Integrand size = 23, antiderivative size = 222 \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^2} \, dx=\frac {x \left (a d+a e x-b c x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {\left (2 \sqrt [3]{b} d+\sqrt [3]{a} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{2/3}}+\frac {c \log (x)}{a^2}+\frac {\left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac {\left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}}-\frac {c \log \left (a+b x^3\right )}{3 a^2} \] Output:
1/3*x*(-b*c*x^2+a*e*x+a*d)/a^2/(b*x^3+a)-1/9*(2*b^(1/3)*d+a^(1/3)*e)*arcta n(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(5/3)/b^(2/3)+c*ln( x)/a^2+1/9*(2*b^(1/3)*d-a^(1/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/b^(2/3)-1 /18*(2*b^(1/3)*d-a^(1/3)*e)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5 /3)/b^(2/3)-1/3*c*ln(b*x^3+a)/a^2
Time = 0.10 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^2} \, dx=\frac {\frac {6 a (c+x (d+e x))}{a+b x^3}-\frac {2 \sqrt {3} \sqrt [3]{a} \left (2 \sqrt [3]{b} d+\sqrt [3]{a} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+18 c \log (x)+\frac {2 \left (2 \sqrt [3]{a} \sqrt [3]{b} d-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {\left (-2 \sqrt [3]{a} \sqrt [3]{b} d+a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}-6 c \log \left (a+b x^3\right )}{18 a^2} \] Input:
Integrate[(c + d*x + e*x^2)/(x*(a + b*x^3)^2),x]
Output:
((6*a*(c + x*(d + e*x)))/(a + b*x^3) - (2*Sqrt[3]*a^(1/3)*(2*b^(1/3)*d + a ^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + 18*c*Log[ x] + (2*(2*a^(1/3)*b^(1/3)*d - a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3 ) + ((-2*a^(1/3)*b^(1/3)*d + a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) - 6*c*Log[a + b*x^3])/(18*a^2)
Time = 0.92 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2368, 25, 2373, 2009}
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.
| \(\displaystyle \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {x \left (a d+a e x-b c x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {\int -\frac {b e x^2+2 b d x+3 b c}{x \left (b x^3+a\right )}dx}{3 a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b e x^2+2 b d x+3 b c}{x \left (b x^3+a\right )}dx}{3 a b}+\frac {x \left (a d+a e x-b c x^2\right )}{3 a^2 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \frac {\int \left (\frac {3 b c}{a x}+\frac {b \left (-3 b c x^2+a e x+2 a d\right )}{a \left (b x^3+a\right )}\right )dx}{3 a b}+\frac {x \left (a d+a e x-b c x^2\right )}{3 a^2 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (\sqrt [3]{a} e+2 \sqrt [3]{b} d\right )}{\sqrt {3} a^{2/3}}-\frac {\sqrt [3]{b} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3}}+\frac {\sqrt [3]{b} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3}}-\frac {b c \log \left (a+b x^3\right )}{a}+\frac {3 b c \log (x)}{a}}{3 a b}+\frac {x \left (a d+a e x-b c x^2\right )}{3 a^2 \left (a+b x^3\right )}\) |
Input:
Int[(c + d*x + e*x^2)/(x*(a + b*x^3)^2),x]
Output:
(x*(a*d + a*e*x - b*c*x^2))/(3*a^2*(a + b*x^3)) + (-((b^(1/3)*(2*b^(1/3)*d + a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]* a^(2/3))) + (3*b*c*Log[x])/a + (b^(1/3)*(2*b^(1/3)*d - a^(1/3)*e)*Log[a^(1 /3) + b^(1/3)*x])/(3*a^(2/3)) - (b^(1/3)*(2*b^(1/3)*d - a^(1/3)*e)*Log[a^( 2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)) - (b*c*Log[a + b*x^3] )/a)/(3*a*b)
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m *Pq, a + b*x^n, x], i}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x]]] /; F reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[E xpandIntegrand[(c*x)^m*(Pq/(a + b*x^n)), x], x] /; FreeQ[{a, b, c, m}, x] & & PolyQ[Pq, x] && IntegerQ[n] && !IGtQ[m, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(\frac {\frac {e \,x^{2}}{3 a}+\frac {d x}{3 a}+\frac {c}{3 a}}{b \,x^{3}+a}+\frac {c \ln \left (-x \right )}{a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{6} b^{2} \textit {\_Z}^{3}+9 a^{4} b^{2} c \,\textit {\_Z}^{2}+\left (6 a^{3} b d e +27 a^{2} b^{2} c^{2}\right ) \textit {\_Z} +a^{2} e^{3}+18 a b c d e -8 a b \,d^{3}+27 c^{3} b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{5} b^{2}-24 \textit {\_R}^{2} a^{3} b^{2} c +\left (-20 a^{2} b d e -36 a \,c^{2} b^{2}\right ) \textit {\_R} -3 a \,e^{3}-36 b c d e +24 b \,d^{3}\right ) x +a^{4} b e \,\textit {\_R}^{2}+\left (-6 a^{2} b c e -4 a^{2} b \,d^{2}\right ) \textit {\_R} -27 b \,c^{2} e +36 b c \,d^{2}\right )\right )}{9}\) | \(235\) |
| default | \(\frac {\frac {\frac {1}{3} a e \,x^{2}+\frac {1}{3} a d x +\frac {1}{3} a c}{b \,x^{3}+a}+\frac {2 a d \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {a e \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}-\frac {c \ln \left (b \,x^{3}+a \right )}{3}}{a^{2}}+\frac {c \ln \left (x \right )}{a^{2}}\) | \(240\) |
Input:
int((e*x^2+d*x+c)/x/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
(1/3*e/a*x^2+1/3*d/a*x+1/3*c/a)/(b*x^3+a)+c/a^2*ln(-x)+1/9*sum(_R*ln((-4*_ R^3*a^5*b^2-24*_R^2*a^3*b^2*c+(-20*a^2*b*d*e-36*a*b^2*c^2)*_R-3*a*e^3-36*b *c*d*e+24*b*d^3)*x+a^4*b*e*_R^2+(-6*a^2*b*c*e-4*a^2*b*d^2)*_R-27*b*c^2*e+3 6*b*c*d^2),_R=RootOf(a^6*b^2*_Z^3+9*a^4*b^2*c*_Z^2+(6*a^3*b*d*e+27*a^2*b^2 *c^2)*_Z+a^2*e^3+18*a*b*c*d*e-8*a*b*d^3+27*c^3*b^2))
Result contains complex when optimal does not.
Time = 0.91 (sec) , antiderivative size = 5018, normalized size of antiderivative = 22.60 \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d*x+c)/x/(b*x^3+a)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x**2+d*x+c)/x/(b*x**3+a)**2,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.91 \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^2} \, dx=\frac {e x^{2} + d x + c}{3 \, {\left (a b x^{3} + a^{2}\right )}} + \frac {c \log \left (x\right )}{a^{2}} + \frac {\sqrt {3} {\left (a e \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3}} - \frac {{\left (6 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (3 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} + a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((e*x^2+d*x+c)/x/(b*x^3+a)^2,x, algorithm="maxima")
Output:
1/3*(e*x^2 + d*x + c)/(a*b*x^3 + a^2) + c*log(x)/a^2 + 1/9*sqrt(3)*(a*e*(a /b)^(2/3) + 2*a*d*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b )^(1/3))/a^3 - 1/18*(6*b*c*(a/b)^(2/3) - a*e*(a/b)^(1/3) + 2*a*d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b*(a/b)^(2/3)) - 1/9*(3*b*c*(a/b)^(2/3 ) + a*e*(a/b)^(1/3) - 2*a*d)*log(x + (a/b)^(1/3))/(a^2*b*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^2} \, dx=-\frac {\sqrt {3} {\left (2 \, b d - \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} - \frac {{\left (2 \, b d + \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} - \frac {c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac {c \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {a e x^{2} + a d x + a c}{3 \, {\left (b x^{3} + a\right )} a^{2}} - \frac {{\left (a^{3} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{3} b d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{5} b} \] Input:
integrate((e*x^2+d*x+c)/x/(b*x^3+a)^2,x, algorithm="giac")
Output:
-1/9*sqrt(3)*(2*b*d - (-a*b^2)^(1/3)*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^( 1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a) - 1/18*(2*b*d + (-a*b^2)^(1/3)*e)*l og(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a) - 1/3*c*log(abs (b*x^3 + a))/a^2 + c*log(abs(x))/a^2 + 1/3*(a*e*x^2 + a*d*x + a*c)/((b*x^3 + a)*a^2) - 1/9*(a^3*b*e*(-a/b)^(1/3) + 2*a^3*b*d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^5*b)
Time = 6.95 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.21 \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^2} \, dx=\frac {\frac {c}{3\,a}+\frac {e\,x^2}{3\,a}+\frac {d\,x}{3\,a}}{b\,x^3+a}+\left (\sum _{k=1}^3\ln \left (\frac {4\,b^2\,c\,d^2-3\,b^2\,c^2\,e}{9\,a^3}-\mathrm {root}\left (729\,a^6\,b^2\,z^3+729\,a^4\,b^2\,c\,z^2+54\,a^3\,b\,d\,e\,z+243\,a^2\,b^2\,c^2\,z+18\,a\,b\,c\,d\,e-8\,a\,b\,d^3+27\,b^2\,c^3+a^2\,e^3,z,k\right )\,\left (\mathrm {root}\left (729\,a^6\,b^2\,z^3+729\,a^4\,b^2\,c\,z^2+54\,a^3\,b\,d\,e\,z+243\,a^2\,b^2\,c^2\,z+18\,a\,b\,c\,d\,e-8\,a\,b\,d^3+27\,b^2\,c^3+a^2\,e^3,z,k\right )\,\left (-a\,b^2\,e+24\,b^3\,c\,x+\mathrm {root}\left (729\,a^6\,b^2\,z^3+729\,a^4\,b^2\,c\,z^2+54\,a^3\,b\,d\,e\,z+243\,a^2\,b^2\,c^2\,z+18\,a\,b\,c\,d\,e-8\,a\,b\,d^3+27\,b^2\,c^3+a^2\,e^3,z,k\right )\,a^2\,b^3\,x\,36\right )+\frac {4\,a^2\,b^2\,d^2+6\,c\,e\,a^2\,b^2}{9\,a^3}+\frac {x\,\left (60\,d\,e\,a^2\,b^2+108\,a\,b^3\,c^2\right )}{27\,a^3}\right )-\frac {x\,\left (-8\,b^2\,d^3+12\,c\,b^2\,d\,e+a\,b\,e^3\right )}{27\,a^3}\right )\,\mathrm {root}\left (729\,a^6\,b^2\,z^3+729\,a^4\,b^2\,c\,z^2+54\,a^3\,b\,d\,e\,z+243\,a^2\,b^2\,c^2\,z+18\,a\,b\,c\,d\,e-8\,a\,b\,d^3+27\,b^2\,c^3+a^2\,e^3,z,k\right )\right )+\frac {c\,\ln \left (x\right )}{a^2} \] Input:
int((c + d*x + e*x^2)/(x*(a + b*x^3)^2),x)
Output:
(c/(3*a) + (e*x^2)/(3*a) + (d*x)/(3*a))/(a + b*x^3) + symsum(log((4*b^2*c* d^2 - 3*b^2*c^2*e)/(9*a^3) - root(729*a^6*b^2*z^3 + 729*a^4*b^2*c*z^2 + 54 *a^3*b*d*e*z + 243*a^2*b^2*c^2*z + 18*a*b*c*d*e - 8*a*b*d^3 + 27*b^2*c^3 + a^2*e^3, z, k)*(root(729*a^6*b^2*z^3 + 729*a^4*b^2*c*z^2 + 54*a^3*b*d*e*z + 243*a^2*b^2*c^2*z + 18*a*b*c*d*e - 8*a*b*d^3 + 27*b^2*c^3 + a^2*e^3, z, k)*(24*b^3*c*x - a*b^2*e + 36*root(729*a^6*b^2*z^3 + 729*a^4*b^2*c*z^2 + 54*a^3*b*d*e*z + 243*a^2*b^2*c^2*z + 18*a*b*c*d*e - 8*a*b*d^3 + 27*b^2*c^3 + a^2*e^3, z, k)*a^2*b^3*x) + (4*a^2*b^2*d^2 + 6*a^2*b^2*c*e)/(9*a^3) + ( x*(108*a*b^3*c^2 + 60*a^2*b^2*d*e))/(27*a^3)) - (x*(a*b*e^3 - 8*b^2*d^3 + 12*b^2*c*d*e))/(27*a^3))*root(729*a^6*b^2*z^3 + 729*a^4*b^2*c*z^2 + 54*a^3 *b*d*e*z + 243*a^2*b^2*c^2*z + 18*a*b*c*d*e - 8*a*b*d^3 + 27*b^2*c^3 + a^2 *e^3, z, k), k, 1, 3) + (c*log(x))/a^2
Time = 0.15 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.22 \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Input:
int((e*x^2+d*x+c)/x/(b*x^3+a)^2,x)
Output:
( - 4*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*s qrt(3)))*a*d - 4*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/ (a**(1/3)*sqrt(3)))*b*d*x**3 - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a **(1/3)*sqrt(3)))*a**2*e - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1 /3)*sqrt(3)))*a*b*e*x**3 - 2*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a** (1/3)*x + b**(2/3)*x**2)*a*d - 2*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3) *a**(1/3)*x + b**(2/3)*x**2)*b*d*x**3 + 4*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*a*d + 4*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*b*d*x**3 - 6*b**(2/3)*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2) *a*c - 6*b**(2/3)*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x **2)*b*c*x**3 - 6*b**(2/3)*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a*c - 6*b** (2/3)*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*b*c*x**3 + 18*b**(2/3)*a**(1/3)* log(x)*a*c + 18*b**(2/3)*a**(1/3)*log(x)*b*c*x**3 + 6*b**(2/3)*a**(1/3)*a* d*x + 6*b**(2/3)*a**(1/3)*a*e*x**2 - 6*b**(2/3)*a**(1/3)*b*c*x**3 + log(a* *(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*e + log(a**(2/3) - b**( 1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b*e*x**3 - 2*log(a**(1/3) + b**(1/3)*x) *a**2*e - 2*log(a**(1/3) + b**(1/3)*x)*a*b*e*x**3)/(18*b**(2/3)*a**(1/3)*a **2*(a + b*x**3))