Integrand size = 38, antiderivative size = 223 \[ \int x^3 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{4} a^3 c x^4+\frac {1}{5} a^3 d x^5+\frac {1}{6} a^3 e x^6+\frac {1}{7} a^2 (3 b c+a f) x^7+\frac {1}{8} a^2 (3 b d+a g) x^8+\frac {1}{9} a^2 (3 b e+a h) x^9+\frac {3}{10} a b (b c+a f) x^{10}+\frac {3}{11} a b (b d+a g) x^{11}+\frac {1}{4} a b (b e+a h) x^{12}+\frac {1}{13} b^2 (b c+3 a f) x^{13}+\frac {1}{14} b^2 (b d+3 a g) x^{14}+\frac {1}{15} b^2 (b e+3 a h) x^{15}+\frac {1}{16} b^3 f x^{16}+\frac {1}{17} b^3 g x^{17}+\frac {1}{18} b^3 h x^{18} \] Output:
1/4*a^3*c*x^4+1/5*a^3*d*x^5+1/6*a^3*e*x^6+1/7*a^2*(a*f+3*b*c)*x^7+1/8*a^2* (a*g+3*b*d)*x^8+1/9*a^2*(a*h+3*b*e)*x^9+3/10*a*b*(a*f+b*c)*x^10+3/11*a*b*( a*g+b*d)*x^11+1/4*a*b*(a*h+b*e)*x^12+1/13*b^2*(3*a*f+b*c)*x^13+1/14*b^2*(3 *a*g+b*d)*x^14+1/15*b^2*(3*a*h+b*e)*x^15+1/16*b^3*f*x^16+1/17*b^3*g*x^17+1 /18*b^3*h*x^18
Time = 0.03 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00 \[ \int x^3 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{4} a^3 c x^4+\frac {1}{5} a^3 d x^5+\frac {1}{6} a^3 e x^6+\frac {1}{7} a^2 (3 b c+a f) x^7+\frac {1}{8} a^2 (3 b d+a g) x^8+\frac {1}{9} a^2 (3 b e+a h) x^9+\frac {3}{10} a b (b c+a f) x^{10}+\frac {3}{11} a b (b d+a g) x^{11}+\frac {1}{4} a b (b e+a h) x^{12}+\frac {1}{13} b^2 (b c+3 a f) x^{13}+\frac {1}{14} b^2 (b d+3 a g) x^{14}+\frac {1}{15} b^2 (b e+3 a h) x^{15}+\frac {1}{16} b^3 f x^{16}+\frac {1}{17} b^3 g x^{17}+\frac {1}{18} b^3 h x^{18} \] Input:
Integrate[x^3*(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]
Output:
(a^3*c*x^4)/4 + (a^3*d*x^5)/5 + (a^3*e*x^6)/6 + (a^2*(3*b*c + a*f)*x^7)/7 + (a^2*(3*b*d + a*g)*x^8)/8 + (a^2*(3*b*e + a*h)*x^9)/9 + (3*a*b*(b*c + a* f)*x^10)/10 + (3*a*b*(b*d + a*g)*x^11)/11 + (a*b*(b*e + a*h)*x^12)/4 + (b^ 2*(b*c + 3*a*f)*x^13)/13 + (b^2*(b*d + 3*a*g)*x^14)/14 + (b^2*(b*e + 3*a*h )*x^15)/15 + (b^3*f*x^16)/16 + (b^3*g*x^17)/17 + (b^3*h*x^18)/18
Time = 0.81 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2360, 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.
| \(\displaystyle \int x^3 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx\) |
| \(\Big \downarrow \) 2360 |
| \(\displaystyle \int \left (a^3 c x^3+a^3 d x^4+a^3 e x^5+a^2 x^6 (a f+3 b c)+a^2 x^7 (a g+3 b d)+a^2 x^8 (a h+3 b e)+b^2 x^{12} (3 a f+b c)+b^2 x^{13} (3 a g+b d)+b^2 x^{14} (3 a h+b e)+3 a b x^9 (a f+b c)+3 a b x^{10} (a g+b d)+3 a b x^{11} (a h+b e)+b^3 f x^{15}+b^3 g x^{16}+b^3 h x^{17}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} a^3 c x^4+\frac {1}{5} a^3 d x^5+\frac {1}{6} a^3 e x^6+\frac {1}{7} a^2 x^7 (a f+3 b c)+\frac {1}{8} a^2 x^8 (a g+3 b d)+\frac {1}{9} a^2 x^9 (a h+3 b e)+\frac {1}{13} b^2 x^{13} (3 a f+b c)+\frac {1}{14} b^2 x^{14} (3 a g+b d)+\frac {1}{15} b^2 x^{15} (3 a h+b e)+\frac {3}{10} a b x^{10} (a f+b c)+\frac {3}{11} a b x^{11} (a g+b d)+\frac {1}{4} a b x^{12} (a h+b e)+\frac {1}{16} b^3 f x^{16}+\frac {1}{17} b^3 g x^{17}+\frac {1}{18} b^3 h x^{18}\) |
Input:
Int[x^3*(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]
Output:
(a^3*c*x^4)/4 + (a^3*d*x^5)/5 + (a^3*e*x^6)/6 + (a^2*(3*b*c + a*f)*x^7)/7 + (a^2*(3*b*d + a*g)*x^8)/8 + (a^2*(3*b*e + a*h)*x^9)/9 + (3*a*b*(b*c + a* f)*x^10)/10 + (3*a*b*(b*d + a*g)*x^11)/11 + (a*b*(b*e + a*h)*x^12)/4 + (b^ 2*(b*c + 3*a*f)*x^13)/13 + (b^2*(b*d + 3*a*g)*x^14)/14 + (b^2*(b*e + 3*a*h )*x^15)/15 + (b^3*f*x^16)/16 + (b^3*g*x^17)/17 + (b^3*h*x^18)/18
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])
Time = 1.30 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.99
| method | result | size |
| norman | \(\frac {a^{3} c \,x^{4}}{4}+\frac {a^{3} d \,x^{5}}{5}+\frac {a^{3} e \,x^{6}}{6}+\left (\frac {1}{7} f \,a^{3}+\frac {3}{7} c \,a^{2} b \right ) x^{7}+\left (\frac {1}{8} a^{3} g +\frac {3}{8} a^{2} b d \right ) x^{8}+\left (\frac {1}{9} a^{3} h +\frac {1}{3} e \,a^{2} b \right ) x^{9}+\left (\frac {3}{10} f \,a^{2} b +\frac {3}{10} c a \,b^{2}\right ) x^{10}+\left (\frac {3}{11} a^{2} b g +\frac {3}{11} d a \,b^{2}\right ) x^{11}+\left (\frac {1}{4} a^{2} b h +\frac {1}{4} e a \,b^{2}\right ) x^{12}+\left (\frac {3}{13} f a \,b^{2}+\frac {1}{13} b^{3} c \right ) x^{13}+\left (\frac {3}{14} a \,b^{2} g +\frac {1}{14} b^{3} d \right ) x^{14}+\left (\frac {1}{5} a \,b^{2} h +\frac {1}{15} e \,b^{3}\right ) x^{15}+\frac {f \,b^{3} x^{16}}{16}+\frac {b^{3} g \,x^{17}}{17}+\frac {b^{3} h \,x^{18}}{18}\) | \(221\) |
| default | \(\frac {b^{3} h \,x^{18}}{18}+\frac {b^{3} g \,x^{17}}{17}+\frac {f \,b^{3} x^{16}}{16}+\frac {\left (3 a \,b^{2} h +e \,b^{3}\right ) x^{15}}{15}+\frac {\left (3 a \,b^{2} g +b^{3} d \right ) x^{14}}{14}+\frac {\left (3 f a \,b^{2}+b^{3} c \right ) x^{13}}{13}+\frac {\left (3 a^{2} b h +3 e a \,b^{2}\right ) x^{12}}{12}+\frac {\left (3 a^{2} b g +3 d a \,b^{2}\right ) x^{11}}{11}+\frac {\left (3 f \,a^{2} b +3 c a \,b^{2}\right ) x^{10}}{10}+\frac {\left (a^{3} h +3 e \,a^{2} b \right ) x^{9}}{9}+\frac {\left (a^{3} g +3 a^{2} b d \right ) x^{8}}{8}+\frac {\left (f \,a^{3}+3 c \,a^{2} b \right ) x^{7}}{7}+\frac {a^{3} e \,x^{6}}{6}+\frac {a^{3} d \,x^{5}}{5}+\frac {a^{3} c \,x^{4}}{4}\) | \(224\) |
| gosper | \(\frac {1}{4} a^{3} c \,x^{4}+\frac {1}{5} a^{3} d \,x^{5}+\frac {1}{6} a^{3} e \,x^{6}+\frac {1}{7} x^{7} f \,a^{3}+\frac {3}{7} x^{7} c \,a^{2} b +\frac {1}{8} x^{8} a^{3} g +\frac {3}{8} a^{2} b d \,x^{8}+\frac {1}{9} x^{9} a^{3} h +\frac {1}{3} x^{9} e \,a^{2} b +\frac {3}{10} x^{10} f \,a^{2} b +\frac {3}{10} x^{10} c a \,b^{2}+\frac {3}{11} x^{11} a^{2} b g +\frac {3}{11} x^{11} d a \,b^{2}+\frac {1}{4} x^{12} a^{2} b h +\frac {1}{4} x^{12} e a \,b^{2}+\frac {3}{13} x^{13} f a \,b^{2}+\frac {1}{13} b^{3} c \,x^{13}+\frac {3}{14} x^{14} a \,b^{2} g +\frac {1}{14} b^{3} d \,x^{14}+\frac {1}{5} x^{15} a \,b^{2} h +\frac {1}{15} b^{3} e \,x^{15}+\frac {1}{16} f \,b^{3} x^{16}+\frac {1}{17} b^{3} g \,x^{17}+\frac {1}{18} b^{3} h \,x^{18}\) | \(230\) |
| risch | \(\frac {1}{4} a^{3} c \,x^{4}+\frac {1}{5} a^{3} d \,x^{5}+\frac {1}{6} a^{3} e \,x^{6}+\frac {1}{7} x^{7} f \,a^{3}+\frac {3}{7} x^{7} c \,a^{2} b +\frac {1}{8} x^{8} a^{3} g +\frac {3}{8} a^{2} b d \,x^{8}+\frac {1}{9} x^{9} a^{3} h +\frac {1}{3} x^{9} e \,a^{2} b +\frac {3}{10} x^{10} f \,a^{2} b +\frac {3}{10} x^{10} c a \,b^{2}+\frac {3}{11} x^{11} a^{2} b g +\frac {3}{11} x^{11} d a \,b^{2}+\frac {1}{4} x^{12} a^{2} b h +\frac {1}{4} x^{12} e a \,b^{2}+\frac {3}{13} x^{13} f a \,b^{2}+\frac {1}{13} b^{3} c \,x^{13}+\frac {3}{14} x^{14} a \,b^{2} g +\frac {1}{14} b^{3} d \,x^{14}+\frac {1}{5} x^{15} a \,b^{2} h +\frac {1}{15} b^{3} e \,x^{15}+\frac {1}{16} f \,b^{3} x^{16}+\frac {1}{17} b^{3} g \,x^{17}+\frac {1}{18} b^{3} h \,x^{18}\) | \(230\) |
| parallelrisch | \(\frac {1}{4} a^{3} c \,x^{4}+\frac {1}{5} a^{3} d \,x^{5}+\frac {1}{6} a^{3} e \,x^{6}+\frac {1}{7} x^{7} f \,a^{3}+\frac {3}{7} x^{7} c \,a^{2} b +\frac {1}{8} x^{8} a^{3} g +\frac {3}{8} a^{2} b d \,x^{8}+\frac {1}{9} x^{9} a^{3} h +\frac {1}{3} x^{9} e \,a^{2} b +\frac {3}{10} x^{10} f \,a^{2} b +\frac {3}{10} x^{10} c a \,b^{2}+\frac {3}{11} x^{11} a^{2} b g +\frac {3}{11} x^{11} d a \,b^{2}+\frac {1}{4} x^{12} a^{2} b h +\frac {1}{4} x^{12} e a \,b^{2}+\frac {3}{13} x^{13} f a \,b^{2}+\frac {1}{13} b^{3} c \,x^{13}+\frac {3}{14} x^{14} a \,b^{2} g +\frac {1}{14} b^{3} d \,x^{14}+\frac {1}{5} x^{15} a \,b^{2} h +\frac {1}{15} b^{3} e \,x^{15}+\frac {1}{16} f \,b^{3} x^{16}+\frac {1}{17} b^{3} g \,x^{17}+\frac {1}{18} b^{3} h \,x^{18}\) | \(230\) |
| orering | \(\frac {x^{4} \left (680680 b^{3} h \,x^{14}+720720 b^{3} g \,x^{13}+765765 f \,x^{12} b^{3}+2450448 a \,b^{2} h \,x^{11}+816816 b^{3} e \,x^{11}+2625480 a \,b^{2} g \,x^{10}+875160 b^{3} d \,x^{10}+2827440 a \,b^{2} f \,x^{9}+942480 b^{3} c \,x^{9}+3063060 a^{2} b h \,x^{8}+3063060 a \,b^{2} e \,x^{8}+3341520 a^{2} b g \,x^{7}+3341520 a \,b^{2} d \,x^{7}+3675672 a^{2} b f \,x^{6}+3675672 a \,b^{2} c \,x^{6}+1361360 a^{3} h \,x^{5}+4084080 a^{2} b e \,x^{5}+1531530 a^{3} g \,x^{4}+4594590 a^{2} b d \,x^{4}+1750320 a^{3} f \,x^{3}+5250960 a^{2} b c \,x^{3}+2042040 a^{3} e \,x^{2}+2450448 a^{3} d x +3063060 c \,a^{3}\right )}{12252240}\) | \(230\) |
Input:
int(x^3*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x,method=_RETURNVERBOS E)
Output:
1/4*a^3*c*x^4+1/5*a^3*d*x^5+1/6*a^3*e*x^6+(1/7*f*a^3+3/7*c*a^2*b)*x^7+(1/8 *a^3*g+3/8*a^2*b*d)*x^8+(1/9*a^3*h+1/3*e*a^2*b)*x^9+(3/10*f*a^2*b+3/10*c*a *b^2)*x^10+(3/11*a^2*b*g+3/11*d*a*b^2)*x^11+(1/4*a^2*b*h+1/4*e*a*b^2)*x^12 +(3/13*f*a*b^2+1/13*b^3*c)*x^13+(3/14*a*b^2*g+1/14*b^3*d)*x^14+(1/5*a*b^2* h+1/15*e*b^3)*x^15+1/16*f*b^3*x^16+1/17*b^3*g*x^17+1/18*b^3*h*x^18
Time = 0.08 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.97 \[ \int x^3 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{18} \, b^{3} h x^{18} + \frac {1}{17} \, b^{3} g x^{17} + \frac {1}{16} \, b^{3} f x^{16} + \frac {1}{15} \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{15} + \frac {1}{14} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{14} + \frac {1}{13} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{13} + \frac {1}{4} \, {\left (a b^{2} e + a^{2} b h\right )} x^{12} + \frac {3}{11} \, {\left (a b^{2} d + a^{2} b g\right )} x^{11} + \frac {3}{10} \, {\left (a b^{2} c + a^{2} b f\right )} x^{10} + \frac {1}{6} \, a^{3} e x^{6} + \frac {1}{9} \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{9} + \frac {1}{5} \, a^{3} d x^{5} + \frac {1}{8} \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{8} + \frac {1}{4} \, a^{3} c x^{4} + \frac {1}{7} \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{7} \] Input:
integrate(x^3*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="fr icas")
Output:
1/18*b^3*h*x^18 + 1/17*b^3*g*x^17 + 1/16*b^3*f*x^16 + 1/15*(b^3*e + 3*a*b^ 2*h)*x^15 + 1/14*(b^3*d + 3*a*b^2*g)*x^14 + 1/13*(b^3*c + 3*a*b^2*f)*x^13 + 1/4*(a*b^2*e + a^2*b*h)*x^12 + 3/11*(a*b^2*d + a^2*b*g)*x^11 + 3/10*(a*b ^2*c + a^2*b*f)*x^10 + 1/6*a^3*e*x^6 + 1/9*(3*a^2*b*e + a^3*h)*x^9 + 1/5*a ^3*d*x^5 + 1/8*(3*a^2*b*d + a^3*g)*x^8 + 1/4*a^3*c*x^4 + 1/7*(3*a^2*b*c + a^3*f)*x^7
Time = 0.04 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.10 \[ \int x^3 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {a^{3} c x^{4}}{4} + \frac {a^{3} d x^{5}}{5} + \frac {a^{3} e x^{6}}{6} + \frac {b^{3} f x^{16}}{16} + \frac {b^{3} g x^{17}}{17} + \frac {b^{3} h x^{18}}{18} + x^{15} \left (\frac {a b^{2} h}{5} + \frac {b^{3} e}{15}\right ) + x^{14} \cdot \left (\frac {3 a b^{2} g}{14} + \frac {b^{3} d}{14}\right ) + x^{13} \cdot \left (\frac {3 a b^{2} f}{13} + \frac {b^{3} c}{13}\right ) + x^{12} \left (\frac {a^{2} b h}{4} + \frac {a b^{2} e}{4}\right ) + x^{11} \cdot \left (\frac {3 a^{2} b g}{11} + \frac {3 a b^{2} d}{11}\right ) + x^{10} \cdot \left (\frac {3 a^{2} b f}{10} + \frac {3 a b^{2} c}{10}\right ) + x^{9} \left (\frac {a^{3} h}{9} + \frac {a^{2} b e}{3}\right ) + x^{8} \left (\frac {a^{3} g}{8} + \frac {3 a^{2} b d}{8}\right ) + x^{7} \left (\frac {a^{3} f}{7} + \frac {3 a^{2} b c}{7}\right ) \] Input:
integrate(x**3*(b*x**3+a)**3*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c),x)
Output:
a**3*c*x**4/4 + a**3*d*x**5/5 + a**3*e*x**6/6 + b**3*f*x**16/16 + b**3*g*x **17/17 + b**3*h*x**18/18 + x**15*(a*b**2*h/5 + b**3*e/15) + x**14*(3*a*b* *2*g/14 + b**3*d/14) + x**13*(3*a*b**2*f/13 + b**3*c/13) + x**12*(a**2*b*h /4 + a*b**2*e/4) + x**11*(3*a**2*b*g/11 + 3*a*b**2*d/11) + x**10*(3*a**2*b *f/10 + 3*a*b**2*c/10) + x**9*(a**3*h/9 + a**2*b*e/3) + x**8*(a**3*g/8 + 3 *a**2*b*d/8) + x**7*(a**3*f/7 + 3*a**2*b*c/7)
Time = 0.04 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.97 \[ \int x^3 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{18} \, b^{3} h x^{18} + \frac {1}{17} \, b^{3} g x^{17} + \frac {1}{16} \, b^{3} f x^{16} + \frac {1}{15} \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{15} + \frac {1}{14} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{14} + \frac {1}{13} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{13} + \frac {1}{4} \, {\left (a b^{2} e + a^{2} b h\right )} x^{12} + \frac {3}{11} \, {\left (a b^{2} d + a^{2} b g\right )} x^{11} + \frac {3}{10} \, {\left (a b^{2} c + a^{2} b f\right )} x^{10} + \frac {1}{6} \, a^{3} e x^{6} + \frac {1}{9} \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{9} + \frac {1}{5} \, a^{3} d x^{5} + \frac {1}{8} \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{8} + \frac {1}{4} \, a^{3} c x^{4} + \frac {1}{7} \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{7} \] Input:
integrate(x^3*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="ma xima")
Output:
1/18*b^3*h*x^18 + 1/17*b^3*g*x^17 + 1/16*b^3*f*x^16 + 1/15*(b^3*e + 3*a*b^ 2*h)*x^15 + 1/14*(b^3*d + 3*a*b^2*g)*x^14 + 1/13*(b^3*c + 3*a*b^2*f)*x^13 + 1/4*(a*b^2*e + a^2*b*h)*x^12 + 3/11*(a*b^2*d + a^2*b*g)*x^11 + 3/10*(a*b ^2*c + a^2*b*f)*x^10 + 1/6*a^3*e*x^6 + 1/9*(3*a^2*b*e + a^3*h)*x^9 + 1/5*a ^3*d*x^5 + 1/8*(3*a^2*b*d + a^3*g)*x^8 + 1/4*a^3*c*x^4 + 1/7*(3*a^2*b*c + a^3*f)*x^7
Time = 0.12 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.03 \[ \int x^3 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{18} \, b^{3} h x^{18} + \frac {1}{17} \, b^{3} g x^{17} + \frac {1}{16} \, b^{3} f x^{16} + \frac {1}{15} \, b^{3} e x^{15} + \frac {1}{5} \, a b^{2} h x^{15} + \frac {1}{14} \, b^{3} d x^{14} + \frac {3}{14} \, a b^{2} g x^{14} + \frac {1}{13} \, b^{3} c x^{13} + \frac {3}{13} \, a b^{2} f x^{13} + \frac {1}{4} \, a b^{2} e x^{12} + \frac {1}{4} \, a^{2} b h x^{12} + \frac {3}{11} \, a b^{2} d x^{11} + \frac {3}{11} \, a^{2} b g x^{11} + \frac {3}{10} \, a b^{2} c x^{10} + \frac {3}{10} \, a^{2} b f x^{10} + \frac {1}{3} \, a^{2} b e x^{9} + \frac {1}{9} \, a^{3} h x^{9} + \frac {3}{8} \, a^{2} b d x^{8} + \frac {1}{8} \, a^{3} g x^{8} + \frac {3}{7} \, a^{2} b c x^{7} + \frac {1}{7} \, a^{3} f x^{7} + \frac {1}{6} \, a^{3} e x^{6} + \frac {1}{5} \, a^{3} d x^{5} + \frac {1}{4} \, a^{3} c x^{4} \] Input:
integrate(x^3*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="gi ac")
Output:
1/18*b^3*h*x^18 + 1/17*b^3*g*x^17 + 1/16*b^3*f*x^16 + 1/15*b^3*e*x^15 + 1/ 5*a*b^2*h*x^15 + 1/14*b^3*d*x^14 + 3/14*a*b^2*g*x^14 + 1/13*b^3*c*x^13 + 3 /13*a*b^2*f*x^13 + 1/4*a*b^2*e*x^12 + 1/4*a^2*b*h*x^12 + 3/11*a*b^2*d*x^11 + 3/11*a^2*b*g*x^11 + 3/10*a*b^2*c*x^10 + 3/10*a^2*b*f*x^10 + 1/3*a^2*b*e *x^9 + 1/9*a^3*h*x^9 + 3/8*a^2*b*d*x^8 + 1/8*a^3*g*x^8 + 3/7*a^2*b*c*x^7 + 1/7*a^3*f*x^7 + 1/6*a^3*e*x^6 + 1/5*a^3*d*x^5 + 1/4*a^3*c*x^4
Time = 6.05 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.92 \[ \int x^3 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=x^7\,\left (\frac {f\,a^3}{7}+\frac {3\,b\,c\,a^2}{7}\right )+x^{13}\,\left (\frac {c\,b^3}{13}+\frac {3\,a\,f\,b^2}{13}\right )+x^8\,\left (\frac {g\,a^3}{8}+\frac {3\,b\,d\,a^2}{8}\right )+x^{14}\,\left (\frac {d\,b^3}{14}+\frac {3\,a\,g\,b^2}{14}\right )+x^9\,\left (\frac {h\,a^3}{9}+\frac {b\,e\,a^2}{3}\right )+x^{15}\,\left (\frac {e\,b^3}{15}+\frac {a\,h\,b^2}{5}\right )+\frac {a^3\,c\,x^4}{4}+\frac {a^3\,d\,x^5}{5}+\frac {a^3\,e\,x^6}{6}+\frac {b^3\,f\,x^{16}}{16}+\frac {b^3\,g\,x^{17}}{17}+\frac {b^3\,h\,x^{18}}{18}+\frac {3\,a\,b\,x^{10}\,\left (b\,c+a\,f\right )}{10}+\frac {3\,a\,b\,x^{11}\,\left (b\,d+a\,g\right )}{11}+\frac {a\,b\,x^{12}\,\left (b\,e+a\,h\right )}{4} \] Input:
int(x^3*(a + b*x^3)^3*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x)
Output:
x^7*((a^3*f)/7 + (3*a^2*b*c)/7) + x^13*((b^3*c)/13 + (3*a*b^2*f)/13) + x^8 *((a^3*g)/8 + (3*a^2*b*d)/8) + x^14*((b^3*d)/14 + (3*a*b^2*g)/14) + x^9*(( a^3*h)/9 + (a^2*b*e)/3) + x^15*((b^3*e)/15 + (a*b^2*h)/5) + (a^3*c*x^4)/4 + (a^3*d*x^5)/5 + (a^3*e*x^6)/6 + (b^3*f*x^16)/16 + (b^3*g*x^17)/17 + (b^3 *h*x^18)/18 + (3*a*b*x^10*(b*c + a*f))/10 + (3*a*b*x^11*(b*d + a*g))/11 + (a*b*x^12*(b*e + a*h))/4
Time = 0.15 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.03 \[ \int x^3 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {x^{4} \left (680680 b^{3} h \,x^{14}+720720 b^{3} g \,x^{13}+765765 b^{3} f \,x^{12}+2450448 a \,b^{2} h \,x^{11}+816816 b^{3} e \,x^{11}+2625480 a \,b^{2} g \,x^{10}+875160 b^{3} d \,x^{10}+2827440 a \,b^{2} f \,x^{9}+942480 b^{3} c \,x^{9}+3063060 a^{2} b h \,x^{8}+3063060 a \,b^{2} e \,x^{8}+3341520 a^{2} b g \,x^{7}+3341520 a \,b^{2} d \,x^{7}+3675672 a^{2} b f \,x^{6}+3675672 a \,b^{2} c \,x^{6}+1361360 a^{3} h \,x^{5}+4084080 a^{2} b e \,x^{5}+1531530 a^{3} g \,x^{4}+4594590 a^{2} b d \,x^{4}+1750320 a^{3} f \,x^{3}+5250960 a^{2} b c \,x^{3}+2042040 a^{3} e \,x^{2}+2450448 a^{3} d x +3063060 a^{3} c \right )}{12252240} \] Input:
int(x^3*(b*x^3+a)^3*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x)
Output:
(x**4*(3063060*a**3*c + 2450448*a**3*d*x + 2042040*a**3*e*x**2 + 1750320*a **3*f*x**3 + 1531530*a**3*g*x**4 + 1361360*a**3*h*x**5 + 5250960*a**2*b*c* x**3 + 4594590*a**2*b*d*x**4 + 4084080*a**2*b*e*x**5 + 3675672*a**2*b*f*x* *6 + 3341520*a**2*b*g*x**7 + 3063060*a**2*b*h*x**8 + 3675672*a*b**2*c*x**6 + 3341520*a*b**2*d*x**7 + 3063060*a*b**2*e*x**8 + 2827440*a*b**2*f*x**9 + 2625480*a*b**2*g*x**10 + 2450448*a*b**2*h*x**11 + 942480*b**3*c*x**9 + 87 5160*b**3*d*x**10 + 816816*b**3*e*x**11 + 765765*b**3*f*x**12 + 720720*b** 3*g*x**13 + 680680*b**3*h*x**14))/12252240