Integrand size = 30, antiderivative size = 244 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=-\frac {c}{8 a x^8}+\frac {b c-a d}{5 a^2 x^5}-\frac {b^2 c-a b d+a^2 e}{2 a^3 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{11/3} \sqrt [3]{b}}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{11/3} \sqrt [3]{b}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{11/3} \sqrt [3]{b}} \]
-1/8*c/a/x^8+1/5*(-a*d+b*c)/a^2/x^5+1/2*(-a^2*e+a*b*d-b^2*c)/a^3/x^2-1/3*( -a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(11/3)/b^(1/3)+1/6*( -a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^ (11/3)/b^(1/3)+1/3*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*arctan(1/3*(a^(1/3)-2*b^ (1/3)*x)/a^(1/3)*3^(1/2))/a^(11/3)/b^(1/3)*3^(1/2)
Time = 0.13 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=\frac {-\frac {15 a^{8/3} c}{x^8}+\frac {24 a^{5/3} (b c-a d)}{x^5}-\frac {60 a^{2/3} \left (b^2 c-a b d+a^2 e\right )}{x^2}+\frac {40 \sqrt {3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {40 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {20 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}}{120 a^{11/3}} \]
((-15*a^(8/3)*c)/x^8 + (24*a^(5/3)*(b*c - a*d))/x^5 - (60*a^(2/3)*(b^2*c - a*b*d + a^2*e))/x^2 + (40*Sqrt[3]*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Arc Tan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (40*(-(b^3*c) + a*b^2* d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + (20*(b^3*c - a*b^ 2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^( 1/3))/(120*a^(11/3))
Time = 0.39 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \int \left (\frac {a d-b c}{a^2 x^6}+\frac {a^2 e-a b d+b^2 c}{a^3 x^3}+\frac {a^3 f-a^2 b e+a b^2 d-b^3 c}{a^3 \left (a+b x^3\right )}+\frac {c}{a x^9}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b c-a d}{5 a^2 x^5}-\frac {a^2 e-a b d+b^2 c}{2 a^3 x^2}+\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{11/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{11/3} \sqrt [3]{b}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{11/3} \sqrt [3]{b}}-\frac {c}{8 a x^8}\) |
-1/8*c/(a*x^8) + (b*c - a*d)/(5*a^2*x^5) - (b^2*c - a*b*d + a^2*e)/(2*a^3* x^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x) /(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(11/3)*b^(1/3)) - ((b^3*c - a*b^2*d + a^2* b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(11/3)*b^(1/3)) + ((b^3*c - a* b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/( 6*a^(11/3)*b^(1/3))
3.3.45.3.1 Defintions of rubi rules used
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]
Time = 1.55 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.70
| method | result | size |
| default | \(-\frac {c}{8 a \,x^{8}}-\frac {a d -b c}{5 a^{2} x^{5}}-\frac {a^{2} e -a b d +b^{2} c}{2 a^{3} x^{2}}+\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{a^{3}}\) | \(170\) |
| risch | \(\frac {-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{6}}{2 a^{3}}-\frac {\left (a d -b c \right ) x^{3}}{5 a^{2}}-\frac {c}{8 a}}{x^{8}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} b \,\textit {\_Z}^{3}-a^{9} f^{3}+3 a^{8} b e \,f^{2}-3 a^{7} b^{2} d \,f^{2}-3 a^{7} b^{2} e^{2} f +3 a^{6} b^{3} c \,f^{2}+6 a^{6} b^{3} d e f +a^{6} b^{3} e^{3}-6 a^{5} b^{4} c e f -3 a^{5} b^{4} d^{2} f -3 a^{5} b^{4} d \,e^{2}+6 a^{4} b^{5} c d f +3 a^{4} b^{5} c \,e^{2}+3 a^{4} b^{5} d^{2} e -3 a^{3} b^{6} c^{2} f -6 a^{3} b^{6} c d e -a^{3} b^{6} d^{3}+3 a^{2} b^{7} c^{2} e +3 a^{2} b^{7} c \,d^{2}-3 a \,b^{8} c^{2} d +c^{3} b^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{11} b +3 a^{9} f^{3}-9 a^{8} b e \,f^{2}+9 a^{7} b^{2} d \,f^{2}+9 a^{7} b^{2} e^{2} f -9 a^{6} b^{3} c \,f^{2}-18 a^{6} b^{3} d e f -3 a^{6} b^{3} e^{3}+18 a^{5} b^{4} c e f +9 a^{5} b^{4} d^{2} f +9 a^{5} b^{4} d \,e^{2}-18 a^{4} b^{5} c d f -9 a^{4} b^{5} c \,e^{2}-9 a^{4} b^{5} d^{2} e +9 a^{3} b^{6} c^{2} f +18 a^{3} b^{6} c d e +3 a^{3} b^{6} d^{3}-9 a^{2} b^{7} c^{2} e -9 a^{2} b^{7} c \,d^{2}+9 a \,b^{8} c^{2} d -3 c^{3} b^{9}\right ) x +\left (-a^{10} f^{2}+2 a^{9} b e f -2 a^{8} b^{2} d f -a^{8} b^{2} e^{2}+2 a^{7} b^{3} c f +2 a^{7} b^{3} d e -2 a^{6} b^{4} c e -a^{6} b^{4} d^{2}+2 a^{5} b^{5} c d -a^{4} b^{6} c^{2}\right ) \textit {\_R} \right )\right )}{3}\) | \(628\) |
-1/8*c/a/x^8-1/5*(a*d-b*c)/a^2/x^5-1/2*(a^2*e-a*b*d+b^2*c)/a^3/x^2+(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))) /a^3*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)
Time = 0.31 (sec) , antiderivative size = 595, normalized size of antiderivative = 2.44 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=\left [-\frac {60 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{8} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 20 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a^{2} b\right )^{\frac {2}{3}} x^{8} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 40 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a^{2} b\right )^{\frac {2}{3}} x^{8} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 60 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e\right )} x^{6} + 15 \, a^{4} b c - 24 \, {\left (a^{3} b^{2} c - a^{4} b d\right )} x^{3}}{120 \, a^{5} b x^{8}}, -\frac {120 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{8} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 20 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a^{2} b\right )^{\frac {2}{3}} x^{8} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 40 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a^{2} b\right )^{\frac {2}{3}} x^{8} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 60 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e\right )} x^{6} + 15 \, a^{4} b c - 24 \, {\left (a^{3} b^{2} c - a^{4} b d\right )} x^{3}}{120 \, a^{5} b x^{8}}\right ] \]
[-1/120*(60*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^8*sqrt (-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3 )*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b))/ (b*x^3 + a)) - 20*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^8*lo g(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 40*(b^3*c - a*b^2*d + a^2 *b*e - a^3*f)*(a^2*b)^(2/3)*x^8*log(a*b*x + (a^2*b)^(2/3)) + 60*(a^2*b^3*c - a^3*b^2*d + a^4*b*e)*x^6 + 15*a^4*b*c - 24*(a^3*b^2*c - a^4*b*d)*x^3)/( a^5*b*x^8), -1/120*(120*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b *f)*x^8*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x - (a^2*b )^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 20*(b^3*c - a*b^2*d + a^2*b*e - a^ 3*f)*(a^2*b)^(2/3)*x^8*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 40*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^8*log(a*b*x + (a^2* b)^(2/3)) + 60*(a^2*b^3*c - a^3*b^2*d + a^4*b*e)*x^6 + 15*a^4*b*c - 24*(a^ 3*b^2*c - a^4*b*d)*x^3)/(a^5*b*x^8)]
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=-\frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {20 \, {\left (b^{2} c - a b d + a^{2} e\right )} x^{6} - 8 \, {\left (a b c - a^{2} d\right )} x^{3} + 5 \, a^{2} c}{40 \, a^{3} x^{8}} \]
-1/3*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*b*(a/b)^(2/3)) + 1/6*(b^3*c - a*b^2*d + a^ 2*b*e - a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b*(a/b)^(2/3)) - 1/3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*log(x + (a/b)^(1/3))/(a^3*b*(a/b )^(2/3)) - 1/40*(20*(b^2*c - a*b*d + a^2*e)*x^6 - 8*(a*b*c - a^2*d)*x^3 + 5*a^2*c)/(a^3*x^8)
Time = 0.27 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.20 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=\frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{4}} - \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{4} b} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{4} b} - \frac {20 \, b^{2} c x^{6} - 20 \, a b d x^{6} + 20 \, a^{2} e x^{6} - 8 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{3} x^{8}} \]
1/3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1 /3)))/a^4 - 1/3*sqrt(3)*((-a*b^2)^(1/3)*b^3*c - (-a*b^2)^(1/3)*a*b^2*d + ( -a*b^2)^(1/3)*a^2*b*e - (-a*b^2)^(1/3)*a^3*f)*arctan(1/3*sqrt(3)*(2*x + (- a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b) - 1/6*((-a*b^2)^(1/3)*b^3*c - (-a*b^2)^( 1/3)*a*b^2*d + (-a*b^2)^(1/3)*a^2*b*e - (-a*b^2)^(1/3)*a^3*f)*log(x^2 + x* (-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b) - 1/40*(20*b^2*c*x^6 - 20*a*b*d*x^6 + 20*a^2*e*x^6 - 8*a*b*c*x^3 + 8*a^2*d*x^3 + 5*a^2*c)/(a^3*x^8)
Time = 9.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^9 \left (a+b x^3\right )} \, dx=-\frac {\frac {c}{8\,a}+\frac {x^3\,\left (a\,d-b\,c\right )}{5\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{2\,a^3}}{x^8}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{11/3}\,b^{1/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{11/3}\,b^{1/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{11/3}\,b^{1/3}} \]
(log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(b ^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a^(11/3)*b^(1/3)) - (log(b^(1/3)*x + a^(1/3))*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a^(11/3)*b^(1/3)) - (log (3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a^(11/3)*b^(1/3)) - (c/(8*a) + (x^3*(a*d - b*c))/(5*a^2) + (x^6*(b^2*c + a^2*e - a*b*d))/(2*a^3))/x^8