Integrand size = 22, antiderivative size = 238 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^3 \, dx=a^3 x+\frac {3}{2} a^2 b x^2+a \left (b^2+a c\right ) x^3+\frac {1}{4} b \left (3 a^2+b^2+6 a c\right ) x^4+\frac {3}{5} \left (a^3+b^2 c+a \left (2 b^2+c^2\right )\right ) x^5+\frac {1}{2} b \left (2 a^2+b^2+2 a c+c^2\right ) x^6+\frac {1}{7} \left (6 a b^2+6 a^2 c+6 b^2 c+c^3\right ) x^7+\frac {3}{8} b \left (2 a^2+b^2+2 a c+c^2\right ) x^8+\frac {1}{3} \left (a^3+b^2 c+a \left (2 b^2+c^2\right )\right ) x^9+\frac {1}{10} b \left (3 a^2+b^2+6 a c\right ) x^{10}+\frac {3}{11} a \left (b^2+a c\right ) x^{11}+\frac {1}{4} a^2 b x^{12}+\frac {a^3 x^{13}}{13} \] Output:
a^3*x+3/2*a^2*b*x^2+a*(a*c+b^2)*x^3+1/4*b*(3*a^2+6*a*c+b^2)*x^4+3/5*(a^3+b ^2*c+a*(2*b^2+c^2))*x^5+1/2*b*(2*a^2+2*a*c+b^2+c^2)*x^6+1/7*(6*a^2*c+6*a*b ^2+6*b^2*c+c^3)*x^7+3/8*b*(2*a^2+2*a*c+b^2+c^2)*x^8+1/3*(a^3+b^2*c+a*(2*b^ 2+c^2))*x^9+1/10*b*(3*a^2+6*a*c+b^2)*x^10+3/11*a*(a*c+b^2)*x^11+1/4*a^2*b* x^12+1/13*a^3*x^13
Time = 0.06 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^3 \, dx=a^3 x+\frac {3}{2} a^2 b x^2+a \left (b^2+a c\right ) x^3+\frac {1}{4} b \left (3 a^2+b^2+6 a c\right ) x^4+\frac {3}{5} \left (a^3+2 a b^2+b^2 c+a c^2\right ) x^5+\frac {1}{2} b \left (2 a^2+b^2+2 a c+c^2\right ) x^6+\frac {1}{7} \left (6 a b^2+6 a^2 c+6 b^2 c+c^3\right ) x^7+\frac {3}{8} b \left (2 a^2+b^2+2 a c+c^2\right ) x^8+\frac {1}{3} \left (a^3+2 a b^2+b^2 c+a c^2\right ) x^9+\frac {1}{10} b \left (3 a^2+b^2+6 a c\right ) x^{10}+\frac {3}{11} a \left (b^2+a c\right ) x^{11}+\frac {1}{4} a^2 b x^{12}+\frac {a^3 x^{13}}{13} \] Input:
Integrate[(a + b*x + c*x^2 + b*x^3 + a*x^4)^3,x]
Output:
a^3*x + (3*a^2*b*x^2)/2 + a*(b^2 + a*c)*x^3 + (b*(3*a^2 + b^2 + 6*a*c)*x^4 )/4 + (3*(a^3 + 2*a*b^2 + b^2*c + a*c^2)*x^5)/5 + (b*(2*a^2 + b^2 + 2*a*c + c^2)*x^6)/2 + ((6*a*b^2 + 6*a^2*c + 6*b^2*c + c^3)*x^7)/7 + (3*b*(2*a^2 + b^2 + 2*a*c + c^2)*x^8)/8 + ((a^3 + 2*a*b^2 + b^2*c + a*c^2)*x^9)/3 + (b *(3*a^2 + b^2 + 6*a*c)*x^10)/10 + (3*a*(b^2 + a*c)*x^11)/11 + (a^2*b*x^12) /4 + (a^3*x^13)/13
Time = 0.44 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2465, 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.
| \(\displaystyle \int \left (a x^4+a+b x^3+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 2465 |
| \(\displaystyle \int \left (3 x^8 \left (a^3+a \left (2 b^2+c^2\right )+b^2 c\right )+3 x^4 \left (a^3+a \left (2 b^2+c^2\right )+b^2 c\right )+a^3 x^{12}+a^3+x^6 \left (6 a^2 c+6 a b^2+6 b^2 c+c^3\right )+3 b x^7 \left (2 a^2+2 a c+b^2+c^2\right )+3 b x^5 \left (2 a^2+2 a c+b^2+c^2\right )+b x^9 \left (3 a^2+6 a c+b^2\right )+b x^3 \left (3 a^2+6 a c+b^2\right )+3 a^2 b x^{11}+3 a^2 b x+3 a x^{10} \left (a c+b^2\right )+3 a x^2 \left (a c+b^2\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} x^9 \left (a^3+a \left (2 b^2+c^2\right )+b^2 c\right )+\frac {3}{5} x^5 \left (a^3+a \left (2 b^2+c^2\right )+b^2 c\right )+\frac {a^3 x^{13}}{13}+a^3 x+\frac {1}{7} x^7 \left (6 a^2 c+6 a b^2+6 b^2 c+c^3\right )+\frac {3}{8} b x^8 \left (2 a^2+2 a c+b^2+c^2\right )+\frac {1}{2} b x^6 \left (2 a^2+2 a c+b^2+c^2\right )+\frac {1}{10} b x^{10} \left (3 a^2+6 a c+b^2\right )+\frac {1}{4} b x^4 \left (3 a^2+6 a c+b^2\right )+\frac {1}{4} a^2 b x^{12}+\frac {3}{2} a^2 b x^2+\frac {3}{11} a x^{11} \left (a c+b^2\right )+a x^3 \left (a c+b^2\right )\) |
Input:
Int[(a + b*x + c*x^2 + b*x^3 + a*x^4)^3,x]
Output:
a^3*x + (3*a^2*b*x^2)/2 + a*(b^2 + a*c)*x^3 + (b*(3*a^2 + b^2 + 6*a*c)*x^4 )/4 + (3*(a^3 + b^2*c + a*(2*b^2 + c^2))*x^5)/5 + (b*(2*a^2 + b^2 + 2*a*c + c^2)*x^6)/2 + ((6*a*b^2 + 6*a^2*c + 6*b^2*c + c^3)*x^7)/7 + (3*b*(2*a^2 + b^2 + 2*a*c + c^2)*x^8)/8 + ((a^3 + b^2*c + a*(2*b^2 + c^2))*x^9)/3 + (b *(3*a^2 + b^2 + 6*a*c)*x^10)/10 + (3*a*(b^2 + a*c)*x^11)/11 + (a^2*b*x^12) /4 + (a^3*x^13)/13
Int[(u_.)*(Px_)^(p_), x_Symbol] :> Int[ExpandToSum[u, Px^p, x], x] /; PolyQ [Px, x] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x ] && IGtQ[p, 0]
Time = 0.08 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.02
| method | result | size |
| norman | \(\frac {a^{3} x^{13}}{13}+\frac {a^{2} b \,x^{12}}{4}+\left (\frac {3}{11} a^{2} c +\frac {3}{11} b^{2} a \right ) x^{11}+\left (\frac {3}{10} b \,a^{2}+\frac {3}{5} a b c +\frac {1}{10} b^{3}\right ) x^{10}+\left (\frac {1}{3} a^{3}+\frac {2}{3} b^{2} a +\frac {1}{3} a \,c^{2}+\frac {1}{3} c \,b^{2}\right ) x^{9}+\left (\frac {3}{4} b \,a^{2}+\frac {3}{4} a b c +\frac {3}{8} b^{3}+\frac {3}{8} b \,c^{2}\right ) x^{8}+\left (\frac {6}{7} a^{2} c +\frac {6}{7} b^{2} a +\frac {6}{7} c \,b^{2}+\frac {1}{7} c^{3}\right ) x^{7}+\left (b \,a^{2}+a b c +\frac {1}{2} b^{3}+\frac {1}{2} b \,c^{2}\right ) x^{6}+\left (\frac {3}{5} a^{3}+\frac {6}{5} b^{2} a +\frac {3}{5} a \,c^{2}+\frac {3}{5} c \,b^{2}\right ) x^{5}+\left (\frac {3}{4} b \,a^{2}+\frac {3}{2} a b c +\frac {1}{4} b^{3}\right ) x^{4}+\left (a^{2} c +b^{2} a \right ) x^{3}+\frac {3 b \,a^{2} x^{2}}{2}+a^{3} x\) | \(243\) |
| gosper | \(\frac {3}{8} b^{3} x^{8}+\frac {3}{4} a^{2} b \,x^{4}+\frac {6}{7} a \,b^{2} x^{7}+\frac {3}{5} x^{10} a b c +\frac {3}{4} x^{8} a b c +x^{6} a b c +\frac {3}{2} x^{4} a b c +\frac {1}{10} b^{3} x^{10}+\frac {6}{5} b^{2} a \,x^{5}+a^{3} x +\frac {2}{3} a \,b^{2} x^{9}+\frac {3}{2} b \,a^{2} x^{2}+a \,b^{2} x^{3}+\frac {3}{10} x^{10} b \,a^{2}+\frac {1}{4} a^{2} b \,x^{12}+\frac {1}{2} x^{6} b^{3}+\frac {1}{7} x^{7} c^{3}+\frac {1}{4} b^{3} x^{4}+\frac {3}{11} x^{11} a^{2} c +\frac {3}{11} x^{11} b^{2} a +\frac {3}{5} x^{5} a^{3}+\frac {1}{3} x^{9} a^{3}+a^{2} b \,x^{6}+\frac {1}{3} x^{9} a \,c^{2}+a^{2} c \,x^{3}+\frac {1}{13} a^{3} x^{13}+\frac {6}{7} x^{7} a^{2} c +\frac {6}{7} x^{7} c \,b^{2}+\frac {1}{2} x^{6} b \,c^{2}+\frac {3}{5} x^{5} a \,c^{2}+\frac {3}{5} x^{5} c \,b^{2}+\frac {1}{3} x^{9} c \,b^{2}+\frac {3}{4} x^{8} b \,a^{2}+\frac {3}{8} x^{8} b \,c^{2}\) | \(288\) |
| risch | \(\frac {3}{8} b^{3} x^{8}+\frac {3}{4} a^{2} b \,x^{4}+\frac {6}{7} a \,b^{2} x^{7}+\frac {3}{5} x^{10} a b c +\frac {3}{4} x^{8} a b c +x^{6} a b c +\frac {3}{2} x^{4} a b c +\frac {1}{10} b^{3} x^{10}+\frac {6}{5} b^{2} a \,x^{5}+a^{3} x +\frac {2}{3} a \,b^{2} x^{9}+\frac {3}{2} b \,a^{2} x^{2}+a \,b^{2} x^{3}+\frac {3}{10} x^{10} b \,a^{2}+\frac {1}{4} a^{2} b \,x^{12}+\frac {1}{2} x^{6} b^{3}+\frac {1}{7} x^{7} c^{3}+\frac {1}{4} b^{3} x^{4}+\frac {3}{11} x^{11} a^{2} c +\frac {3}{11} x^{11} b^{2} a +\frac {3}{5} x^{5} a^{3}+\frac {1}{3} x^{9} a^{3}+a^{2} b \,x^{6}+\frac {1}{3} x^{9} a \,c^{2}+a^{2} c \,x^{3}+\frac {1}{13} a^{3} x^{13}+\frac {6}{7} x^{7} a^{2} c +\frac {6}{7} x^{7} c \,b^{2}+\frac {1}{2} x^{6} b \,c^{2}+\frac {3}{5} x^{5} a \,c^{2}+\frac {3}{5} x^{5} c \,b^{2}+\frac {1}{3} x^{9} c \,b^{2}+\frac {3}{4} x^{8} b \,a^{2}+\frac {3}{8} x^{8} b \,c^{2}\) | \(288\) |
| parallelrisch | \(\frac {3}{8} b^{3} x^{8}+\frac {3}{4} a^{2} b \,x^{4}+\frac {6}{7} a \,b^{2} x^{7}+\frac {3}{5} x^{10} a b c +\frac {3}{4} x^{8} a b c +x^{6} a b c +\frac {3}{2} x^{4} a b c +\frac {1}{10} b^{3} x^{10}+\frac {6}{5} b^{2} a \,x^{5}+a^{3} x +\frac {2}{3} a \,b^{2} x^{9}+\frac {3}{2} b \,a^{2} x^{2}+a \,b^{2} x^{3}+\frac {3}{10} x^{10} b \,a^{2}+\frac {1}{4} a^{2} b \,x^{12}+\frac {1}{2} x^{6} b^{3}+\frac {1}{7} x^{7} c^{3}+\frac {1}{4} b^{3} x^{4}+\frac {3}{11} x^{11} a^{2} c +\frac {3}{11} x^{11} b^{2} a +\frac {3}{5} x^{5} a^{3}+\frac {1}{3} x^{9} a^{3}+a^{2} b \,x^{6}+\frac {1}{3} x^{9} a \,c^{2}+a^{2} c \,x^{3}+\frac {1}{13} a^{3} x^{13}+\frac {6}{7} x^{7} a^{2} c +\frac {6}{7} x^{7} c \,b^{2}+\frac {1}{2} x^{6} b \,c^{2}+\frac {3}{5} x^{5} a \,c^{2}+\frac {3}{5} x^{5} c \,b^{2}+\frac {1}{3} x^{9} c \,b^{2}+\frac {3}{4} x^{8} b \,a^{2}+\frac {3}{8} x^{8} b \,c^{2}\) | \(288\) |
| orering | \(\frac {x \left (9240 a^{3} x^{12}+30030 b \,a^{2} x^{11}+32760 a^{2} c \,x^{10}+32760 a \,b^{2} x^{10}+36036 a^{2} b \,x^{9}+72072 a b c \,x^{9}+12012 x^{9} b^{3}+40040 a^{3} x^{8}+80080 a \,b^{2} x^{8}+40040 a \,c^{2} x^{8}+40040 b^{2} c \,x^{8}+90090 a^{2} b \,x^{7}+90090 a b c \,x^{7}+45045 b^{3} x^{7}+45045 b \,c^{2} x^{7}+102960 a^{2} c \,x^{6}+102960 a \,x^{6} b^{2}+102960 b^{2} c \,x^{6}+17160 c^{3} x^{6}+120120 a^{2} b \,x^{5}+120120 a b c \,x^{5}+60060 b^{3} x^{5}+60060 b \,c^{2} x^{5}+72072 a^{3} x^{4}+144144 a \,b^{2} x^{4}+72072 a \,c^{2} x^{4}+72072 b^{2} c \,x^{4}+90090 a^{2} x^{3} b +180180 a b c \,x^{3}+30030 b^{3} x^{3}+120120 a^{2} c \,x^{2}+120120 a \,b^{2} x^{2}+180180 b \,a^{2} x +120120 a^{3}\right )}{120120}\) | \(293\) |
| default | \(\frac {a^{3} x^{13}}{13}+\frac {a^{2} b \,x^{12}}{4}+\frac {\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +a^{2} c \right ) x^{11}}{11}+\frac {\left (a \left (2 a b +2 b c \right )+b \left (2 a c +b^{2}\right )+2 a b c +b \,a^{2}\right ) x^{10}}{10}+\frac {\left (a \left (2 a^{2}+2 b^{2}+c^{2}\right )+b \left (2 a b +2 b c \right )+c \left (2 a c +b^{2}\right )+2 b^{2} a +a^{3}\right ) x^{9}}{9}+\frac {\left (a \left (2 a b +2 b c \right )+b \left (2 a^{2}+2 b^{2}+c^{2}\right )+c \left (2 a b +2 b c \right )+b \left (2 a c +b^{2}\right )+2 b \,a^{2}\right ) x^{8}}{8}+\frac {\left (2 a \left (2 a c +b^{2}\right )+2 b \left (2 a b +2 b c \right )+c \left (2 a^{2}+2 b^{2}+c^{2}\right )\right ) x^{7}}{7}+\frac {\left (a \left (2 a b +2 b c \right )+b \left (2 a^{2}+2 b^{2}+c^{2}\right )+c \left (2 a b +2 b c \right )+b \left (2 a c +b^{2}\right )+2 b \,a^{2}\right ) x^{6}}{6}+\frac {\left (a \left (2 a^{2}+2 b^{2}+c^{2}\right )+b \left (2 a b +2 b c \right )+c \left (2 a c +b^{2}\right )+2 b^{2} a +a^{3}\right ) x^{5}}{5}+\frac {\left (a \left (2 a b +2 b c \right )+b \left (2 a c +b^{2}\right )+2 a b c +b \,a^{2}\right ) x^{4}}{4}+\frac {\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +a^{2} c \right ) x^{3}}{3}+\frac {3 b \,a^{2} x^{2}}{2}+a^{3} x\) | \(430\) |
Input:
int((a*x^4+b*x^3+c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/13*a^3*x^13+1/4*a^2*b*x^12+(3/11*a^2*c+3/11*b^2*a)*x^11+(3/10*b*a^2+3/5* a*b*c+1/10*b^3)*x^10+(1/3*a^3+2/3*b^2*a+1/3*a*c^2+1/3*c*b^2)*x^9+(3/4*b*a^ 2+3/4*a*b*c+3/8*b^3+3/8*b*c^2)*x^8+(6/7*a^2*c+6/7*b^2*a+6/7*c*b^2+1/7*c^3) *x^7+(b*a^2+a*b*c+1/2*b^3+1/2*b*c^2)*x^6+(3/5*a^3+6/5*b^2*a+3/5*a*c^2+3/5* c*b^2)*x^5+(3/4*b*a^2+3/2*a*b*c+1/4*b^3)*x^4+(a^2*c+a*b^2)*x^3+3/2*b*a^2*x ^2+a^3*x
Time = 0.06 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.96 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^3 \, dx=\frac {1}{13} \, a^{3} x^{13} + \frac {1}{4} \, a^{2} b x^{12} + \frac {3}{11} \, {\left (a b^{2} + a^{2} c\right )} x^{11} + \frac {1}{10} \, {\left (3 \, a^{2} b + b^{3} + 6 \, a b c\right )} x^{10} + \frac {1}{3} \, {\left (a^{3} + 2 \, a b^{2} + b^{2} c + a c^{2}\right )} x^{9} + \frac {3}{8} \, {\left (2 \, a^{2} b + b^{3} + 2 \, a b c + b c^{2}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, a b^{2} + c^{3} + 6 \, {\left (a^{2} + b^{2}\right )} c\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a^{2} b + b^{3} + 2 \, a b c + b c^{2}\right )} x^{6} + \frac {3}{5} \, {\left (a^{3} + 2 \, a b^{2} + b^{2} c + a c^{2}\right )} x^{5} + \frac {3}{2} \, a^{2} b x^{2} + \frac {1}{4} \, {\left (3 \, a^{2} b + b^{3} + 6 \, a b c\right )} x^{4} + a^{3} x + {\left (a b^{2} + a^{2} c\right )} x^{3} \] Input:
integrate((a*x^4+b*x^3+c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
1/13*a^3*x^13 + 1/4*a^2*b*x^12 + 3/11*(a*b^2 + a^2*c)*x^11 + 1/10*(3*a^2*b + b^3 + 6*a*b*c)*x^10 + 1/3*(a^3 + 2*a*b^2 + b^2*c + a*c^2)*x^9 + 3/8*(2* a^2*b + b^3 + 2*a*b*c + b*c^2)*x^8 + 1/7*(6*a*b^2 + c^3 + 6*(a^2 + b^2)*c) *x^7 + 1/2*(2*a^2*b + b^3 + 2*a*b*c + b*c^2)*x^6 + 3/5*(a^3 + 2*a*b^2 + b^ 2*c + a*c^2)*x^5 + 3/2*a^2*b*x^2 + 1/4*(3*a^2*b + b^3 + 6*a*b*c)*x^4 + a^3 *x + (a*b^2 + a^2*c)*x^3
Time = 0.04 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.18 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^3 \, dx=\frac {a^{3} x^{13}}{13} + a^{3} x + \frac {a^{2} b x^{12}}{4} + \frac {3 a^{2} b x^{2}}{2} + x^{11} \cdot \left (\frac {3 a^{2} c}{11} + \frac {3 a b^{2}}{11}\right ) + x^{10} \cdot \left (\frac {3 a^{2} b}{10} + \frac {3 a b c}{5} + \frac {b^{3}}{10}\right ) + x^{9} \left (\frac {a^{3}}{3} + \frac {2 a b^{2}}{3} + \frac {a c^{2}}{3} + \frac {b^{2} c}{3}\right ) + x^{8} \cdot \left (\frac {3 a^{2} b}{4} + \frac {3 a b c}{4} + \frac {3 b^{3}}{8} + \frac {3 b c^{2}}{8}\right ) + x^{7} \cdot \left (\frac {6 a^{2} c}{7} + \frac {6 a b^{2}}{7} + \frac {6 b^{2} c}{7} + \frac {c^{3}}{7}\right ) + x^{6} \left (a^{2} b + a b c + \frac {b^{3}}{2} + \frac {b c^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 a^{3}}{5} + \frac {6 a b^{2}}{5} + \frac {3 a c^{2}}{5} + \frac {3 b^{2} c}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} b}{4} + \frac {3 a b c}{2} + \frac {b^{3}}{4}\right ) + x^{3} \left (a^{2} c + a b^{2}\right ) \] Input:
integrate((a*x**4+b*x**3+c*x**2+b*x+a)**3,x)
Output:
a**3*x**13/13 + a**3*x + a**2*b*x**12/4 + 3*a**2*b*x**2/2 + x**11*(3*a**2* c/11 + 3*a*b**2/11) + x**10*(3*a**2*b/10 + 3*a*b*c/5 + b**3/10) + x**9*(a* *3/3 + 2*a*b**2/3 + a*c**2/3 + b**2*c/3) + x**8*(3*a**2*b/4 + 3*a*b*c/4 + 3*b**3/8 + 3*b*c**2/8) + x**7*(6*a**2*c/7 + 6*a*b**2/7 + 6*b**2*c/7 + c**3 /7) + x**6*(a**2*b + a*b*c + b**3/2 + b*c**2/2) + x**5*(3*a**3/5 + 6*a*b** 2/5 + 3*a*c**2/5 + 3*b**2*c/5) + x**4*(3*a**2*b/4 + 3*a*b*c/2 + b**3/4) + x**3*(a**2*c + a*b**2)
Time = 0.04 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.19 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^3 \, dx=\frac {1}{13} \, a^{3} x^{13} + \frac {1}{4} \, a^{2} b x^{12} + \frac {3}{11} \, a b^{2} x^{11} + \frac {1}{10} \, b^{3} x^{10} + \frac {1}{7} \, c^{3} x^{7} + \frac {1}{4} \, b^{3} x^{4} + a^{3} x + \frac {1}{20} \, {\left (12 \, a x^{5} + 15 \, b x^{4} + 20 \, c x^{3} + 30 \, b x^{2}\right )} a^{2} + \frac {1}{70} \, {\left (30 \, a x^{7} + 35 \, b x^{6} + 42 \, c x^{5}\right )} b^{2} + \frac {1}{24} \, {\left (8 \, a x^{9} + 9 \, b x^{8}\right )} c^{2} + \frac {1}{420} \, {\left (140 \, a^{2} x^{9} + 315 \, a b x^{8} + 180 \, b^{2} x^{7} + 252 \, c^{2} x^{5} + 420 \, b^{2} x^{3} + 42 \, {\left (10 \, a x^{6} + 12 \, b x^{5} + 15 \, c x^{4}\right )} b + 60 \, {\left (6 \, a x^{7} + 7 \, b x^{6}\right )} c\right )} a + \frac {1}{840} \, {\left (252 \, a^{2} x^{10} + 560 \, a b x^{9} + 720 \, b c x^{7} + 315 \, {\left (b^{2} + 2 \, a c\right )} x^{8} + 420 \, c^{2} x^{6}\right )} b + \frac {1}{165} \, {\left (45 \, a^{2} x^{11} + 99 \, a b x^{10} + 55 \, b^{2} x^{9}\right )} c \] Input:
integrate((a*x^4+b*x^3+c*x^2+b*x+a)^3,x, algorithm="maxima")
Output:
1/13*a^3*x^13 + 1/4*a^2*b*x^12 + 3/11*a*b^2*x^11 + 1/10*b^3*x^10 + 1/7*c^3 *x^7 + 1/4*b^3*x^4 + a^3*x + 1/20*(12*a*x^5 + 15*b*x^4 + 20*c*x^3 + 30*b*x ^2)*a^2 + 1/70*(30*a*x^7 + 35*b*x^6 + 42*c*x^5)*b^2 + 1/24*(8*a*x^9 + 9*b* x^8)*c^2 + 1/420*(140*a^2*x^9 + 315*a*b*x^8 + 180*b^2*x^7 + 252*c^2*x^5 + 420*b^2*x^3 + 42*(10*a*x^6 + 12*b*x^5 + 15*c*x^4)*b + 60*(6*a*x^7 + 7*b*x^ 6)*c)*a + 1/840*(252*a^2*x^10 + 560*a*b*x^9 + 720*b*c*x^7 + 315*(b^2 + 2*a *c)*x^8 + 420*c^2*x^6)*b + 1/165*(45*a^2*x^11 + 99*a*b*x^10 + 55*b^2*x^9)* c
Time = 0.13 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.21 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^3 \, dx=\frac {1}{13} \, a^{3} x^{13} + \frac {1}{4} \, a^{2} b x^{12} + \frac {3}{11} \, a b^{2} x^{11} + \frac {3}{11} \, a^{2} c x^{11} + \frac {3}{10} \, a^{2} b x^{10} + \frac {1}{10} \, b^{3} x^{10} + \frac {3}{5} \, a b c x^{10} + \frac {1}{3} \, a^{3} x^{9} + \frac {2}{3} \, a b^{2} x^{9} + \frac {1}{3} \, b^{2} c x^{9} + \frac {1}{3} \, a c^{2} x^{9} + \frac {3}{4} \, a^{2} b x^{8} + \frac {3}{8} \, b^{3} x^{8} + \frac {3}{4} \, a b c x^{8} + \frac {3}{8} \, b c^{2} x^{8} + \frac {6}{7} \, a b^{2} x^{7} + \frac {6}{7} \, a^{2} c x^{7} + \frac {6}{7} \, b^{2} c x^{7} + \frac {1}{7} \, c^{3} x^{7} + a^{2} b x^{6} + \frac {1}{2} \, b^{3} x^{6} + a b c x^{6} + \frac {1}{2} \, b c^{2} x^{6} + \frac {3}{5} \, a^{3} x^{5} + \frac {6}{5} \, a b^{2} x^{5} + \frac {3}{5} \, b^{2} c x^{5} + \frac {3}{5} \, a c^{2} x^{5} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{4} \, b^{3} x^{4} + \frac {3}{2} \, a b c x^{4} + a b^{2} x^{3} + a^{2} c x^{3} + \frac {3}{2} \, a^{2} b x^{2} + a^{3} x \] Input:
integrate((a*x^4+b*x^3+c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
1/13*a^3*x^13 + 1/4*a^2*b*x^12 + 3/11*a*b^2*x^11 + 3/11*a^2*c*x^11 + 3/10* a^2*b*x^10 + 1/10*b^3*x^10 + 3/5*a*b*c*x^10 + 1/3*a^3*x^9 + 2/3*a*b^2*x^9 + 1/3*b^2*c*x^9 + 1/3*a*c^2*x^9 + 3/4*a^2*b*x^8 + 3/8*b^3*x^8 + 3/4*a*b*c* x^8 + 3/8*b*c^2*x^8 + 6/7*a*b^2*x^7 + 6/7*a^2*c*x^7 + 6/7*b^2*c*x^7 + 1/7* c^3*x^7 + a^2*b*x^6 + 1/2*b^3*x^6 + a*b*c*x^6 + 1/2*b*c^2*x^6 + 3/5*a^3*x^ 5 + 6/5*a*b^2*x^5 + 3/5*b^2*c*x^5 + 3/5*a*c^2*x^5 + 3/4*a^2*b*x^4 + 1/4*b^ 3*x^4 + 3/2*a*b*c*x^4 + a*b^2*x^3 + a^2*c*x^3 + 3/2*a^2*b*x^2 + a^3*x
Time = 0.15 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.94 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^3 \, dx=x^9\,\left (\frac {a^3}{3}+\frac {2\,a\,b^2}{3}+\frac {a\,c^2}{3}+\frac {b^2\,c}{3}\right )+x^5\,\left (\frac {3\,a^3}{5}+\frac {6\,a\,b^2}{5}+\frac {3\,a\,c^2}{5}+\frac {3\,b^2\,c}{5}\right )+x^7\,\left (\frac {6\,a^2\,c}{7}+\frac {6\,a\,b^2}{7}+\frac {6\,b^2\,c}{7}+\frac {c^3}{7}\right )+a^3\,x+\frac {a^3\,x^{13}}{13}+\frac {3\,a^2\,b\,x^2}{2}+\frac {a^2\,b\,x^{12}}{4}+\frac {b\,x^6\,\left (2\,a^2+2\,a\,c+b^2+c^2\right )}{2}+\frac {3\,b\,x^8\,\left (2\,a^2+2\,a\,c+b^2+c^2\right )}{8}+a\,x^3\,\left (b^2+a\,c\right )+\frac {3\,a\,x^{11}\,\left (b^2+a\,c\right )}{11}+\frac {b\,x^4\,\left (3\,a^2+6\,c\,a+b^2\right )}{4}+\frac {b\,x^{10}\,\left (3\,a^2+6\,c\,a+b^2\right )}{10} \] Input:
int((a + b*x + a*x^4 + b*x^3 + c*x^2)^3,x)
Output:
x^9*((2*a*b^2)/3 + (a*c^2)/3 + (b^2*c)/3 + a^3/3) + x^5*((6*a*b^2)/5 + (3* a*c^2)/5 + (3*b^2*c)/5 + (3*a^3)/5) + x^7*((6*a*b^2)/7 + (6*a^2*c)/7 + (6* b^2*c)/7 + c^3/7) + a^3*x + (a^3*x^13)/13 + (3*a^2*b*x^2)/2 + (a^2*b*x^12) /4 + (b*x^6*(2*a*c + 2*a^2 + b^2 + c^2))/2 + (3*b*x^8*(2*a*c + 2*a^2 + b^2 + c^2))/8 + a*x^3*(a*c + b^2) + (3*a*x^11*(a*c + b^2))/11 + (b*x^4*(6*a*c + 3*a^2 + b^2))/4 + (b*x^10*(6*a*c + 3*a^2 + b^2))/10
Time = 0.14 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.23 \[ \int \left (a+b x+c x^2+b x^3+a x^4\right )^3 \, dx=\frac {x \left (9240 a^{3} x^{12}+30030 a^{2} b \,x^{11}+32760 a^{2} c \,x^{10}+32760 a \,b^{2} x^{10}+36036 a^{2} b \,x^{9}+72072 a b c \,x^{9}+12012 b^{3} x^{9}+40040 a^{3} x^{8}+80080 a \,b^{2} x^{8}+40040 a \,c^{2} x^{8}+40040 b^{2} c \,x^{8}+90090 a^{2} b \,x^{7}+90090 a b c \,x^{7}+45045 b^{3} x^{7}+45045 b \,c^{2} x^{7}+102960 a^{2} c \,x^{6}+102960 a \,b^{2} x^{6}+102960 b^{2} c \,x^{6}+17160 c^{3} x^{6}+120120 a^{2} b \,x^{5}+120120 a b c \,x^{5}+60060 b^{3} x^{5}+60060 b \,c^{2} x^{5}+72072 a^{3} x^{4}+144144 a \,b^{2} x^{4}+72072 a \,c^{2} x^{4}+72072 b^{2} c \,x^{4}+90090 a^{2} b \,x^{3}+180180 a b c \,x^{3}+30030 b^{3} x^{3}+120120 a^{2} c \,x^{2}+120120 a \,b^{2} x^{2}+180180 a^{2} b x +120120 a^{3}\right )}{120120} \] Input:
int((a*x^4+b*x^3+c*x^2+b*x+a)^3,x)
Output:
(x*(9240*a**3*x**12 + 40040*a**3*x**8 + 72072*a**3*x**4 + 120120*a**3 + 30 030*a**2*b*x**11 + 36036*a**2*b*x**9 + 90090*a**2*b*x**7 + 120120*a**2*b*x **5 + 90090*a**2*b*x**3 + 180180*a**2*b*x + 32760*a**2*c*x**10 + 102960*a* *2*c*x**6 + 120120*a**2*c*x**2 + 32760*a*b**2*x**10 + 80080*a*b**2*x**8 + 102960*a*b**2*x**6 + 144144*a*b**2*x**4 + 120120*a*b**2*x**2 + 72072*a*b*c *x**9 + 90090*a*b*c*x**7 + 120120*a*b*c*x**5 + 180180*a*b*c*x**3 + 40040*a *c**2*x**8 + 72072*a*c**2*x**4 + 12012*b**3*x**9 + 45045*b**3*x**7 + 60060 *b**3*x**5 + 30030*b**3*x**3 + 40040*b**2*c*x**8 + 102960*b**2*c*x**6 + 72 072*b**2*c*x**4 + 45045*b*c**2*x**7 + 60060*b*c**2*x**5 + 17160*c**3*x**6) )/120120