Integrand size = 30, antiderivative size = 224 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )} \, dx=-\frac {c}{2 a x^2}+\frac {(b e-a f) x}{b^2}+\frac {f x^4}{4 b}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} b^{7/3}}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} b^{7/3}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} b^{7/3}} \]
-1/2*c/a/x^2+(-a*f+b*e)*x/b^2+1/4*f*x^4/b-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^(5/3)/b^(7/3)+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^(5/3)/b^(7/3)+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^( 5/3)/b^(7/3)*3^(1/2)
Time = 0.11 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )} \, dx=\frac {1}{12} \left (-\frac {6 c}{a x^2}+\frac {12 (b e-a f) x}{b^2}+\frac {3 f x^4}{b}+\frac {4 \sqrt {3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3} b^{7/3}}+\frac {4 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3} b^{7/3}}+\frac {2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3} b^{7/3}}\right ) \]
((-6*c)/(a*x^2) + (12*(b*e - a*f)*x)/b^2 + (3*f*x^4)/b + (4*Sqrt[3]*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]]) /(a^(5/3)*b^(7/3)) + (4*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(a^(5/3)*b^(7/3)) + (2*(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])/(a^(5/3)*b^(7/3)))/12
Time = 0.38 (sec) , antiderivative size = 224, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \int \left (\frac {a^3 f-a^2 b e+a b^2 d-b^3 c}{a b^2 \left (a+b x^3\right )}+\frac {b e-a f}{b^2}+\frac {c}{a x^3}+\frac {f x^3}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{5/3} b^{7/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{5/3} b^{7/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{5/3} b^{7/3}}+\frac {x (b e-a f)}{b^2}-\frac {c}{2 a x^2}+\frac {f x^4}{4 b}\) |
-1/2*c/(a*x^2) + ((b*e - a*f)*x)/b^2 + (f*x^4)/(4*b) + ((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^(5/3)*b^(7/3)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(5/3)*b^(7/3)) + ((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^(5/3)*b^(7/3))
3.3.41.3.1 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 = 155, normalized size of antiderivative = 0.69
| method | result | size |
| default | \(-\frac {-\frac {1}{4} b f \,x^{4}+a f x -b e x}{b^{2}}+\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{a \,b^{2}}-\frac {c}{2 a \,x^{2}}\) | \(155\) |
| risch | \(\frac {f \,x^{4}}{4 b}-\frac {a f x}{b^{2}}+\frac {e x}{b}-\frac {c}{2 a \,x^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (-a^{9} f^{3}+3 a^{8} b e \,f^{2}-3 a^{7} b^{2} d \,f^{2}-3 a^{7} b^{2} e^{2} f +3 a^{6} b^{3} c \,f^{2}+6 a^{6} b^{3} d e f +a^{6} b^{3} e^{3}-6 a^{5} b^{4} c e f -3 a^{5} b^{4} d^{2} f -3 a^{5} b^{4} d \,e^{2}+6 a^{4} b^{5} c d f +3 a^{4} b^{5} c \,e^{2}+3 a^{4} b^{5} d^{2} e -3 a^{3} b^{6} c^{2} f -6 a^{3} b^{6} c d e -a^{3} b^{6} d^{3}+3 a^{2} b^{7} c^{2} e +3 a^{2} b^{7} c \,d^{2}-3 a \,b^{8} c^{2} d +c^{3} b^{9}+\textit {\_Z}^{3} b \,a^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (3 a^{9} f^{3}-9 a^{8} b e \,f^{2}+9 a^{7} b^{2} d \,f^{2}+9 a^{7} b^{2} e^{2} f -9 a^{6} b^{3} c \,f^{2}-18 a^{6} b^{3} d e f -3 a^{6} b^{3} e^{3}+18 a^{5} b^{4} c e f +9 a^{5} b^{4} d^{2} f +9 a^{5} b^{4} d \,e^{2}-18 a^{4} b^{5} c d f -9 a^{4} b^{5} c \,e^{2}-9 a^{4} b^{5} d^{2} e +9 a^{3} b^{6} c^{2} f +18 a^{3} b^{6} c d e +3 a^{3} b^{6} d^{3}-9 a^{2} b^{7} c^{2} e -9 a^{2} b^{7} c \,d^{2}+9 a \,b^{8} c^{2} d -3 c^{3} b^{9}-4 \textit {\_R}^{3} a^{5} b \right ) x +\left (-a^{8} f^{2}+2 a^{7} b e f -2 a^{6} b^{2} d f -a^{6} b^{2} e^{2}+2 a^{5} b^{3} c f +2 a^{5} b^{3} d e -2 a^{4} b^{4} c e -a^{4} b^{4} d^{2}+2 a^{3} b^{5} c d -a^{2} b^{6} c^{2}\right ) \textit {\_R} \right )}{3 b^{2}}\) | \(612\) |
-1/b^2*(-1/4*b*f*x^4+a*f*x-b*e*x)+(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/b^2*(a^3*f-a^2*b*e+a*b^2*d-b^3 *c)-1/2*c/a/x^2
Time = 0.31 (sec) , antiderivative size = 565, normalized size of antiderivative = 2.52 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )} \, dx=\left [\frac {3 \, a^{3} b^{2} f x^{6} - 6 \, a^{2} b^{3} c - 6 \, \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^{2} \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 ) + 2 \, {\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^{2} \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 ) - 4 \, {\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^{2} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 12 \, {\left (a^{3} b^{2} e - a^{4} b f\right )} x^{3}}{12 \, a^{3} b^{3} x^{2}}, \frac {3 \, a^{3} b^{2} f x^{6} - 6 \, a^{2} b^{3} c - 12 \, \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^{2} \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 ) + 2 \, {\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^{2} \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 ) - 4 \, {\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^{2} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 12 \, {\left (a^{3} b^{2} e - a^{4} b f\right )} x^{3}}{12 \, a^{3} b^{3} x^{2}}\right ] \]
[1/12*(3*a^3*b^2*f*x^6 - 6*a^2*b^3*c - 6*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^2*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)) + 2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^2*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) - 4*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^2*log(a*b*x + (a^ 2*b)^(2/3)) + 12*(a^3*b^2*e - a^4*b*f)*x^3)/(a^3*b^3*x^2), 1/12*(3*a^3*b^2 *f*x^6 - 6*a^2*b^3*c - 12*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4 *b*f)*x^2*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) + 2*(b^3*c - a*b^2*d + a^2*b*e - a ^3*f)*(a^2*b)^(2/3)*x^2*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) - 4*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(a^2*b)^(2/3)*x^2*log(a*b*x + (a^2* b)^(2/3)) + 12*(a^3*b^2*e - a^4*b*f)*x^3)/(a^3*b^3*x^2)]
Time = 1.46 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.46 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )} \, dx=x \left (- \frac {a f}{b^{2}} + \frac {e}{b}\right ) + \operatorname {RootSum} {\left (27 t^{3} a^{5} b^{7} - a^{9} f^{3} + 3 a^{8} b e f^{2} - 3 a^{7} b^{2} d f^{2} - 3 a^{7} b^{2} e^{2} f + 3 a^{6} b^{3} c f^{2} + 6 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} - 6 a^{5} b^{4} c e f - 3 a^{5} b^{4} d^{2} f - 3 a^{5} b^{4} d e^{2} + 6 a^{4} b^{5} c d f + 3 a^{4} b^{5} c e^{2} + 3 a^{4} b^{5} d^{2} e - 3 a^{3} b^{6} c^{2} f - 6 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 3 a^{2} b^{7} c^{2} e + 3 a^{2} b^{7} c d^{2} - 3 a b^{8} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {3 t a^{2} b^{2}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac {f x^{4}}{4 b} - \frac {c}{2 a x^{2}} \]
x*(-a*f/b**2 + e/b) + RootSum(27*_t**3*a**5*b**7 - 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 + b**9*c**3, Lambda(_t, _t*log(3*_t*a**2*b**2/(a**3*f - a**2*b*e + a*b**2*d - b**3*c) + x))) + f* x**4/(4*b) - c/(2*a*x**2)
Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )} \, dx=\frac {b f x^{4} + 4 \, {\left (b e - a f\right )} x}{4 \, b^{2}} - \frac {c}{2 \, a x^{2}} - \frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/4*(b*f*x^4 + 4*(b*e - a*f)*x)/b^2 - 1/2*c/(a*x^2) - 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*b^3*(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*b^3*(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*b^3*(a/b)^(2/3))
Time = 0.27 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.02 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )} \, dx=\frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2} b^{2}} - \frac {c}{2 \, a x^{2}} + \frac {b^{3} f x^{4} + 4 \, b^{3} e x - 4 \, a b^{2} f x}{4 \, b^{4}} \]
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*b^2)^(2/3)*a*b) + 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*b^2)^(2/3)* a*b) + 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^2*b^2) - 1/2*c/(a*x^2) + 1/4*(b^3*f*x^4 + 4*b^3*e*x - 4*a* b^2*f*x)/b^4
Time = 0.29 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^3 \left (a+b x^3\right )} \, dx=x\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )-\frac {c}{2\,a\,x^2}+\frac {f\,x^4}{4\,b}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{5/3}\,b^{7/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{5/3}\,b^{7/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{5/3}\,b^{7/3}} \]
x*(e/b - (a*f)/b^2) - c/(2*a*x^2) + (f*x^4)/(4*b) - (log(b^(1/3)*x + a^(1/ 3))*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a^(5/3)*b^(7/3)) - (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^(5/3)*b^(7/3)) + (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^(5/3)*b^(7/3))