Integrand size = 30, antiderivative size = 266 \[ \int \frac {x^{14} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {a \left (3 b^3 c-6 a b^2 d+10 a^2 b e-15 a^3 f\right ) x^3}{3 b^7}+\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^6}{6 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^9}{9 b^5}+\frac {(b e-3 a f) x^{12}}{12 b^4}+\frac {f x^{15}}{15 b^3}-\frac {a^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 b^8 \left (a+b x^3\right )^2}+\frac {a^3 \left (4 b^3 c-5 a b^2 d+6 a^2 b e-7 a^3 f\right )}{3 b^8 \left (a+b x^3\right )}+\frac {a^2 \left (6 b^3 c-10 a b^2 d+15 a^2 b e-21 a^3 f\right ) \log \left (a+b x^3\right )}{3 b^8} \]
-1/3*a*(-15*a^3*f+10*a^2*b*e-6*a*b^2*d+3*b^3*c)*x^3/b^7+1/6*(-10*a^3*f+6*a ^2*b*e-3*a*b^2*d+b^3*c)*x^6/b^6+1/9*(6*a^2*f-3*a*b*e+b^2*d)*x^9/b^5+1/12*( -3*a*f+b*e)*x^12/b^4+1/15*f*x^15/b^3-1/6*a^4*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c )/b^8/(b*x^3+a)^2+1/3*a^3*(-7*a^3*f+6*a^2*b*e-5*a*b^2*d+4*b^3*c)/b^8/(b*x^ 3+a)+1/3*a^2*(-21*a^3*f+15*a^2*b*e-10*a*b^2*d+6*b^3*c)*ln(b*x^3+a)/b^8
Time = 0.15 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.92 \[ \int \frac {x^{14} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {60 a b \left (-3 b^3 c+6 a b^2 d-10 a^2 b e+15 a^3 f\right ) x^3+30 b^2 \left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^6+20 b^3 \left (b^2 d-3 a b e+6 a^2 f\right ) x^9+15 b^4 (b e-3 a f) x^{12}+12 b^5 f x^{15}+\frac {30 a^4 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{\left (a+b x^3\right )^2}-\frac {60 a^3 \left (-4 b^3 c+5 a b^2 d-6 a^2 b e+7 a^3 f\right )}{a+b x^3}+60 a^2 \left (6 b^3 c-10 a b^2 d+15 a^2 b e-21 a^3 f\right ) \log \left (a+b x^3\right )}{180 b^8} \]
(60*a*b*(-3*b^3*c + 6*a*b^2*d - 10*a^2*b*e + 15*a^3*f)*x^3 + 30*b^2*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^6 + 20*b^3*(b^2*d - 3*a*b*e + 6*a^2 *f)*x^9 + 15*b^4*(b*e - 3*a*f)*x^12 + 12*b^5*f*x^15 + (30*a^4*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3)^2 - (60*a^3*(-4*b^3*c + 5*a*b^2*d - 6*a^2*b*e + 7*a^3*f))/(a + b*x^3) + 60*a^2*(6*b^3*c - 10*a*b^2*d + 15*a^ 2*b*e - 21*a^3*f)*Log[a + b*x^3])/(180*b^8)
Time = 0.62 (sec) , antiderivative size = 262, 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.
| \(\displaystyle \int \frac {x^{14} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2361 |
| \(\displaystyle \frac {1}{3} \int \frac {x^{12} \left (f x^9+e x^6+d x^3+c\right )}{\left (b x^3+a\right )^3}dx^3\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \frac {1}{3} \int \left (\frac {f x^{12}}{b^3}+\frac {(b e-3 a f) x^9}{b^4}+\frac {\left (6 f a^2-3 b e a+b^2 d\right ) x^6}{b^5}+\frac {\left (-10 f a^3+6 b e a^2-3 b^2 d a+b^3 c\right ) x^3}{b^6}+\frac {a \left (15 f a^3-10 b e a^2+6 b^2 d a-3 b^3 c\right )}{b^7}-\frac {a^2 \left (21 f a^3-15 b e a^2+10 b^2 d a-6 b^3 c\right )}{b^7 \left (b x^3+a\right )}+\frac {a^3 \left (7 f a^3-6 b e a^2+5 b^2 d a-4 b^3 c\right )}{b^7 \left (b x^3+a\right )^2}-\frac {a^4 \left (f a^3-b e a^2+b^2 d a-b^3 c\right )}{b^7 \left (b x^3+a\right )^3}\right )dx^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (\frac {x^9 \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac {a^3 \left (-7 a^3 f+6 a^2 b e-5 a b^2 d+4 b^3 c\right )}{b^8 \left (a+b x^3\right )}+\frac {a^2 \log \left (a+b x^3\right ) \left (-21 a^3 f+15 a^2 b e-10 a b^2 d+6 b^3 c\right )}{b^8}-\frac {a x^3 \left (-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right )}{b^7}+\frac {x^6 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{2 b^6}-\frac {a^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 b^8 \left (a+b x^3\right )^2}+\frac {x^{12} (b e-3 a f)}{4 b^4}+\frac {f x^{15}}{5 b^3}\right )\) |
(-((a*(3*b^3*c - 6*a*b^2*d + 10*a^2*b*e - 15*a^3*f)*x^3)/b^7) + ((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^6)/(2*b^6) + ((b^2*d - 3*a*b*e + 6*a^2 *f)*x^9)/(3*b^5) + ((b*e - 3*a*f)*x^12)/(4*b^4) + (f*x^15)/(5*b^3) - (a^4* (b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(2*b^8*(a + b*x^3)^2) + (a^3*(4*b^3*c - 5*a*b^2*d + 6*a^2*b*e - 7*a^3*f))/(b^8*(a + b*x^3)) + (a^2*(6*b^3*c - 1 0*a*b^2*d + 15*a^2*b*e - 21*a^3*f)*Log[a + b*x^3])/b^8)/3
3.3.76.3.1 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 = 1.53 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.98
| method | result | size |
| norman | \(\frac {-\frac {a^{2} \left (21 f \,a^{5}-15 a^{4} e b +10 a^{3} d \,b^{2}-6 a^{2} c \,b^{3}\right )}{2 b^{8}}+\frac {f \,x^{21}}{15 b}-\frac {\left (7 a f -5 b e \right ) x^{18}}{60 b^{2}}+\frac {\left (21 a^{2} f -15 a e b +10 b^{2} d \right ) x^{15}}{90 b^{3}}-\frac {\left (21 f \,a^{3}-15 a^{2} b e +10 a \,b^{2} d -6 b^{3} c \right ) x^{12}}{36 b^{4}}+\frac {a \left (21 f \,a^{3}-15 a^{2} b e +10 a \,b^{2} d -6 b^{3} c \right ) x^{9}}{9 b^{5}}-\frac {2 a \left (21 f \,a^{5}-15 a^{4} e b +10 a^{3} d \,b^{2}-6 a^{2} c \,b^{3}\right ) x^{3}}{3 b^{7}}}{\left (b \,x^{3}+a \right )^{2}}-\frac {a^{2} \left (21 f \,a^{3}-15 a^{2} b e +10 a \,b^{2} d -6 b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 b^{8}}\) | \(260\) |
| default | \(\frac {\frac {f \,x^{15} b^{4}}{15}+\frac {\left (-3 a \,b^{3} f +b^{4} e \right ) x^{12}}{12}+\frac {\left (6 a^{2} b^{2} f -3 a \,b^{3} e +b^{4} d \right ) x^{9}}{9}+\frac {\left (-10 a^{3} b f +6 a^{2} e \,b^{2}-3 a \,b^{3} d +b^{4} c \right ) x^{6}}{6}+\frac {\left (15 a^{4} f -10 a^{3} b e +6 a^{2} b^{2} d -3 a \,b^{3} c \right ) x^{3}}{3}}{b^{7}}-\frac {a^{2} \left (\frac {\left (21 f \,a^{3}-15 a^{2} b e +10 a \,b^{2} d -6 b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{b}-\frac {a^{2} \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{2 b \left (b \,x^{3}+a \right )^{2}}+\frac {a \left (7 f \,a^{3}-6 a^{2} b e +5 a \,b^{2} d -4 b^{3} c \right )}{b \left (b \,x^{3}+a \right )}\right )}{3 b^{7}}\) | \(261\) |
| risch | \(\frac {f \,x^{15}}{15 b^{3}}-\frac {x^{12} a f}{4 b^{4}}+\frac {e \,x^{12}}{12 b^{3}}+\frac {2 x^{9} a^{2} f}{3 b^{5}}-\frac {a e \,x^{9}}{3 b^{4}}+\frac {x^{9} d}{9 b^{3}}-\frac {5 x^{6} a^{3} f}{3 b^{6}}+\frac {a^{2} e \,x^{6}}{b^{5}}-\frac {x^{6} a d}{2 b^{4}}+\frac {x^{6} c}{6 b^{3}}+\frac {5 a^{4} f \,x^{3}}{b^{7}}-\frac {10 a^{3} e \,x^{3}}{3 b^{6}}+\frac {2 a^{2} d \,x^{3}}{b^{5}}-\frac {a c \,x^{3}}{b^{4}}+\frac {\left (-\frac {7}{3} a^{6} f +2 a^{5} b e -\frac {5}{3} a^{4} b^{2} d +\frac {4}{3} a^{3} b^{3} c \right ) x^{3}-\frac {a^{4} \left (13 f \,a^{3}-11 a^{2} b e +9 a \,b^{2} d -7 b^{3} c \right )}{6 b}}{b^{7} \left (b \,x^{3}+a \right )^{2}}-\frac {7 a^{5} \ln \left (b \,x^{3}+a \right ) f}{b^{8}}+\frac {5 a^{4} \ln \left (b \,x^{3}+a \right ) e}{b^{7}}-\frac {10 a^{3} \ln \left (b \,x^{3}+a \right ) d}{3 b^{6}}+\frac {2 a^{2} \ln \left (b \,x^{3}+a \right ) c}{b^{5}}\) | \(302\) |
| parallelrisch | \(-\frac {1200 x^{3} a^{4} b^{3} d -720 x^{3} a^{3} b^{4} c +21 x^{18} a \,b^{6} f -42 x^{15} a^{2} b^{5} f +30 x^{15} a \,b^{6} e +1260 \ln \left (b \,x^{3}+a \right ) x^{6} a^{5} b^{2} f -900 \ln \left (b \,x^{3}+a \right ) x^{6} a^{4} b^{3} e +600 \ln \left (b \,x^{3}+a \right ) x^{6} a^{3} b^{4} d -360 \ln \left (b \,x^{3}+a \right ) x^{6} a^{2} b^{5} c +2520 \ln \left (b \,x^{3}+a \right ) x^{3} a^{6} b f -1800 \ln \left (b \,x^{3}+a \right ) x^{3} a^{5} b^{2} e -75 x^{12} a^{2} b^{5} e +50 x^{12} a \,b^{6} d +2520 x^{3} a^{6} b f -1800 x^{3} a^{5} b^{2} e +1890 a^{7} f -540 a^{4} b^{3} c +1200 \ln \left (b \,x^{3}+a \right ) x^{3} a^{4} b^{3} d -720 \ln \left (b \,x^{3}+a \right ) x^{3} a^{3} b^{4} c +900 a^{5} b^{2} d -30 x^{12} b^{7} c -15 x^{18} b^{7} e -1350 a^{6} b e -420 x^{9} a^{4} b^{3} f +300 x^{9} a^{3} b^{4} e -200 x^{9} a^{2} b^{5} d +120 x^{9} a \,b^{6} c -900 \ln \left (b \,x^{3}+a \right ) a^{6} b e +600 \ln \left (b \,x^{3}+a \right ) a^{5} b^{2} d -360 \ln \left (b \,x^{3}+a \right ) a^{4} b^{3} c +105 x^{12} a^{3} b^{4} f -20 x^{15} b^{7} d +1260 \ln \left (b \,x^{3}+a \right ) a^{7} f -12 f \,x^{21} b^{7}}{180 b^{8} \left (b \,x^{3}+a \right )^{2}}\) | \(462\) |
(-1/2*a^2*(21*a^5*f-15*a^4*b*e+10*a^3*b^2*d-6*a^2*b^3*c)/b^8+1/15*f/b*x^21 -1/60*(7*a*f-5*b*e)/b^2*x^18+1/90*(21*a^2*f-15*a*b*e+10*b^2*d)/b^3*x^15-1/ 36*(21*a^3*f-15*a^2*b*e+10*a*b^2*d-6*b^3*c)/b^4*x^12+1/9*a/b^5*(21*a^3*f-1 5*a^2*b*e+10*a*b^2*d-6*b^3*c)*x^9-2/3*a*(21*a^5*f-15*a^4*b*e+10*a^3*b^2*d- 6*a^2*b^3*c)/b^7*x^3)/(b*x^3+a)^2-1/3*a^2*(21*a^3*f-15*a^2*b*e+10*a*b^2*d- 6*b^3*c)/b^8*ln(b*x^3+a)
Time = 0.25 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.49 \[ \int \frac {x^{14} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {12 \, b^{7} f x^{21} + 3 \, {\left (5 \, b^{7} e - 7 \, a b^{6} f\right )} x^{18} + 2 \, {\left (10 \, b^{7} d - 15 \, a b^{6} e + 21 \, a^{2} b^{5} f\right )} x^{15} + 5 \, {\left (6 \, b^{7} c - 10 \, a b^{6} d + 15 \, a^{2} b^{5} e - 21 \, a^{3} b^{4} f\right )} x^{12} - 20 \, {\left (6 \, a b^{6} c - 10 \, a^{2} b^{5} d + 15 \, a^{3} b^{4} e - 21 \, a^{4} b^{3} f\right )} x^{9} + 210 \, a^{4} b^{3} c - 270 \, a^{5} b^{2} d + 330 \, a^{6} b e - 390 \, a^{7} f - 30 \, {\left (11 \, a^{2} b^{5} c - 21 \, a^{3} b^{4} d + 34 \, a^{4} b^{3} e - 50 \, a^{5} b^{2} f\right )} x^{6} + 60 \, {\left (a^{3} b^{4} c + a^{4} b^{3} d - 4 \, a^{5} b^{2} e + 8 \, a^{6} b f\right )} x^{3} + 60 \, {\left (6 \, a^{4} b^{3} c - 10 \, a^{5} b^{2} d + 15 \, a^{6} b e - 21 \, a^{7} f + {\left (6 \, a^{2} b^{5} c - 10 \, a^{3} b^{4} d + 15 \, a^{4} b^{3} e - 21 \, a^{5} b^{2} f\right )} x^{6} + 2 \, {\left (6 \, a^{3} b^{4} c - 10 \, a^{4} b^{3} d + 15 \, a^{5} b^{2} e - 21 \, a^{6} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{180 \, {\left (b^{10} x^{6} + 2 \, a b^{9} x^{3} + a^{2} b^{8}\right )}} \]
1/180*(12*b^7*f*x^21 + 3*(5*b^7*e - 7*a*b^6*f)*x^18 + 2*(10*b^7*d - 15*a*b ^6*e + 21*a^2*b^5*f)*x^15 + 5*(6*b^7*c - 10*a*b^6*d + 15*a^2*b^5*e - 21*a^ 3*b^4*f)*x^12 - 20*(6*a*b^6*c - 10*a^2*b^5*d + 15*a^3*b^4*e - 21*a^4*b^3*f )*x^9 + 210*a^4*b^3*c - 270*a^5*b^2*d + 330*a^6*b*e - 390*a^7*f - 30*(11*a ^2*b^5*c - 21*a^3*b^4*d + 34*a^4*b^3*e - 50*a^5*b^2*f)*x^6 + 60*(a^3*b^4*c + a^4*b^3*d - 4*a^5*b^2*e + 8*a^6*b*f)*x^3 + 60*(6*a^4*b^3*c - 10*a^5*b^2 *d + 15*a^6*b*e - 21*a^7*f + (6*a^2*b^5*c - 10*a^3*b^4*d + 15*a^4*b^3*e - 21*a^5*b^2*f)*x^6 + 2*(6*a^3*b^4*c - 10*a^4*b^3*d + 15*a^5*b^2*e - 21*a^6* b*f)*x^3)*log(b*x^3 + a))/(b^10*x^6 + 2*a*b^9*x^3 + a^2*b^8)
Timed out. \[ \int \frac {x^{14} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.03 \[ \int \frac {x^{14} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {7 \, a^{4} b^{3} c - 9 \, a^{5} b^{2} d + 11 \, a^{6} b e - 13 \, a^{7} f + 2 \, {\left (4 \, a^{3} b^{4} c - 5 \, a^{4} b^{3} d + 6 \, a^{5} b^{2} e - 7 \, a^{6} b f\right )} x^{3}}{6 \, {\left (b^{10} x^{6} + 2 \, a b^{9} x^{3} + a^{2} b^{8}\right )}} + \frac {12 \, b^{4} f x^{15} + 15 \, {\left (b^{4} e - 3 \, a b^{3} f\right )} x^{12} + 20 \, {\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{9} + 30 \, {\left (b^{4} c - 3 \, a b^{3} d + 6 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{6} - 60 \, {\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d + 10 \, a^{3} b e - 15 \, a^{4} f\right )} x^{3}}{180 \, b^{7}} + \frac {{\left (6 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 15 \, a^{4} b e - 21 \, a^{5} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{8}} \]
1/6*(7*a^4*b^3*c - 9*a^5*b^2*d + 11*a^6*b*e - 13*a^7*f + 2*(4*a^3*b^4*c - 5*a^4*b^3*d + 6*a^5*b^2*e - 7*a^6*b*f)*x^3)/(b^10*x^6 + 2*a*b^9*x^3 + a^2* b^8) + 1/180*(12*b^4*f*x^15 + 15*(b^4*e - 3*a*b^3*f)*x^12 + 20*(b^4*d - 3* a*b^3*e + 6*a^2*b^2*f)*x^9 + 30*(b^4*c - 3*a*b^3*d + 6*a^2*b^2*e - 10*a^3* b*f)*x^6 - 60*(3*a*b^3*c - 6*a^2*b^2*d + 10*a^3*b*e - 15*a^4*f)*x^3)/b^7 + 1/3*(6*a^2*b^3*c - 10*a^3*b^2*d + 15*a^4*b*e - 21*a^5*f)*log(b*x^3 + a)/b ^8
Time = 0.28 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.28 \[ \int \frac {x^{14} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {{\left (6 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 15 \, a^{4} b e - 21 \, a^{5} f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{8}} - \frac {18 \, a^{2} b^{5} c x^{6} - 30 \, a^{3} b^{4} d x^{6} + 45 \, a^{4} b^{3} e x^{6} - 63 \, a^{5} b^{2} f x^{6} + 28 \, a^{3} b^{4} c x^{3} - 50 \, a^{4} b^{3} d x^{3} + 78 \, a^{5} b^{2} e x^{3} - 112 \, a^{6} b f x^{3} + 11 \, a^{4} b^{3} c - 21 \, a^{5} b^{2} d + 34 \, a^{6} b e - 50 \, a^{7} f}{6 \, {\left (b x^{3} + a\right )}^{2} b^{8}} + \frac {12 \, b^{12} f x^{15} + 15 \, b^{12} e x^{12} - 45 \, a b^{11} f x^{12} + 20 \, b^{12} d x^{9} - 60 \, a b^{11} e x^{9} + 120 \, a^{2} b^{10} f x^{9} + 30 \, b^{12} c x^{6} - 90 \, a b^{11} d x^{6} + 180 \, a^{2} b^{10} e x^{6} - 300 \, a^{3} b^{9} f x^{6} - 180 \, a b^{11} c x^{3} + 360 \, a^{2} b^{10} d x^{3} - 600 \, a^{3} b^{9} e x^{3} + 900 \, a^{4} b^{8} f x^{3}}{180 \, b^{15}} \]
1/3*(6*a^2*b^3*c - 10*a^3*b^2*d + 15*a^4*b*e - 21*a^5*f)*log(abs(b*x^3 + a ))/b^8 - 1/6*(18*a^2*b^5*c*x^6 - 30*a^3*b^4*d*x^6 + 45*a^4*b^3*e*x^6 - 63* a^5*b^2*f*x^6 + 28*a^3*b^4*c*x^3 - 50*a^4*b^3*d*x^3 + 78*a^5*b^2*e*x^3 - 1 12*a^6*b*f*x^3 + 11*a^4*b^3*c - 21*a^5*b^2*d + 34*a^6*b*e - 50*a^7*f)/((b* x^3 + a)^2*b^8) + 1/180*(12*b^12*f*x^15 + 15*b^12*e*x^12 - 45*a*b^11*f*x^1 2 + 20*b^12*d*x^9 - 60*a*b^11*e*x^9 + 120*a^2*b^10*f*x^9 + 30*b^12*c*x^6 - 90*a*b^11*d*x^6 + 180*a^2*b^10*e*x^6 - 300*a^3*b^9*f*x^6 - 180*a*b^11*c*x ^3 + 360*a^2*b^10*d*x^3 - 600*a^3*b^9*e*x^3 + 900*a^4*b^8*f*x^3)/b^15
Time = 9.24 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.69 \[ \int \frac {x^{14} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=x^{12}\,\left (\frac {e}{12\,b^3}-\frac {a\,f}{4\,b^4}\right )+x^6\,\left (\frac {c}{6\,b^3}-\frac {a^3\,f}{6\,b^6}-\frac {a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{2\,b^2}+\frac {a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{2\,b}\right )-x^9\,\left (\frac {a^2\,f}{3\,b^5}-\frac {d}{9\,b^3}+\frac {a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{3\,b}\right )-\frac {\frac {13\,f\,a^7-11\,e\,a^6\,b+9\,d\,a^5\,b^2-7\,c\,a^4\,b^3}{6\,b}+x^3\,\left (\frac {7\,f\,a^6}{3}-2\,e\,a^5\,b+\frac {5\,d\,a^4\,b^2}{3}-\frac {4\,c\,a^3\,b^3}{3}\right )}{a^2\,b^7+2\,a\,b^8\,x^3+b^9\,x^6}-x^3\,\left (\frac {a\,\left (\frac {c}{b^3}-\frac {a^3\,f}{b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b}\right )}{b}-\frac {a^2\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b^2}+\frac {a^3\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{3\,b^3}\right )-\frac {\ln \left (b\,x^3+a\right )\,\left (21\,f\,a^5-15\,e\,a^4\,b+10\,d\,a^3\,b^2-6\,c\,a^2\,b^3\right )}{3\,b^8}+\frac {f\,x^{15}}{15\,b^3} \]
x^12*(e/(12*b^3) - (a*f)/(4*b^4)) + x^6*(c/(6*b^3) - (a^3*f)/(6*b^6) - (a^ 2*(e/b^3 - (3*a*f)/b^4))/(2*b^2) + (a*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b))/(2*b)) - x^9*((a^2*f)/(3*b^5) - d/(9*b^3) + (a*(e/b^3 - (3*a*f)/b^4))/(3*b)) - ((13*a^7*f - 7*a^4*b^3*c + 9*a^5*b^2*d - 11*a^6* b*e)/(6*b) + x^3*((7*a^6*f)/3 - (4*a^3*b^3*c)/3 + (5*a^4*b^2*d)/3 - 2*a^5* b*e))/(a^2*b^7 + b^9*x^6 + 2*a*b^8*x^3) - x^3*((a*(c/b^3 - (a^3*f)/b^6 - ( 3*a^2*(e/b^3 - (3*a*f)/b^4))/b^2 + (3*a*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b ^3 - (3*a*f)/b^4))/b))/b))/b - (a^2*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b))/b^2 + (a^3*(e/b^3 - (3*a*f)/b^4))/(3*b^3)) - (log(a + b *x^3)*(21*a^5*f - 6*a^2*b^3*c + 10*a^3*b^2*d - 15*a^4*b*e))/(3*b^8) + (f*x ^15)/(15*b^3)