-\sqrt [3]{a} C\right ) \int \frac {1}{-\left (\frac {2 \sqrt [3]{-b} x}{\sqrt [3]{a}}+1\right )^2-3}d\left (\frac {2 \sqrt [3]{-b} x}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a} \sqrt [3]{-b}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \arctan \left (\frac {\frac {2 \sqrt [3]{-b} x}{\sqrt [3]{a}}+1}{\sqrt {3}}\right ) \left (\sqrt [3]{-b} B-\sqrt [3]{a} C\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{-b}}+\frac {C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}\) |
(2*((-b)^(1/3)*B - a^(1/3)*C)*ArcTan[(1 + (2*(-b)^(1/3)*x)/a^(1/3))/Sqrt[3 ]])/(Sqrt[3]*a^(1/3)*(-b)^(1/3)) + (C*Log[a^(1/3) - (-b)^(1/3)*x])/(-b)^(1 /3)
3.1.41.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = a^(1/3)/(-b)^(1/3)}, Sim p[-C/b Int[1/(q - x), x], x] + Simp[(B - C*q)/b Int[1/(q^2 + q*x + x^2) , x], x]] /; EqQ[A*(-b)^(2/3) + a^(1/3)*(-b)^(1/3)*B - 2*a^(2/3)*C, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
Leaf count of result is larger than twice the leaf count of optimal. \(226\) vs. \(2(67)=134\).
Time = 1.59 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.58
| method | result | size |
| default | \(\left (a^{\frac {1}{3}} \left (-b \right )^{\frac {1}{3}} B -2 a^{\frac {2}{3}} C \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-\left (-b \right )^{\frac {2}{3}} B \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-\frac {C \left (-b \right )^{\frac {2}{3}} \ln \left (b \,x^{3}+a \right )}{3 b}\) | \(227\) |
int((a^(1/3)*(-b)^(1/3)*B-2*a^(2/3)*C-(-b)^(2/3)*B*x-(-b)^(2/3)*C*x^2)/(b* x^3+a),x,method=_RETURNVERBOSE)
(a^(1/3)*(-b)^(1/3)*B-2*a^(2/3)*C)*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/ 6/b/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2 )*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))-(-b)^(2/3)*B*(-1/3/b/(a/b)^(1/3 )*ln(x+(a/b)^(1/3))+1/6/b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/ 3*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))-1/3*C*(-b )^(2/3)*ln(b*x^3+a)/b
Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (67) = 134\).
Time = 1.66 (sec) , antiderivative size = 470, normalized size of antiderivative = 5.34 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-2 a^{2/3} C-(-b)^{2/3} B x-(-b)^{2/3} C x^2}{a+b x^3} \, dx=\left [\frac {\sqrt {\frac {1}{3}} b \sqrt {\frac {C^{2} a \left (-b\right )^{\frac {1}{3}} - 2 \, B C a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} - B^{2} a^{\frac {1}{3}} b}{a b}} \log \left (-\frac {C^{3} a^{2} + B^{3} a b - 2 \, {\left (C^{3} a b + B^{3} b^{2}\right )} x^{3} - 3 \, {\left (C^{3} a + B^{3} b\right )} a^{\frac {2}{3}} \left (-b\right )^{\frac {1}{3}} x + 3 \, \sqrt {\frac {1}{3}} {\left ({\left (2 \, B^{2} b x^{2} + C^{2} a x + B C a\right )} a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} + {\left (2 \, C^{2} a b x^{2} - B C a b x - B^{2} a b\right )} a^{\frac {1}{3}} + {\left (2 \, B C a b x^{2} - B^{2} a b x + C^{2} a^{2}\right )} \left (-b\right )^{\frac {1}{3}}\right )} \sqrt {\frac {C^{2} a \left (-b\right )^{\frac {1}{3}} - 2 \, B C a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} - B^{2} a^{\frac {1}{3}} b}{a b}}}{b x^{3} + a}\right ) - C \left (-b\right )^{\frac {2}{3}} \log \left (b x + a^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}}\right )}{b}, -\frac {2 \, \sqrt {\frac {1}{3}} b \sqrt {-\frac {C^{2} a \left (-b\right )^{\frac {1}{3}} - 2 \, B C a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} - B^{2} a^{\frac {1}{3}} b}{a b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left ({\left (2 \, C^{2} x + B C\right )} a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} - {\left (2 \, B C b x + B^{2} b\right )} a^{\frac {1}{3}} - {\left (2 \, B^{2} b x - C^{2} a\right )} \left (-b\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {C^{2} a \left (-b\right )^{\frac {1}{3}} - 2 \, B C a^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} - B^{2} a^{\frac {1}{3}} b}{a b}}}{C^{3} a + B^{3} b}\right ) + C \left (-b\right )^{\frac {2}{3}} \log \left (b x + a^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}}\right )}{b}\right ] \]
integrate((a^(1/3)*(-b)^(1/3)*B-2*a^(2/3)*C-(-b)^(2/3)*B*x-(-b)^(2/3)*C*x^ 2)/(b*x^3+a),x, algorithm="fricas")
[(sqrt(1/3)*b*sqrt((C^2*a*(-b)^(1/3) - 2*B*C*a^(2/3)*(-b)^(2/3) - B^2*a^(1 /3)*b)/(a*b))*log(-(C^3*a^2 + B^3*a*b - 2*(C^3*a*b + B^3*b^2)*x^3 - 3*(C^3 *a + B^3*b)*a^(2/3)*(-b)^(1/3)*x + 3*sqrt(1/3)*((2*B^2*b*x^2 + C^2*a*x + B *C*a)*a^(2/3)*(-b)^(2/3) + (2*C^2*a*b*x^2 - B*C*a*b*x - B^2*a*b)*a^(1/3) + (2*B*C*a*b*x^2 - B^2*a*b*x + C^2*a^2)*(-b)^(1/3))*sqrt((C^2*a*(-b)^(1/3) - 2*B*C*a^(2/3)*(-b)^(2/3) - B^2*a^(1/3)*b)/(a*b)))/(b*x^3 + a)) - C*(-b)^ (2/3)*log(b*x + a^(1/3)*(-b)^(2/3)))/b, -(2*sqrt(1/3)*b*sqrt(-(C^2*a*(-b)^ (1/3) - 2*B*C*a^(2/3)*(-b)^(2/3) - B^2*a^(1/3)*b)/(a*b))*arctan(sqrt(1/3)* ((2*C^2*x + B*C)*a^(2/3)*(-b)^(2/3) - (2*B*C*b*x + B^2*b)*a^(1/3) - (2*B^2 *b*x - C^2*a)*(-b)^(1/3))*sqrt(-(C^2*a*(-b)^(1/3) - 2*B*C*a^(2/3)*(-b)^(2/ 3) - B^2*a^(1/3)*b)/(a*b))/(C^3*a + B^3*b)) + C*(-b)^(2/3)*log(b*x + a^(1/ 3)*(-b)^(2/3)))/b]
Timed out. \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-2 a^{2/3} C-(-b)^{2/3} B x-(-b)^{2/3} C x^2}{a+b x^3} \, dx=\text {Timed out} \]
integrate((a**(1/3)*(-b)**(1/3)*B-2*a**(2/3)*C-(-b)**(2/3)*B*x-(-b)**(2/3) *C*x**2)/(b*x**3+a),x)
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (67) = 134\).
Time = 0.31 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.86 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-2 a^{2/3} C-(-b)^{2/3} B x-(-b)^{2/3} C x^2}{a+b x^3} \, dx=\frac {\sqrt {3} {\left (2 \, C a \left (-b\right )^{\frac {2}{3}} - {\left (6 \, C a^{\frac {2}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} - 3 \, B a^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + {\left (3 \, B \left (\frac {a}{b}\right )^{\frac {2}{3}} + \frac {2 \, C a}{b}\right )} \left (-b\right )^{\frac {2}{3}}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} + \frac {{\left (2 \, C a^{\frac {2}{3}} - B a^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} - {\left (2 \, C \left (\frac {a}{b}\right )^{\frac {2}{3}} + B \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-b\right )^{\frac {2}{3}}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, C a^{\frac {2}{3}} - B a^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} + {\left (C \left (\frac {a}{b}\right )^{\frac {2}{3}} - B \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-b\right )^{\frac {2}{3}}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
integrate((a^(1/3)*(-b)^(1/3)*B-2*a^(2/3)*C-(-b)^(2/3)*B*x-(-b)^(2/3)*C*x^ 2)/(b*x^3+a),x, algorithm="maxima")
1/9*sqrt(3)*(2*C*a*(-b)^(2/3) - (6*C*a^(2/3)*(a/b)^(1/3) - 3*B*a^(1/3)*(-b )^(1/3)*(a/b)^(1/3) + (3*B*(a/b)^(2/3) + 2*C*a/b)*(-b)^(2/3))*b)*arctan(1/ 3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b) + 1/6*(2*C*a^(2/3) - B*a^ (1/3)*(-b)^(1/3) - (2*C*(a/b)^(2/3) + B*(a/b)^(1/3))*(-b)^(2/3))*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b*(a/b)^(2/3)) - 1/3*(2*C*a^(2/3) - B*a^(1/ 3)*(-b)^(1/3) + (C*(a/b)^(2/3) - B*(a/b)^(1/3))*(-b)^(2/3))*log(x + (a/b)^ (1/3))/(b*(a/b)^(2/3))
Timed out. \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-2 a^{2/3} C-(-b)^{2/3} B x-(-b)^{2/3} C x^2}{a+b x^3} \, dx=\text {Timed out} \]
integrate((a^(1/3)*(-b)^(1/3)*B-2*a^(2/3)*C-(-b)^(2/3)*B*x-(-b)^(2/3)*C*x^ 2)/(b*x^3+a),x, algorithm="giac")
Time = 11.27 (sec) , antiderivative size = 444, normalized size of antiderivative = 5.05 \[ \int \frac {\sqrt [3]{a} \sqrt [3]{-b} B-2 a^{2/3} C-(-b)^{2/3} B x-(-b)^{2/3} C x^2}{a+b x^3} \, dx=\sum _{k=1}^3\ln \left (\mathrm {root}\left (27\,a^2\,b^3\,z^3+27\,C\,a^2\,{\left (-b\right )}^{8/3}\,z^2+18\,B\,C\,a^{5/3}\,{\left (-b\right )}^{8/3}\,z+9\,B^2\,a^{4/3}\,b^3\,z-9\,C^2\,a^2\,{\left (-b\right )}^{7/3}\,z-18\,B\,C^2\,a^{5/3}\,{\left (-b\right )}^{7/3}+9\,B^2\,C\,a^{4/3}\,{\left (-b\right )}^{8/3}+9\,C^3\,a^2\,b^2,z,k\right )\,\left (\frac {6\,C\,a}{{\left (-b\right )}^{4/3}}-\frac {x\,\left (3\,B\,a^{1/3}\,{\left (-b\right )}^{4/3}+6\,C\,a^{2/3}\,b\right )}{b^2}+\frac {\mathrm {root}\left (27\,a^2\,b^3\,z^3+27\,C\,a^2\,{\left (-b\right )}^{8/3}\,z^2+18\,B\,C\,a^{5/3}\,{\left (-b\right )}^{8/3}\,z+9\,B^2\,a^{4/3}\,b^3\,z-9\,C^2\,a^2\,{\left (-b\right )}^{7/3}\,z-18\,B\,C^2\,a^{5/3}\,{\left (-b\right )}^{7/3}+9\,B^2\,C\,a^{4/3}\,{\left (-b\right )}^{8/3}+9\,C^3\,a^2\,b^2,z,k\right )\,a\,9}{b}\right )+\frac {B^2\,a^{1/3}\,b^2+C^2\,a\,{\left (-b\right )}^{4/3}-2\,B\,C\,a^{2/3}\,{\left (-b\right )}^{5/3}}{b^3}-\frac {x\,\left (2\,C^2\,a^{2/3}\,{\left (-b\right )}^{2/3}-B^2\,{\left (-b\right )}^{4/3}+B\,C\,a^{1/3}\,b\right )}{b^2}\right )\,\mathrm {root}\left (27\,a^2\,b^3\,z^3+27\,C\,a^2\,{\left (-b\right )}^{8/3}\,z^2+18\,B\,C\,a^{5/3}\,{\left (-b\right )}^{8/3}\,z+9\,B^2\,a^{4/3}\,b^3\,z-9\,C^2\,a^2\,{\left (-b\right )}^{7/3}\,z-18\,B\,C^2\,a^{5/3}\,{\left (-b\right )}^{7/3}+9\,B^2\,C\,a^{4/3}\,{\left (-b\right )}^{8/3}+9\,C^3\,a^2\,b^2,z,k\right ) \]
symsum(log(root(27*a^2*b^3*z^3 + 27*C*a^2*(-b)^(8/3)*z^2 + 18*B*C*a^(5/3)* (-b)^(8/3)*z + 9*B^2*a^(4/3)*b^3*z - 9*C^2*a^2*(-b)^(7/3)*z - 18*B*C^2*a^( 5/3)*(-b)^(7/3) + 9*B^2*C*a^(4/3)*(-b)^(8/3) + 9*C^3*a^2*b^2, z, k)*((6*C* a)/(-b)^(4/3) - (x*(3*B*a^(1/3)*(-b)^(4/3) + 6*C*a^(2/3)*b))/b^2 + (9*root (27*a^2*b^3*z^3 + 27*C*a^2*(-b)^(8/3)*z^2 + 18*B*C*a^(5/3)*(-b)^(8/3)*z + 9*B^2*a^(4/3)*b^3*z - 9*C^2*a^2*(-b)^(7/3)*z - 18*B*C^2*a^(5/3)*(-b)^(7/3) + 9*B^2*C*a^(4/3)*(-b)^(8/3) + 9*C^3*a^2*b^2, z, k)*a)/b) + (B^2*a^(1/3)* b^2 + C^2*a*(-b)^(4/3) - 2*B*C*a^(2/3)*(-b)^(5/3))/b^3 - (x*(2*C^2*a^(2/3) *(-b)^(2/3) - B^2*(-b)^(4/3) + B*C*a^(1/3)*b))/b^2)*root(27*a^2*b^3*z^3 + 27*C*a^2*(-b)^(8/3)*z^2 + 18*B*C*a^(5/3)*(-b)^(8/3)*z + 9*B^2*a^(4/3)*b^3* z - 9*C^2*a^2*(-b)^(7/3)*z - 18*B*C^2*a^(5/3)*(-b)^(7/3) + 9*B^2*C*a^(4/3) *(-b)^(8/3) + 9*C^3*a^2*b^2, z, k), k, 1, 3)