Integrand size = 30, antiderivative size = 242 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=-\frac {c}{7 a x^7}+\frac {b c-a d}{4 a^2 x^4}-\frac {b^2 c-a b d+a^2 e}{a^3 x}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{10/3} b^{2/3}}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} b^{2/3}}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} b^{2/3}} \]
-1/7*c/a/x^7+1/4*(-a*d+b*c)/a^2/x^4+(-a^2*e+a*b*d-b^2*c)/a^3/x+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^(10/3)/b^(2/3)-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^(10/3) /b^(2/3)+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^(10/3)/b^(2/3)*3^(1/2)
Time = 0.12 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=\frac {-\frac {12 a^{7/3} c}{x^7}+\frac {21 a^{4/3} (b c-a d)}{x^4}-\frac {84 \sqrt [3]{a} \left (b^2 c-a b d+a^2 e\right )}{x}+\frac {28 \sqrt {3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+\frac {28 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {14 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}}{84 a^{10/3}} \]
((-12*a^(7/3)*c)/x^7 + (21*a^(4/3)*(b*c - a*d))/x^4 - (84*a^(1/3)*(b^2*c - a*b*d + a^2*e))/x + (28*Sqrt[3]*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTa n[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (28*(b^3*c - a*b^2*d + a ^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (14*(-(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^(2/ 3))/(84*a^(10/3))
Time = 0.42 (sec) , antiderivative size = 242, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \int \left (\frac {a d-b c}{a^2 x^5}+\frac {a^2 e-a b d+b^2 c}{a^3 x^2}+\frac {x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^3 \left (a+b x^3\right )}+\frac {c}{a x^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b c-a d}{4 a^2 x^4}-\frac {a^2 e-a b d+b^2 c}{a^3 x}+\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{10/3} b^{2/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{10/3} b^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{10/3} b^{2/3}}-\frac {c}{7 a x^7}\) |
-1/7*c/(a*x^7) + (b*c - a*d)/(4*a^2*x^4) - (b^2*c - a*b*d + a^2*e)/(a^3*x) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sq rt[3]*a^(1/3))])/(Sqrt[3]*a^(10/3)*b^(2/3)) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(10/3)*b^(2/3)) - ((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^ (10/3)*b^(2/3))
3.3.44.3.1 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 = 1.53 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.70
| method | result | size |
| default | \(-\frac {c}{7 a \,x^{7}}-\frac {a d -b c}{4 a^{2} x^{4}}-\frac {a^{2} e -a b d +b^{2} c}{a^{3} x}+\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{a^{3}}\) | \(170\) |
| risch | \(\frac {-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{6}}{a^{3}}-\frac {\left (a d -b c \right ) x^{3}}{4 a^{2}}-\frac {c}{7 a}}{x^{7}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{10} b^{2} \textit {\_Z}^{3}+a^{9} f^{3}-3 a^{8} b e \,f^{2}+3 a^{7} b^{2} d \,f^{2}+3 a^{7} b^{2} e^{2} f -3 a^{6} b^{3} c \,f^{2}-6 a^{6} b^{3} d e f -a^{6} b^{3} e^{3}+6 a^{5} b^{4} c e f +3 a^{5} b^{4} d^{2} f +3 a^{5} b^{4} d \,e^{2}-6 a^{4} b^{5} c d f -3 a^{4} b^{5} c \,e^{2}-3 a^{4} b^{5} d^{2} e +3 a^{3} b^{6} c^{2} f +6 a^{3} b^{6} c d e +a^{3} b^{6} d^{3}-3 a^{2} b^{7} c^{2} e -3 a^{2} b^{7} c \,d^{2}+3 a \,b^{8} c^{2} d -c^{3} b^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{10} b^{2}-3 a^{9} f^{3}+9 a^{8} b e \,f^{2}-9 a^{7} b^{2} d \,f^{2}-9 a^{7} b^{2} e^{2} f +9 a^{6} b^{3} c \,f^{2}+18 a^{6} b^{3} d e f +3 a^{6} b^{3} e^{3}-18 a^{5} b^{4} c e f -9 a^{5} b^{4} d^{2} f -9 a^{5} b^{4} d \,e^{2}+18 a^{4} b^{5} c d f +9 a^{4} b^{5} c \,e^{2}+9 a^{4} b^{5} d^{2} e -9 a^{3} b^{6} c^{2} f -18 a^{3} b^{6} c d e -3 a^{3} b^{6} d^{3}+9 a^{2} b^{7} c^{2} e +9 a^{2} b^{7} c \,d^{2}-9 a \,b^{8} c^{2} d +3 c^{3} b^{9}\right ) x +\left (a^{10} b f -a^{9} b^{2} e +a^{8} b^{3} d -a^{7} b^{4} c \right ) \textit {\_R}^{2}\right )\right )}{3}\) | \(567\) |
-1/7*c/a/x^7-1/4*(a*d-b*c)/a^2/x^4-(a^2*e-a*b*d+b^2*c)/a^3/x+(-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* (a^3*f-a^2*b*e+a*b^2*d-b^3*c)
Time = 0.31 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.52 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=\left [-\frac {42 \, \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^{7} \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 ) + 14 \, {\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^{7} \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 ) - 28 \, {\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^{7} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right ) + 84 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{6} + 12 \, a^{3} b^{2} c - 21 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{3}}{84 \, a^{4} b^{2} x^{7}}, -\frac {84 \, \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^{7} \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 ) + 14 \, {\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^{7} \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 ) - 28 \, {\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^{7} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right ) + 84 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{6} + 12 \, a^{3} b^{2} c - 21 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{3}}{84 \, a^{4} b^{2} x^{7}}\right ] \]
[-1/84*(42*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^7*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)) + 14*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a*b^2)^(2/3)*x^7*l og(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 28*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a*b^2)^(2/3)*x^7*log(b*x - (-a*b^2)^(1/3)) + 84*(a*b^4* c - a^2*b^3*d + a^3*b^2*e)*x^6 + 12*a^3*b^2*c - 21*(a^2*b^3*c - a^3*b^2*d) *x^3)/(a^4*b^2*x^7), -1/84*(84*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^7*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) + 14*(b^3*c - a*b^2*d + a^2*b*e - a^3*f) *(-a*b^2)^(2/3)*x^7*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 2 8*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a*b^2)^(2/3)*x^7*log(b*x - (-a*b^2 )^(1/3)) + 84*(a*b^4*c - a^2*b^3*d + a^3*b^2*e)*x^6 + 12*a^3*b^2*c - 21*(a ^2*b^3*c - a^3*b^2*d)*x^3)/(a^4*b^2*x^7)]
Time = 43.35 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.79 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{10} b^{2} + a^{9} f^{3} - 3 a^{8} b e f^{2} + 3 a^{7} b^{2} d f^{2} + 3 a^{7} b^{2} e^{2} f - 3 a^{6} b^{3} c f^{2} - 6 a^{6} b^{3} d e f - a^{6} b^{3} e^{3} + 6 a^{5} b^{4} c e f + 3 a^{5} b^{4} d^{2} f + 3 a^{5} b^{4} d e^{2} - 6 a^{4} b^{5} c d f - 3 a^{4} b^{5} c e^{2} - 3 a^{4} b^{5} d^{2} e + 3 a^{3} b^{6} c^{2} f + 6 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} - 3 a^{2} b^{7} c^{2} e - 3 a^{2} b^{7} c d^{2} + 3 a b^{8} c^{2} d - b^{9} c^{3}, \left ( t \mapsto t \log {\left (\frac {9 t^{2} a^{7} b}{a^{6} f^{2} - 2 a^{5} b e f + 2 a^{4} b^{2} d f + a^{4} b^{2} e^{2} - 2 a^{3} b^{3} c f - 2 a^{3} b^{3} d e + 2 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 2 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac {- 4 a^{2} c + x^{6} \left (- 28 a^{2} e + 28 a b d - 28 b^{2} c\right ) + x^{3} \left (- 7 a^{2} d + 7 a b c\right )}{28 a^{3} x^{7}} \]
RootSum(27*_t**3*a**10*b**2 + 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**7* b/(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))) + (-4*a**2*c + x**6*(-28*a**2*e + 28*a*b*d - 28*b**2* c) + x**3*(-7*a**2*d + 7*a*b*c))/(28*a**3*x**7)
Time = 0.28 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=-\frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {28 \, {\left (b^{2} c - a b d + a^{2} e\right )} x^{6} - 7 \, {\left (a b c - a^{2} d\right )} x^{3} + 4 \, a^{2} c}{28 \, a^{3} x^{7}} \]
-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)^(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^3*b*(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^3*b*(a/b )^(1/3)) - 1/28*(28*(b^2*c - a*b*d + a^2*e)*x^6 - 7*(a*b*c - a^2*d)*x^3 + 4*a^2*c)/(a^3*x^7)
Time = 0.28 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.12 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=-\frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3}} + \frac {{\left (b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b e \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{4}} - \frac {28 \, b^{2} c x^{6} - 28 \, a b d x^{6} + 28 \, a^{2} e x^{6} - 7 \, a b c x^{3} + 7 \, a^{2} d x^{3} + 4 \, a^{2} c}{28 \, a^{3} x^{7}} \]
-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^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^2)^(1/3) *a^3) + 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^4 - 1/ 28*(28*b^2*c*x^6 - 28*a*b*d*x^6 + 28*a^2*e*x^6 - 7*a*b*c*x^3 + 7*a^2*d*x^3 + 4*a^2*c)/(a^3*x^7)
Time = 9.26 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^8 \left (a+b x^3\right )} \, dx=\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{10/3}\,b^{2/3}}-\frac {\frac {c}{7\,a}+\frac {x^3\,\left (a\,d-b\,c\right )}{4\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{a^3}}{x^7}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{10/3}\,b^{2/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{10/3}\,b^{2/3}} \]
(log(b^(1/3)*x + a^(1/3))*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a^(10/3) *b^(2/3)) - (c/(7*a) + (x^3*(a*d - b*c))/(4*a^2) + (x^6*(b^2*c + a^2*e - a *b*d))/a^3)/x^7 - (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^(10/3)*b^(2/3)) + (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^(10/3)*b^(2/3))