Integrand size = 30, antiderivative size = 227 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=-\frac {c}{a x}+\frac {(b e-a f) x^2}{2 b^2}+\frac {f x^5}{5 b}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} b^{8/3}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} b^{8/3}}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} b^{8/3}} \]
-c/a/x+1/2*(-a*f+b*e)*x^2/b^2+1/5*f*x^5/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^(4/3)/b^(8/3)-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^(4/3)/b^(8/3)+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^( 4/3)/b^(8/3)*3^(1/2)
Time = 0.11 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=\frac {-30 \sqrt [3]{a} b^{8/3} c+15 a^{4/3} b^{2/3} (b e-a f) x^3+6 a^{4/3} b^{5/3} f x^6+10 \sqrt {3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+10 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-5 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{30 a^{4/3} b^{8/3} x} \]
(-30*a^(1/3)*b^(8/3)*c + 15*a^(4/3)*b^(2/3)*(b*e - a*f)*x^3 + 6*a^(4/3)*b^ (5/3)*f*x^6 + 10*Sqrt[3]*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 10*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)* x*Log[a^(1/3) + b^(1/3)*x] - 5*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x*Log[a ^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(30*a^(4/3)*b^(8/3)*x)
Time = 0.40 (sec) , antiderivative size = 227, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \int \left (\frac {x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a b^2 \left (a+b x^3\right )}+\frac {x (b e-a f)}{b^2}+\frac {c}{a x^2}+\frac {f x^4}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{4/3} b^{8/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{4/3} b^{8/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{4/3} b^{8/3}}+\frac {x^2 (b e-a f)}{2 b^2}-\frac {c}{a x}+\frac {f x^5}{5 b}\) |
-(c/(a*x)) + ((b*e - a*f)*x^2)/(2*b^2) + (f*x^5)/(5*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))])/(Sq rt[3]*a^(4/3)*b^(8/3)) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(4/3)*b^(8/3)) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*L og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(4/3)*b^(8/3))
3.3.40.3.1 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 = 1.54 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.70
| method | result | size |
| default | \(-\frac {-\frac {b f \,x^{5}}{5}+\frac {\left (a f -b e \right ) x^{2}}{2}}{b^{2}}+\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{a \,b^{2}}-\frac {c}{a x}\) | \(159\) |
| risch | \(\frac {f \,x^{5}}{5 b}-\frac {x^{2} a f}{2 b^{2}}+\frac {e \,x^{2}}{2 b}-\frac {c}{a x}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{9} f^{3}-3 a^{8} b e \,f^{2}+3 a^{7} b^{2} d \,f^{2}+3 a^{7} b^{2} e^{2} f -3 a^{6} b^{3} c \,f^{2}-6 a^{6} b^{3} d e f -a^{6} b^{3} e^{3}+6 a^{5} b^{4} c e f +3 a^{5} b^{4} d^{2} f +3 a^{5} b^{4} d \,e^{2}-6 a^{4} b^{5} c d f -3 a^{4} b^{5} c \,e^{2}-3 a^{4} b^{5} d^{2} e +3 a^{3} b^{6} c^{2} f +6 a^{3} b^{6} c d e +a^{3} b^{6} d^{3}-3 a^{2} b^{7} c^{2} e -3 a^{2} b^{7} c \,d^{2}+3 a \,b^{8} c^{2} d -c^{3} b^{9}+\textit {\_Z}^{3} b^{2} a^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (-3 a^{9} f^{3}+9 a^{8} b e \,f^{2}-9 a^{7} b^{2} d \,f^{2}-9 a^{7} b^{2} e^{2} f +9 a^{6} b^{3} c \,f^{2}+18 a^{6} b^{3} d e f +3 a^{6} b^{3} e^{3}-18 a^{5} b^{4} c e f -9 a^{5} b^{4} d^{2} f -9 a^{5} b^{4} d \,e^{2}+18 a^{4} b^{5} c d f +9 a^{4} b^{5} c \,e^{2}+9 a^{4} b^{5} d^{2} e -9 a^{3} b^{6} c^{2} f -18 a^{3} b^{6} c d e -3 a^{3} b^{6} d^{3}+9 a^{2} b^{7} c^{2} e +9 a^{2} b^{7} c \,d^{2}-9 a \,b^{8} c^{2} d +3 c^{3} b^{9}-4 \textit {\_R}^{3} a^{4} b^{2}\right ) x +\left (a^{6} b f -a^{5} b^{2} e +a^{4} b^{3} d -a^{3} b^{4} c \right ) \textit {\_R}^{2}\right )}{3 b^{2}}\) | \(556\) |
-1/b^2*(-1/5*b*f*x^5+1/2*(a*f-b*e)*x^2)+(-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)))*(a^3*f-a^2*b*e+a*b^2*d-b^ 3*c)/a/b^2-c/a/x
Time = 0.31 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.47 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=\left [\frac {6 \, a^{2} b^{3} f x^{6} - 30 \, a b^{4} c + 15 \, {\left (a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{3} - 15 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) - 5 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a b^{2}\right )^{\frac {2}{3}} x \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 10 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a b^{2}\right )^{\frac {2}{3}} x \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{30 \, a^{2} b^{4} x}, \frac {6 \, a^{2} b^{3} f x^{6} - 30 \, a b^{4} c + 15 \, {\left (a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{3} - 30 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - 5 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a b^{2}\right )^{\frac {2}{3}} x \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 10 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a b^{2}\right )^{\frac {2}{3}} x \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{30 \, a^{2} b^{4} x}\right ] \]
[1/30*(6*a^2*b^3*f*x^6 - 30*a*b^4*c + 15*(a^2*b^3*e - a^3*b^2*f)*x^3 - 15* sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x*sqrt((-a*b^2)^(1/3 )/a)*log((2*b^2*x^3 - a*b + 3*sqrt(1/3)*(a*b*x + 2*(-a*b^2)^(2/3)*x^2 + (- a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2/3)*x)/(b*x^3 + a)) - 5*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a*b^2)^(2/3)*x*log(b^2*x^2 + (-a *b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) + 10*(b^3*c - a*b^2*d + a^2*b*e - a^3*f) *(-a*b^2)^(2/3)*x*log(b*x - (-a*b^2)^(1/3)))/(a^2*b^4*x), 1/30*(6*a^2*b^3* f*x^6 - 30*a*b^4*c + 15*(a^2*b^3*e - a^3*b^2*f)*x^3 - 30*sqrt(1/3)*(a*b^4* c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt (1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) - 5*(b^3*c - a*b ^2*d + a^2*b*e - a^3*f)*(-a*b^2)^(2/3)*x*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) + 10*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a*b^2)^(2/3)* x*log(b*x - (-a*b^2)^(1/3)))/(a^2*b^4*x)]
Time = 1.22 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.80 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=x^{2} \left (- \frac {a f}{2 b^{2}} + \frac {e}{2 b}\right ) + \operatorname {RootSum} {\left (27 t^{3} a^{4} b^{8} + a^{9} f^{3} - 3 a^{8} b e f^{2} + 3 a^{7} b^{2} d f^{2} + 3 a^{7} b^{2} e^{2} f - 3 a^{6} b^{3} c f^{2} - 6 a^{6} b^{3} d e f - a^{6} b^{3} e^{3} + 6 a^{5} b^{4} c e f + 3 a^{5} b^{4} d^{2} f + 3 a^{5} b^{4} d e^{2} - 6 a^{4} b^{5} c d f - 3 a^{4} b^{5} c e^{2} - 3 a^{4} b^{5} d^{2} e + 3 a^{3} b^{6} c^{2} f + 6 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} - 3 a^{2} b^{7} c^{2} e - 3 a^{2} b^{7} c d^{2} + 3 a b^{8} c^{2} d - b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {9 t^{2} a^{3} b^{5}}{a^{6} f^{2} - 2 a^{5} b e f + 2 a^{4} b^{2} d f + a^{4} b^{2} e^{2} - 2 a^{3} b^{3} c f - 2 a^{3} b^{3} d e + 2 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 2 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{5}}{5 b} - \frac {c}{a x} \]
x**2*(-a*f/(2*b**2) + e/(2*b)) + RootSum(27*_t**3*a**4*b**8 + 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(9*_t**2*a**3*b**5/(a**6*f**2 - 2*a**5*b*e*f + 2*a**4*b* *2*d*f + a**4*b**2*e**2 - 2*a**3*b**3*c*f - 2*a**3*b**3*d*e + 2*a**2*b**4* c*e + a**2*b**4*d**2 - 2*a*b**5*c*d + b**6*c**2) + x))) + f*x**5/(5*b) - c /(a*x)
Time = 0.29 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=\frac {2 \, b f x^{5} + 5 \, {\left (b e - a f\right )} x^{2}}{10 \, b^{2}} - \frac {c}{a x} - \frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
1/10*(2*b*f*x^5 + 5*(b*e - a*f)*x^2)/b^2 - c/(a*x) - 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)^(1/3)) - 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)^(1/3)) + 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)^(1/3))
Time = 0.27 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.17 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=-\frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2}} - \frac {c}{a x} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2}} + \frac {{\left (b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2} b^{2}} + \frac {2 \, b^{4} f x^{5} + 5 \, b^{4} e x^{2} - 5 \, a b^{3} f x^{2}}{10 \, b^{5}} \]
-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)^(1/3)*a*b^2) - c/(a*x) + 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)^(1/3)*a*b^2) + 1/3*(b^3*c*(-a/b)^(1/3) - a*b^2*d*(-a/b)^(1/3) + a^2 *b*e*(-a/b)^(1/3) - a^3*f*(-a/b)^(1/3))*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1 /3)))/(a^2*b^2) + 1/10*(2*b^4*f*x^5 + 5*b^4*e*x^2 - 5*a*b^3*f*x^2)/b^5
Time = 9.50 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^2 \left (a+b x^3\right )} \, dx=x^2\,\left (\frac {e}{2\,b}-\frac {a\,f}{2\,b^2}\right )-\frac {c}{a\,x}+\frac {f\,x^5}{5\,b}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{4/3}\,b^{8/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{4/3}\,b^{8/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{4/3}\,b^{8/3}} \]
x^2*(e/(2*b) - (a*f)/(2*b^2)) - c/(a*x) + (f*x^5)/(5*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^(4/3)*b^(8/3)) - (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^(4/3)*b^(8/3)) + (log(3^(1/2)*a^(1/3)*1 i - 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^(4/3)*b^(8/3))