Integrand size = 30, antiderivative size = 220 \[ \int \frac {x^{11} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x^3}{3 b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^6}{6 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^9}{9 b^4}+\frac {(b e-2 a f) x^{12}}{12 b^3}+\frac {f x^{15}}{15 b^2}+\frac {a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 b^7 \left (a+b x^3\right )}+\frac {a^2 \left (3 b^3 c-4 a b^2 d+5 a^2 b e-6 a^3 f\right ) \log \left (a+b x^3\right )}{3 b^7} \]
-1/3*a*(-5*a^3*f+4*a^2*b*e-3*a*b^2*d+2*b^3*c)*x^3/b^6+1/6*(-4*a^3*f+3*a^2* b*e-2*a*b^2*d+b^3*c)*x^6/b^5+1/9*(3*a^2*f-2*a*b*e+b^2*d)*x^9/b^4+1/12*(-2* a*f+b*e)*x^12/b^3+1/15*f*x^15/b^2+1/3*a^3*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/b ^7/(b*x^3+a)+1/3*a^2*(-6*a^3*f+5*a^2*b*e-4*a*b^2*d+3*b^3*c)*ln(b*x^3+a)/b^ 7
Time = 0.14 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.93 \[ \int \frac {x^{11} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {60 a b \left (-2 b^3 c+3 a b^2 d-4 a^2 b e+5 a^3 f\right ) x^3+30 b^2 \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^6+20 b^3 \left (b^2 d-2 a b e+3 a^2 f\right ) x^9+15 b^4 (b e-2 a f) x^{12}+12 b^5 f x^{15}-\frac {60 a^3 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a+b x^3}+60 a^2 \left (3 b^3 c-4 a b^2 d+5 a^2 b e-6 a^3 f\right ) \log \left (a+b x^3\right )}{180 b^7} \]
(60*a*b*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f)*x^3 + 30*b^2*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^6 + 20*b^3*(b^2*d - 2*a*b*e + 3*a^2*f) *x^9 + 15*b^4*(b*e - 2*a*f)*x^12 + 12*b^5*f*x^15 - (60*a^3*(-(b^3*c) + a*b ^2*d - a^2*b*e + a^3*f))/(a + b*x^3) + 60*a^2*(3*b^3*c - 4*a*b^2*d + 5*a^2 *b*e - 6*a^3*f)*Log[a + b*x^3])/(180*b^7)
Time = 0.54 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2361, 2123, 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.
| \(\displaystyle \int \frac {x^{11} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2361 |
| \(\displaystyle \frac {1}{3} \int \frac {x^9 \left (f x^9+e x^6+d x^3+c\right )}{\left (b x^3+a\right )^2}dx^3\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \frac {1}{3} \int \left (\frac {f x^{12}}{b^2}+\frac {(b e-2 a f) x^9}{b^3}+\frac {\left (3 f a^2-2 b e a+b^2 d\right ) x^6}{b^4}+\frac {\left (-4 f a^3+3 b e a^2-2 b^2 d a+b^3 c\right ) x^3}{b^5}+\frac {a \left (5 f a^3-4 b e a^2+3 b^2 d a-2 b^3 c\right )}{b^6}-\frac {a^2 \left (6 f a^3-5 b e a^2+4 b^2 d a-3 b^3 c\right )}{b^6 \left (b x^3+a\right )}+\frac {a^3 \left (f a^3-b e a^2+b^2 d a-b^3 c\right )}{b^6 \left (b x^3+a\right )^2}\right )dx^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (\frac {x^9 \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac {a^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^7 \left (a+b x^3\right )}+\frac {a^2 \log \left (a+b x^3\right ) \left (-6 a^3 f+5 a^2 b e-4 a b^2 d+3 b^3 c\right )}{b^7}-\frac {a x^3 \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{b^6}+\frac {x^6 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{2 b^5}+\frac {x^{12} (b e-2 a f)}{4 b^3}+\frac {f x^{15}}{5 b^2}\right )\) |
(-((a*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f)*x^3)/b^6) + ((b^3*c - 2* a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^6)/(2*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f) *x^9)/(3*b^4) + ((b*e - 2*a*f)*x^12)/(4*b^3) + (f*x^15)/(5*b^2) + (a^3*(b^ 3*c - a*b^2*d + a^2*b*e - a^3*f))/(b^7*(a + b*x^3)) + (a^2*(3*b^3*c - 4*a* b^2*d + 5*a^2*b*e - 6*a^3*f)*Log[a + b*x^3])/b^7)/3
3.3.51.3.1 Defintions of rubi rules used
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*SubstFor[x^n, Pq, x]*(a + b*x)^p, x ], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && PolyQ[Pq, x^n] && IntegerQ[S implify[(m + 1)/n]]
Time = 1.67 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(\frac {\frac {f \,x^{15} b^{4}}{15}+\frac {\left (-2 a \,b^{3} f +b^{4} e \right ) x^{12}}{12}+\frac {\left (3 a^{2} b^{2} f -2 a \,b^{3} e +b^{4} d \right ) x^{9}}{9}+\frac {\left (-4 a^{3} b f +3 a^{2} e \,b^{2}-2 a \,b^{3} d +b^{4} c \right ) x^{6}}{6}+\frac {\left (5 a^{4} f -4 a^{3} b e +3 a^{2} b^{2} d -2 a \,b^{3} c \right ) x^{3}}{3}}{b^{6}}-\frac {a^{2} \left (\frac {\left (6 f \,a^{3}-5 a^{2} b e +4 a \,b^{2} d -3 b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{b}+\frac {a \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{b \left (b \,x^{3}+a \right )}\right )}{3 b^{6}}\) | \(217\) |
| norman | \(\frac {-\frac {a \left (6 f \,a^{5}-5 a^{4} e b +4 a^{3} d \,b^{2}-3 a^{2} c \,b^{3}\right )}{3 b^{7}}+\frac {f \,x^{18}}{15 b}-\frac {\left (6 a f -5 b e \right ) x^{15}}{60 b^{2}}+\frac {\left (6 a^{2} f -5 a e b +4 b^{2} d \right ) x^{12}}{36 b^{3}}-\frac {\left (6 f \,a^{3}-5 a^{2} b e +4 a \,b^{2} d -3 b^{3} c \right ) x^{9}}{18 b^{4}}+\frac {a \left (6 f \,a^{3}-5 a^{2} b e +4 a \,b^{2} d -3 b^{3} c \right ) x^{6}}{6 b^{5}}}{b \,x^{3}+a}-\frac {a^{2} \left (6 f \,a^{3}-5 a^{2} b e +4 a \,b^{2} d -3 b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 b^{7}}\) | \(217\) |
| risch | \(\frac {f \,x^{15}}{15 b^{2}}-\frac {a f \,x^{12}}{6 b^{3}}+\frac {e \,x^{12}}{12 b^{2}}+\frac {a^{2} f \,x^{9}}{3 b^{4}}-\frac {2 a e \,x^{9}}{9 b^{3}}+\frac {d \,x^{9}}{9 b^{2}}-\frac {2 f \,x^{6} a^{3}}{3 b^{5}}+\frac {a^{2} e \,x^{6}}{2 b^{4}}-\frac {a d \,x^{6}}{3 b^{3}}+\frac {c \,x^{6}}{6 b^{2}}+\frac {5 a^{4} f \,x^{3}}{3 b^{6}}-\frac {4 a^{3} e \,x^{3}}{3 b^{5}}+\frac {a^{2} d \,x^{3}}{b^{4}}-\frac {2 a c \,x^{3}}{3 b^{3}}-\frac {a^{6} f}{3 b^{7} \left (b \,x^{3}+a \right )}+\frac {a^{5} e}{3 b^{6} \left (b \,x^{3}+a \right )}-\frac {a^{4} d}{3 b^{5} \left (b \,x^{3}+a \right )}+\frac {a^{3} c}{3 b^{4} \left (b \,x^{3}+a \right )}-\frac {2 a^{5} \ln \left (b \,x^{3}+a \right ) f}{b^{7}}+\frac {5 a^{4} \ln \left (b \,x^{3}+a \right ) e}{3 b^{6}}-\frac {4 a^{3} \ln \left (b \,x^{3}+a \right ) d}{3 b^{5}}+\frac {a^{2} \ln \left (b \,x^{3}+a \right ) c}{b^{4}}\) | \(288\) |
| parallelrisch | \(-\frac {-12 f \,x^{18} b^{6}+360 a^{6} f -15 x^{15} b^{6} e -20 x^{12} b^{6} d -30 x^{9} b^{6} c +360 \ln \left (b \,x^{3}+a \right ) a^{6} f -180 a^{3} b^{3} c -50 x^{9} a^{2} b^{4} e +40 x^{9} a \,b^{5} d -180 x^{6} a^{4} b^{2} f +150 x^{6} a^{3} b^{3} e -120 x^{6} a^{2} b^{4} d +90 x^{6} a \,b^{5} c +240 a^{4} b^{2} d -300 a^{5} b e +360 \ln \left (b \,x^{3}+a \right ) x^{3} a^{5} b f -300 \ln \left (b \,x^{3}+a \right ) x^{3} a^{4} b^{2} e +240 \ln \left (b \,x^{3}+a \right ) x^{3} a^{3} b^{3} d -180 \ln \left (b \,x^{3}+a \right ) x^{3} a^{2} b^{4} c +18 x^{15} a \,b^{5} f -30 x^{12} a^{2} b^{4} f +25 x^{12} a \,b^{5} e +60 x^{9} a^{3} b^{3} f -300 \ln \left (b \,x^{3}+a \right ) a^{5} b e +240 \ln \left (b \,x^{3}+a \right ) a^{4} b^{2} d -180 \ln \left (b \,x^{3}+a \right ) a^{3} b^{3} c}{180 b^{7} \left (b \,x^{3}+a \right )}\) | \(336\) |
1/b^6*(1/15*f*x^15*b^4+1/12*(-2*a*b^3*f+b^4*e)*x^12+1/9*(3*a^2*b^2*f-2*a*b ^3*e+b^4*d)*x^9+1/6*(-4*a^3*b*f+3*a^2*b^2*e-2*a*b^3*d+b^4*c)*x^6+1/3*(5*a^ 4*f-4*a^3*b*e+3*a^2*b^2*d-2*a*b^3*c)*x^3)-1/3*a^2/b^6*((6*a^3*f-5*a^2*b*e+ 4*a*b^2*d-3*b^3*c)/b*ln(b*x^3+a)+a*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/b/(b*x^3+ a))
Time = 0.28 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.38 \[ \int \frac {x^{11} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {12 \, b^{6} f x^{18} + 3 \, {\left (5 \, b^{6} e - 6 \, a b^{5} f\right )} x^{15} + 5 \, {\left (4 \, b^{6} d - 5 \, a b^{5} e + 6 \, a^{2} b^{4} f\right )} x^{12} + 10 \, {\left (3 \, b^{6} c - 4 \, a b^{5} d + 5 \, a^{2} b^{4} e - 6 \, a^{3} b^{3} f\right )} x^{9} + 60 \, a^{3} b^{3} c - 60 \, a^{4} b^{2} d + 60 \, a^{5} b e - 60 \, a^{6} f - 30 \, {\left (3 \, a b^{5} c - 4 \, a^{2} b^{4} d + 5 \, a^{3} b^{3} e - 6 \, a^{4} b^{2} f\right )} x^{6} - 60 \, {\left (2 \, a^{2} b^{4} c - 3 \, a^{3} b^{3} d + 4 \, a^{4} b^{2} e - 5 \, a^{5} b f\right )} x^{3} + 60 \, {\left (3 \, a^{3} b^{3} c - 4 \, a^{4} b^{2} d + 5 \, a^{5} b e - 6 \, a^{6} f + {\left (3 \, a^{2} b^{4} c - 4 \, a^{3} b^{3} d + 5 \, a^{4} b^{2} e - 6 \, a^{5} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{180 \, {\left (b^{8} x^{3} + a b^{7}\right )}} \]
1/180*(12*b^6*f*x^18 + 3*(5*b^6*e - 6*a*b^5*f)*x^15 + 5*(4*b^6*d - 5*a*b^5 *e + 6*a^2*b^4*f)*x^12 + 10*(3*b^6*c - 4*a*b^5*d + 5*a^2*b^4*e - 6*a^3*b^3 *f)*x^9 + 60*a^3*b^3*c - 60*a^4*b^2*d + 60*a^5*b*e - 60*a^6*f - 30*(3*a*b^ 5*c - 4*a^2*b^4*d + 5*a^3*b^3*e - 6*a^4*b^2*f)*x^6 - 60*(2*a^2*b^4*c - 3*a ^3*b^3*d + 4*a^4*b^2*e - 5*a^5*b*f)*x^3 + 60*(3*a^3*b^3*c - 4*a^4*b^2*d + 5*a^5*b*e - 6*a^6*f + (3*a^2*b^4*c - 4*a^3*b^3*d + 5*a^4*b^2*e - 6*a^5*b*f )*x^3)*log(b*x^3 + a))/(b^8*x^3 + a*b^7)
Time = 13.53 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.07 \[ \int \frac {x^{11} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=- \frac {a^{2} \cdot \left (6 a^{3} f - 5 a^{2} b e + 4 a b^{2} d - 3 b^{3} c\right ) \log {\left (a + b x^{3} \right )}}{3 b^{7}} + x^{12} \left (- \frac {a f}{6 b^{3}} + \frac {e}{12 b^{2}}\right ) + x^{9} \left (\frac {a^{2} f}{3 b^{4}} - \frac {2 a e}{9 b^{3}} + \frac {d}{9 b^{2}}\right ) + x^{6} \left (- \frac {2 a^{3} f}{3 b^{5}} + \frac {a^{2} e}{2 b^{4}} - \frac {a d}{3 b^{3}} + \frac {c}{6 b^{2}}\right ) + x^{3} \cdot \left (\frac {5 a^{4} f}{3 b^{6}} - \frac {4 a^{3} e}{3 b^{5}} + \frac {a^{2} d}{b^{4}} - \frac {2 a c}{3 b^{3}}\right ) + \frac {- a^{6} f + a^{5} b e - a^{4} b^{2} d + a^{3} b^{3} c}{3 a b^{7} + 3 b^{8} x^{3}} + \frac {f x^{15}}{15 b^{2}} \]
-a**2*(6*a**3*f - 5*a**2*b*e + 4*a*b**2*d - 3*b**3*c)*log(a + b*x**3)/(3*b **7) + x**12*(-a*f/(6*b**3) + e/(12*b**2)) + x**9*(a**2*f/(3*b**4) - 2*a*e /(9*b**3) + d/(9*b**2)) + x**6*(-2*a**3*f/(3*b**5) + a**2*e/(2*b**4) - a*d /(3*b**3) + c/(6*b**2)) + x**3*(5*a**4*f/(3*b**6) - 4*a**3*e/(3*b**5) + a* *2*d/b**4 - 2*a*c/(3*b**3)) + (-a**6*f + a**5*b*e - a**4*b**2*d + a**3*b** 3*c)/(3*a*b**7 + 3*b**8*x**3) + f*x**15/(15*b**2)
Time = 0.21 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.01 \[ \int \frac {x^{11} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {a^{3} b^{3} c - a^{4} b^{2} d + a^{5} b e - a^{6} f}{3 \, {\left (b^{8} x^{3} + a b^{7}\right )}} + \frac {12 \, b^{4} f x^{15} + 15 \, {\left (b^{4} e - 2 \, a b^{3} f\right )} x^{12} + 20 \, {\left (b^{4} d - 2 \, a b^{3} e + 3 \, a^{2} b^{2} f\right )} x^{9} + 30 \, {\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{6} - 60 \, {\left (2 \, a b^{3} c - 3 \, a^{2} b^{2} d + 4 \, a^{3} b e - 5 \, a^{4} f\right )} x^{3}}{180 \, b^{6}} + \frac {{\left (3 \, a^{2} b^{3} c - 4 \, a^{3} b^{2} d + 5 \, a^{4} b e - 6 \, a^{5} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{7}} \]
1/3*(a^3*b^3*c - a^4*b^2*d + a^5*b*e - a^6*f)/(b^8*x^3 + a*b^7) + 1/180*(1 2*b^4*f*x^15 + 15*(b^4*e - 2*a*b^3*f)*x^12 + 20*(b^4*d - 2*a*b^3*e + 3*a^2 *b^2*f)*x^9 + 30*(b^4*c - 2*a*b^3*d + 3*a^2*b^2*e - 4*a^3*b*f)*x^6 - 60*(2 *a*b^3*c - 3*a^2*b^2*d + 4*a^3*b*e - 5*a^4*f)*x^3)/b^6 + 1/3*(3*a^2*b^3*c - 4*a^3*b^2*d + 5*a^4*b*e - 6*a^5*f)*log(b*x^3 + a)/b^7
Time = 0.27 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.33 \[ \int \frac {x^{11} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=\frac {{\left (3 \, a^{2} b^{3} c - 4 \, a^{3} b^{2} d + 5 \, a^{4} b e - 6 \, a^{5} f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{7}} - \frac {3 \, a^{2} b^{4} c x^{3} - 4 \, a^{3} b^{3} d x^{3} + 5 \, a^{4} b^{2} e x^{3} - 6 \, a^{5} b f x^{3} + 2 \, a^{3} b^{3} c - 3 \, a^{4} b^{2} d + 4 \, a^{5} b e - 5 \, a^{6} f}{3 \, {\left (b x^{3} + a\right )} b^{7}} + \frac {12 \, b^{8} f x^{15} + 15 \, b^{8} e x^{12} - 30 \, a b^{7} f x^{12} + 20 \, b^{8} d x^{9} - 40 \, a b^{7} e x^{9} + 60 \, a^{2} b^{6} f x^{9} + 30 \, b^{8} c x^{6} - 60 \, a b^{7} d x^{6} + 90 \, a^{2} b^{6} e x^{6} - 120 \, a^{3} b^{5} f x^{6} - 120 \, a b^{7} c x^{3} + 180 \, a^{2} b^{6} d x^{3} - 240 \, a^{3} b^{5} e x^{3} + 300 \, a^{4} b^{4} f x^{3}}{180 \, b^{10}} \]
1/3*(3*a^2*b^3*c - 4*a^3*b^2*d + 5*a^4*b*e - 6*a^5*f)*log(abs(b*x^3 + a))/ b^7 - 1/3*(3*a^2*b^4*c*x^3 - 4*a^3*b^3*d*x^3 + 5*a^4*b^2*e*x^3 - 6*a^5*b*f *x^3 + 2*a^3*b^3*c - 3*a^4*b^2*d + 4*a^5*b*e - 5*a^6*f)/((b*x^3 + a)*b^7) + 1/180*(12*b^8*f*x^15 + 15*b^8*e*x^12 - 30*a*b^7*f*x^12 + 20*b^8*d*x^9 - 40*a*b^7*e*x^9 + 60*a^2*b^6*f*x^9 + 30*b^8*c*x^6 - 60*a*b^7*d*x^6 + 90*a^2 *b^6*e*x^6 - 120*a^3*b^5*f*x^6 - 120*a*b^7*c*x^3 + 180*a^2*b^6*d*x^3 - 240 *a^3*b^5*e*x^3 + 300*a^4*b^4*f*x^3)/b^10
Time = 9.65 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.62 \[ \int \frac {x^{11} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx=x^{12}\,\left (\frac {e}{12\,b^2}-\frac {a\,f}{6\,b^3}\right )-x^3\,\left (\frac {2\,a\,\left (\frac {c}{b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )}{3\,b}-\frac {a^2\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{3\,b^2}\right )-x^9\,\left (\frac {a^2\,f}{9\,b^4}-\frac {d}{9\,b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{9\,b}\right )+x^6\,\left (\frac {c}{6\,b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{6\,b^2}+\frac {a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{3\,b}\right )-\frac {\ln \left (b\,x^3+a\right )\,\left (6\,f\,a^5-5\,e\,a^4\,b+4\,d\,a^3\,b^2-3\,c\,a^2\,b^3\right )}{3\,b^7}+\frac {f\,x^{15}}{15\,b^2}-\frac {f\,a^6-e\,a^5\,b+d\,a^4\,b^2-c\,a^3\,b^3}{3\,b\,\left (b^7\,x^3+a\,b^6\right )} \]
x^12*(e/(12*b^2) - (a*f)/(6*b^3)) - x^3*((2*a*(c/b^2 - (a^2*(e/b^2 - (2*a* f)/b^3))/b^2 + (2*a*((a^2*f)/b^4 - d/b^2 + (2*a*(e/b^2 - (2*a*f)/b^3))/b)) /b))/(3*b) - (a^2*((a^2*f)/b^4 - d/b^2 + (2*a*(e/b^2 - (2*a*f)/b^3))/b))/( 3*b^2)) - x^9*((a^2*f)/(9*b^4) - d/(9*b^2) + (2*a*(e/b^2 - (2*a*f)/b^3))/( 9*b)) + x^6*(c/(6*b^2) - (a^2*(e/b^2 - (2*a*f)/b^3))/(6*b^2) + (a*((a^2*f) /b^4 - d/b^2 + (2*a*(e/b^2 - (2*a*f)/b^3))/b))/(3*b)) - (log(a + b*x^3)*(6 *a^5*f - 3*a^2*b^3*c + 4*a^3*b^2*d - 5*a^4*b*e))/(3*b^7) + (f*x^15)/(15*b^ 2) - (a^6*f - a^3*b^3*c + a^4*b^2*d - a^5*b*e)/(3*b*(a*b^6 + b^7*x^3))