Integrand size = 18, antiderivative size = 160 \[ \int x^2 (a+b x)^n \left (c+d x^3\right ) \, dx=\frac {a^2 \left (b^3 c-a^3 d\right ) (a+b x)^{1+n}}{b^6 (1+n)}-\frac {a \left (2 b^3 c-5 a^3 d\right ) (a+b x)^{2+n}}{b^6 (2+n)}+\frac {\left (b^3 c-10 a^3 d\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {10 a^2 d (a+b x)^{4+n}}{b^6 (4+n)}-\frac {5 a d (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d (a+b x)^{6+n}}{b^6 (6+n)} \]
a^2*(-a^3*d+b^3*c)*(b*x+a)^(1+n)/b^6/(1+n)-a*(-5*a^3*d+2*b^3*c)*(b*x+a)^(2 +n)/b^6/(2+n)+(-10*a^3*d+b^3*c)*(b*x+a)^(3+n)/b^6/(3+n)+10*a^2*d*(b*x+a)^( 4+n)/b^6/(4+n)-5*a*d*(b*x+a)^(5+n)/b^6/(5+n)+d*(b*x+a)^(6+n)/b^6/(6+n)
Time = 0.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.83 \[ \int x^2 (a+b x)^n \left (c+d x^3\right ) \, dx=\frac {(a+b x)^{1+n} \left (\frac {a^2 b^3 c-a^5 d}{1+n}+\frac {a \left (-2 b^3 c+5 a^3 d\right ) (a+b x)}{2+n}+\frac {\left (b^3 c-10 a^3 d\right ) (a+b x)^2}{3+n}+\frac {10 a^2 d (a+b x)^3}{4+n}-\frac {5 a d (a+b x)^4}{5+n}+\frac {d (a+b x)^5}{6+n}\right )}{b^6} \]
((a + b*x)^(1 + n)*((a^2*b^3*c - a^5*d)/(1 + n) + (a*(-2*b^3*c + 5*a^3*d)* (a + b*x))/(2 + n) + ((b^3*c - 10*a^3*d)*(a + b*x)^2)/(3 + n) + (10*a^2*d* (a + b*x)^3)/(4 + n) - (5*a*d*(a + b*x)^4)/(5 + n) + (d*(a + b*x)^5)/(6 + n)))/b^6
Time = 0.34 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {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.
| \(\displaystyle \int x^2 \left (c+d x^3\right ) (a+b x)^n \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {a \left (5 a^3 d-2 b^3 c\right ) (a+b x)^{n+1}}{b^5}+\frac {\left (b^3 c-10 a^3 d\right ) (a+b x)^{n+2}}{b^5}+\frac {10 a^2 d (a+b x)^{n+3}}{b^5}+\frac {\left (a^2 b^3 c-a^5 d\right ) (a+b x)^n}{b^5}-\frac {5 a d (a+b x)^{n+4}}{b^5}+\frac {d (a+b x)^{n+5}}{b^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \left (2 b^3 c-5 a^3 d\right ) (a+b x)^{n+2}}{b^6 (n+2)}+\frac {\left (b^3 c-10 a^3 d\right ) (a+b x)^{n+3}}{b^6 (n+3)}+\frac {10 a^2 d (a+b x)^{n+4}}{b^6 (n+4)}+\frac {a^2 \left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^6 (n+1)}-\frac {5 a d (a+b x)^{n+5}}{b^6 (n+5)}+\frac {d (a+b x)^{n+6}}{b^6 (n+6)}\) |
(a^2*(b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^6*(1 + n)) - (a*(2*b^3*c - 5*a^ 3*d)*(a + b*x)^(2 + n))/(b^6*(2 + n)) + ((b^3*c - 10*a^3*d)*(a + b*x)^(3 + n))/(b^6*(3 + n)) + (10*a^2*d*(a + b*x)^(4 + n))/(b^6*(4 + n)) - (5*a*d*( a + b*x)^(5 + n))/(b^6*(5 + n)) + (d*(a + b*x)^(6 + n))/(b^6*(6 + n))
3.2.74.3.1 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])
Leaf count of result is larger than twice the leaf count of optimal. \(414\) vs. \(2(160)=320\).
Time = 0.90 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.59
| method | result | size |
| norman | \(\frac {d \,x^{6} {\mathrm e}^{n \ln \left (b x +a \right )}}{6+n}+\frac {\left (b^{3} c \,n^{3}+15 b^{3} c \,n^{2}+20 a^{3} d n +74 b^{3} c n +120 b^{3} c \right ) x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+18 n^{3}+119 n^{2}+342 n +360\right )}+\frac {a d n \,x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+11 n +30\right )}-\frac {2 a^{3} \left (-b^{3} c \,n^{3}-15 b^{3} c \,n^{2}-74 b^{3} c n +60 a^{3} d -120 b^{3} c \right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}+\frac {2 n \,a^{2} \left (-b^{3} c \,n^{3}-15 b^{3} c \,n^{2}-74 b^{3} c n +60 a^{3} d -120 b^{3} c \right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}-\frac {5 n d \,a^{2} x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+15 n^{2}+74 n +120\right )}-\frac {\left (-b^{3} c \,n^{3}-15 b^{3} c \,n^{2}-74 b^{3} c n +60 a^{3} d -120 b^{3} c \right ) a n \,x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+20 n^{4}+155 n^{3}+580 n^{2}+1044 n +720\right )}\) | \(415\) |
| gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (-b^{5} d \,n^{5} x^{5}-15 b^{5} d \,n^{4} x^{5}+5 a \,b^{4} d \,n^{4} x^{4}-85 b^{5} d \,n^{3} x^{5}+50 a \,b^{4} d \,n^{3} x^{4}-b^{5} c \,n^{5} x^{2}-225 b^{5} d \,n^{2} x^{5}-20 a^{2} b^{3} d \,n^{3} x^{3}+175 a \,b^{4} d \,n^{2} x^{4}-18 b^{5} c \,n^{4} x^{2}-274 b^{5} d n \,x^{5}-120 a^{2} b^{3} d \,n^{2} x^{3}+2 a \,b^{4} c \,n^{4} x +250 a \,b^{4} d n \,x^{4}-121 b^{5} c \,n^{3} x^{2}-120 x^{5} d \,b^{5}+60 a^{3} b^{2} d \,n^{2} x^{2}-220 a^{2} b^{3} d n \,x^{3}+32 a \,b^{4} c \,n^{3} x +120 d a \,x^{4} b^{4}-372 b^{5} c \,n^{2} x^{2}+180 a^{3} b^{2} d n \,x^{2}-2 a^{2} b^{3} c \,n^{3}-120 d \,a^{2} x^{3} b^{3}+178 a \,b^{4} c \,n^{2} x -508 b^{5} c n \,x^{2}-120 a^{4} b d n x +120 a^{3} b^{2} d \,x^{2}-30 a^{2} b^{3} c \,n^{2}+388 a \,b^{4} c n x -240 b^{5} c \,x^{2}-120 a^{4} b d x -148 a^{2} b^{3} c n +240 a \,b^{4} c x +120 d \,a^{5}-240 a^{2} b^{3} c \right )}{b^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}\) | \(451\) |
| risch | \(-\frac {\left (-b^{6} d \,n^{5} x^{6}-a \,b^{5} d \,n^{5} x^{5}-15 b^{6} d \,n^{4} x^{6}-10 a \,b^{5} d \,n^{4} x^{5}-85 b^{6} d \,n^{3} x^{6}+5 a^{2} b^{4} d \,n^{4} x^{4}-35 a \,b^{5} d \,n^{3} x^{5}-b^{6} c \,n^{5} x^{3}-225 b^{6} d \,n^{2} x^{6}+30 a^{2} b^{4} d \,n^{3} x^{4}-a \,b^{5} c \,n^{5} x^{2}-50 a \,b^{5} d \,n^{2} x^{5}-18 b^{6} c \,n^{4} x^{3}-274 b^{6} d n \,x^{6}-20 a^{3} b^{3} d \,n^{3} x^{3}+55 a^{2} b^{4} d \,n^{2} x^{4}-16 a \,b^{5} c \,n^{4} x^{2}-24 a d n \,x^{5} b^{5}-121 b^{6} c \,n^{3} x^{3}-120 d \,x^{6} b^{6}-60 a^{3} b^{3} d \,n^{2} x^{3}+2 a^{2} b^{4} c \,n^{4} x +30 a^{2} d n \,x^{4} b^{4}-89 a \,b^{5} c \,n^{3} x^{2}-372 b^{6} c \,n^{2} x^{3}+60 a^{4} b^{2} d \,n^{2} x^{2}-40 a^{3} b^{3} d n \,x^{3}+30 a^{2} b^{4} c \,n^{3} x -194 a \,b^{5} c \,n^{2} x^{2}-508 b^{6} c n \,x^{3}+60 a^{4} b^{2} d n \,x^{2}-2 a^{3} b^{3} c \,n^{3}+148 a^{2} b^{4} c \,n^{2} x -120 a \,b^{5} c n \,x^{2}-240 b^{6} c \,x^{3}-120 a^{5} b d n x -30 a^{3} b^{3} c \,n^{2}+240 a^{2} b^{4} c n x -148 a^{3} b^{3} c n +120 a^{6} d -240 a^{3} b^{3} c \right ) \left (b x +a \right )^{n}}{\left (5+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{6}}\) | \(541\) |
| parallelrisch | \(\frac {121 x^{3} \left (b x +a \right )^{n} a \,b^{6} c \,n^{3}+16 x^{2} \left (b x +a \right )^{n} a^{2} b^{5} c \,n^{4}-30 x^{4} \left (b x +a \right )^{n} a^{3} b^{4} d n +60 x^{3} \left (b x +a \right )^{n} a^{4} b^{3} d \,n^{2}+372 x^{3} \left (b x +a \right )^{n} a \,b^{6} c \,n^{2}+89 x^{2} \left (b x +a \right )^{n} a^{2} b^{5} c \,n^{3}-2 x \left (b x +a \right )^{n} a^{3} b^{4} c \,n^{4}+40 x^{3} \left (b x +a \right )^{n} a^{4} b^{3} d n +508 x^{3} \left (b x +a \right )^{n} a \,b^{6} c n -60 x^{2} \left (b x +a \right )^{n} a^{5} b^{2} d \,n^{2}+194 x^{2} \left (b x +a \right )^{n} a^{2} b^{5} c \,n^{2}-30 x \left (b x +a \right )^{n} a^{3} b^{4} c \,n^{3}-60 x^{2} \left (b x +a \right )^{n} a^{5} b^{2} d n +120 x^{2} \left (b x +a \right )^{n} a^{2} b^{5} c n -148 x \left (b x +a \right )^{n} a^{3} b^{4} c \,n^{2}+120 x \left (b x +a \right )^{n} a^{6} b d n -240 x \left (b x +a \right )^{n} a^{3} b^{4} c n -120 \left (b x +a \right )^{n} a^{7} d +120 x^{6} \left (b x +a \right )^{n} a \,b^{6} d +x^{6} \left (b x +a \right )^{n} a \,b^{6} d \,n^{5}+15 x^{6} \left (b x +a \right )^{n} a \,b^{6} d \,n^{4}+x^{5} \left (b x +a \right )^{n} a^{2} b^{5} d \,n^{5}+85 x^{6} \left (b x +a \right )^{n} a \,b^{6} d \,n^{3}+10 x^{5} \left (b x +a \right )^{n} a^{2} b^{5} d \,n^{4}+225 x^{6} \left (b x +a \right )^{n} a \,b^{6} d \,n^{2}+35 x^{5} \left (b x +a \right )^{n} a^{2} b^{5} d \,n^{3}-5 x^{4} \left (b x +a \right )^{n} a^{3} b^{4} d \,n^{4}+x^{3} \left (b x +a \right )^{n} a \,b^{6} c \,n^{5}+274 x^{6} \left (b x +a \right )^{n} a \,b^{6} d n +50 x^{5} \left (b x +a \right )^{n} a^{2} b^{5} d \,n^{2}-30 x^{4} \left (b x +a \right )^{n} a^{3} b^{4} d \,n^{3}+18 x^{3} \left (b x +a \right )^{n} a \,b^{6} c \,n^{4}+x^{2} \left (b x +a \right )^{n} a^{2} b^{5} c \,n^{5}+24 x^{5} \left (b x +a \right )^{n} a^{2} b^{5} d n -55 x^{4} \left (b x +a \right )^{n} a^{3} b^{4} d \,n^{2}+20 x^{3} \left (b x +a \right )^{n} a^{4} b^{3} d \,n^{3}+240 x^{3} \left (b x +a \right )^{n} a \,b^{6} c +2 \left (b x +a \right )^{n} a^{4} b^{3} c \,n^{3}+30 \left (b x +a \right )^{n} a^{4} b^{3} c \,n^{2}+148 \left (b x +a \right )^{n} a^{4} b^{3} c n +240 \left (b x +a \right )^{n} a^{4} b^{3} c}{\left (n^{5}+20 n^{4}+155 n^{3}+580 n^{2}+1044 n +720\right ) \left (1+n \right ) a \,b^{6}}\) | \(851\) |
d/(6+n)*x^6*exp(n*ln(b*x+a))+(b^3*c*n^3+15*b^3*c*n^2+20*a^3*d*n+74*b^3*c*n +120*b^3*c)/b^3/(n^4+18*n^3+119*n^2+342*n+360)*x^3*exp(n*ln(b*x+a))+a*d*n/ b/(n^2+11*n+30)*x^5*exp(n*ln(b*x+a))-2*a^3*(-b^3*c*n^3-15*b^3*c*n^2-74*b^3 *c*n+60*a^3*d-120*b^3*c)/b^6/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1764*n+7 20)*exp(n*ln(b*x+a))+2/b^5*n*a^2*(-b^3*c*n^3-15*b^3*c*n^2-74*b^3*c*n+60*a^ 3*d-120*b^3*c)/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1764*n+720)*x*exp(n*ln (b*x+a))-5*n*d*a^2/b^2/(n^3+15*n^2+74*n+120)*x^4*exp(n*ln(b*x+a))-(-b^3*c* n^3-15*b^3*c*n^2-74*b^3*c*n+60*a^3*d-120*b^3*c)*a/b^4*n/(n^5+20*n^4+155*n^ 3+580*n^2+1044*n+720)*x^2*exp(n*ln(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (160) = 320\).
Time = 0.29 (sec) , antiderivative size = 490, normalized size of antiderivative = 3.06 \[ \int x^2 (a+b x)^n \left (c+d x^3\right ) \, dx=\frac {{\left (2 \, a^{3} b^{3} c n^{3} + 30 \, a^{3} b^{3} c n^{2} + 148 \, a^{3} b^{3} c n + 240 \, a^{3} b^{3} c - 120 \, a^{6} d + {\left (b^{6} d n^{5} + 15 \, b^{6} d n^{4} + 85 \, b^{6} d n^{3} + 225 \, b^{6} d n^{2} + 274 \, b^{6} d n + 120 \, b^{6} d\right )} x^{6} + {\left (a b^{5} d n^{5} + 10 \, a b^{5} d n^{4} + 35 \, a b^{5} d n^{3} + 50 \, a b^{5} d n^{2} + 24 \, a b^{5} d n\right )} x^{5} - 5 \, {\left (a^{2} b^{4} d n^{4} + 6 \, a^{2} b^{4} d n^{3} + 11 \, a^{2} b^{4} d n^{2} + 6 \, a^{2} b^{4} d n\right )} x^{4} + {\left (b^{6} c n^{5} + 18 \, b^{6} c n^{4} + 240 \, b^{6} c + {\left (121 \, b^{6} c + 20 \, a^{3} b^{3} d\right )} n^{3} + 12 \, {\left (31 \, b^{6} c + 5 \, a^{3} b^{3} d\right )} n^{2} + 4 \, {\left (127 \, b^{6} c + 10 \, a^{3} b^{3} d\right )} n\right )} x^{3} + {\left (a b^{5} c n^{5} + 16 \, a b^{5} c n^{4} + 89 \, a b^{5} c n^{3} + 2 \, {\left (97 \, a b^{5} c - 30 \, a^{4} b^{2} d\right )} n^{2} + 60 \, {\left (2 \, a b^{5} c - a^{4} b^{2} d\right )} n\right )} x^{2} - 2 \, {\left (a^{2} b^{4} c n^{4} + 15 \, a^{2} b^{4} c n^{3} + 74 \, a^{2} b^{4} c n^{2} + 60 \, {\left (2 \, a^{2} b^{4} c - a^{5} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}} \]
(2*a^3*b^3*c*n^3 + 30*a^3*b^3*c*n^2 + 148*a^3*b^3*c*n + 240*a^3*b^3*c - 12 0*a^6*d + (b^6*d*n^5 + 15*b^6*d*n^4 + 85*b^6*d*n^3 + 225*b^6*d*n^2 + 274*b ^6*d*n + 120*b^6*d)*x^6 + (a*b^5*d*n^5 + 10*a*b^5*d*n^4 + 35*a*b^5*d*n^3 + 50*a*b^5*d*n^2 + 24*a*b^5*d*n)*x^5 - 5*(a^2*b^4*d*n^4 + 6*a^2*b^4*d*n^3 + 11*a^2*b^4*d*n^2 + 6*a^2*b^4*d*n)*x^4 + (b^6*c*n^5 + 18*b^6*c*n^4 + 240*b ^6*c + (121*b^6*c + 20*a^3*b^3*d)*n^3 + 12*(31*b^6*c + 5*a^3*b^3*d)*n^2 + 4*(127*b^6*c + 10*a^3*b^3*d)*n)*x^3 + (a*b^5*c*n^5 + 16*a*b^5*c*n^4 + 89*a *b^5*c*n^3 + 2*(97*a*b^5*c - 30*a^4*b^2*d)*n^2 + 60*(2*a*b^5*c - a^4*b^2*d )*n)*x^2 - 2*(a^2*b^4*c*n^4 + 15*a^2*b^4*c*n^3 + 74*a^2*b^4*c*n^2 + 60*(2* a^2*b^4*c - a^5*b*d)*n)*x)*(b*x + a)^n/(b^6*n^6 + 21*b^6*n^5 + 175*b^6*n^4 + 735*b^6*n^3 + 1624*b^6*n^2 + 1764*b^6*n + 720*b^6)
Leaf count of result is larger than twice the leaf count of optimal. 6397 vs. \(2 (144) = 288\).
Time = 2.03 (sec) , antiderivative size = 6397, normalized size of antiderivative = 39.98 \[ \int x^2 (a+b x)^n \left (c+d x^3\right ) \, dx=\text {Too large to display} \]
Piecewise((a**n*(c*x**3/3 + d*x**6/6), Eq(b, 0)), (60*a**5*d*log(a/b + x)/ (60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 137*a**5*d/(60*a**5*b**6 + 300*a**4* b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b **11*x**5) + 300*a**4*b*d*x*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x** 5) + 625*a**4*b*d*x/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 600*a**3*b**2*d* x**2*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 6 00*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 1100*a**3*b**2*d*x **2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x **3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 2*a**2*b**3*c/(60*a**5*b**6 + 30 0*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 600*a**2*b**3*d*x**3*log(a/b + x)/(60*a**5*b**6 + 300* a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 900*a**2*b**3*d*x**3/(60*a**5*b**6 + 300*a**4*b**7*x + 6 00*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 10*a*b**4*c*x/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 60 0*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 300*a*b**4*d*x**4*l og(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*...
Time = 0.20 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.58 \[ \int x^2 (a+b x)^n \left (c+d x^3\right ) \, dx=\frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \, {\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} \]
((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x + a)^n*c/((n^3 + 6*n^2 + 11*n + 6)*b^3) + ((n^5 + 15*n^4 + 85*n^3 + 225*n ^2 + 274*n + 120)*b^6*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a*b^5* x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^2*b^4*x^4 + 20*(n^3 + 3*n^2 + 2*n)* a^3*b^3*x^3 - 60*(n^2 + n)*a^4*b^2*x^2 + 120*a^5*b*n*x - 120*a^6)*(b*x + a )^n*d/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*b^6)
Leaf count of result is larger than twice the leaf count of optimal. 835 vs. \(2 (160) = 320\).
Time = 0.31 (sec) , antiderivative size = 835, normalized size of antiderivative = 5.22 \[ \int x^2 (a+b x)^n \left (c+d x^3\right ) \, dx=\frac {{\left (b x + a\right )}^{n} b^{6} d n^{5} x^{6} + {\left (b x + a\right )}^{n} a b^{5} d n^{5} x^{5} + 15 \, {\left (b x + a\right )}^{n} b^{6} d n^{4} x^{6} + 10 \, {\left (b x + a\right )}^{n} a b^{5} d n^{4} x^{5} + 85 \, {\left (b x + a\right )}^{n} b^{6} d n^{3} x^{6} + {\left (b x + a\right )}^{n} b^{6} c n^{5} x^{3} - 5 \, {\left (b x + a\right )}^{n} a^{2} b^{4} d n^{4} x^{4} + 35 \, {\left (b x + a\right )}^{n} a b^{5} d n^{3} x^{5} + 225 \, {\left (b x + a\right )}^{n} b^{6} d n^{2} x^{6} + {\left (b x + a\right )}^{n} a b^{5} c n^{5} x^{2} + 18 \, {\left (b x + a\right )}^{n} b^{6} c n^{4} x^{3} - 30 \, {\left (b x + a\right )}^{n} a^{2} b^{4} d n^{3} x^{4} + 50 \, {\left (b x + a\right )}^{n} a b^{5} d n^{2} x^{5} + 274 \, {\left (b x + a\right )}^{n} b^{6} d n x^{6} + 16 \, {\left (b x + a\right )}^{n} a b^{5} c n^{4} x^{2} + 121 \, {\left (b x + a\right )}^{n} b^{6} c n^{3} x^{3} + 20 \, {\left (b x + a\right )}^{n} a^{3} b^{3} d n^{3} x^{3} - 55 \, {\left (b x + a\right )}^{n} a^{2} b^{4} d n^{2} x^{4} + 24 \, {\left (b x + a\right )}^{n} a b^{5} d n x^{5} + 120 \, {\left (b x + a\right )}^{n} b^{6} d x^{6} - 2 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c n^{4} x + 89 \, {\left (b x + a\right )}^{n} a b^{5} c n^{3} x^{2} + 372 \, {\left (b x + a\right )}^{n} b^{6} c n^{2} x^{3} + 60 \, {\left (b x + a\right )}^{n} a^{3} b^{3} d n^{2} x^{3} - 30 \, {\left (b x + a\right )}^{n} a^{2} b^{4} d n x^{4} - 30 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c n^{3} x + 194 \, {\left (b x + a\right )}^{n} a b^{5} c n^{2} x^{2} - 60 \, {\left (b x + a\right )}^{n} a^{4} b^{2} d n^{2} x^{2} + 508 \, {\left (b x + a\right )}^{n} b^{6} c n x^{3} + 40 \, {\left (b x + a\right )}^{n} a^{3} b^{3} d n x^{3} + 2 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c n^{3} - 148 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c n^{2} x + 120 \, {\left (b x + a\right )}^{n} a b^{5} c n x^{2} - 60 \, {\left (b x + a\right )}^{n} a^{4} b^{2} d n x^{2} + 240 \, {\left (b x + a\right )}^{n} b^{6} c x^{3} + 30 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c n^{2} - 240 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c n x + 120 \, {\left (b x + a\right )}^{n} a^{5} b d n x + 148 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c n + 240 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c - 120 \, {\left (b x + a\right )}^{n} a^{6} d}{b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}} \]
((b*x + a)^n*b^6*d*n^5*x^6 + (b*x + a)^n*a*b^5*d*n^5*x^5 + 15*(b*x + a)^n* b^6*d*n^4*x^6 + 10*(b*x + a)^n*a*b^5*d*n^4*x^5 + 85*(b*x + a)^n*b^6*d*n^3* x^6 + (b*x + a)^n*b^6*c*n^5*x^3 - 5*(b*x + a)^n*a^2*b^4*d*n^4*x^4 + 35*(b* x + a)^n*a*b^5*d*n^3*x^5 + 225*(b*x + a)^n*b^6*d*n^2*x^6 + (b*x + a)^n*a*b ^5*c*n^5*x^2 + 18*(b*x + a)^n*b^6*c*n^4*x^3 - 30*(b*x + a)^n*a^2*b^4*d*n^3 *x^4 + 50*(b*x + a)^n*a*b^5*d*n^2*x^5 + 274*(b*x + a)^n*b^6*d*n*x^6 + 16*( b*x + a)^n*a*b^5*c*n^4*x^2 + 121*(b*x + a)^n*b^6*c*n^3*x^3 + 20*(b*x + a)^ n*a^3*b^3*d*n^3*x^3 - 55*(b*x + a)^n*a^2*b^4*d*n^2*x^4 + 24*(b*x + a)^n*a* b^5*d*n*x^5 + 120*(b*x + a)^n*b^6*d*x^6 - 2*(b*x + a)^n*a^2*b^4*c*n^4*x + 89*(b*x + a)^n*a*b^5*c*n^3*x^2 + 372*(b*x + a)^n*b^6*c*n^2*x^3 + 60*(b*x + a)^n*a^3*b^3*d*n^2*x^3 - 30*(b*x + a)^n*a^2*b^4*d*n*x^4 - 30*(b*x + a)^n* a^2*b^4*c*n^3*x + 194*(b*x + a)^n*a*b^5*c*n^2*x^2 - 60*(b*x + a)^n*a^4*b^2 *d*n^2*x^2 + 508*(b*x + a)^n*b^6*c*n*x^3 + 40*(b*x + a)^n*a^3*b^3*d*n*x^3 + 2*(b*x + a)^n*a^3*b^3*c*n^3 - 148*(b*x + a)^n*a^2*b^4*c*n^2*x + 120*(b*x + a)^n*a*b^5*c*n*x^2 - 60*(b*x + a)^n*a^4*b^2*d*n*x^2 + 240*(b*x + a)^n*b ^6*c*x^3 + 30*(b*x + a)^n*a^3*b^3*c*n^2 - 240*(b*x + a)^n*a^2*b^4*c*n*x + 120*(b*x + a)^n*a^5*b*d*n*x + 148*(b*x + a)^n*a^3*b^3*c*n + 240*(b*x + a)^ n*a^3*b^3*c - 120*(b*x + a)^n*a^6*d)/(b^6*n^6 + 21*b^6*n^5 + 175*b^6*n^4 + 735*b^6*n^3 + 1624*b^6*n^2 + 1764*b^6*n + 720*b^6)
Time = 19.82 (sec) , antiderivative size = 495, normalized size of antiderivative = 3.09 \[ \int x^2 (a+b x)^n \left (c+d x^3\right ) \, dx={\left (a+b\,x\right )}^n\,\left (\frac {d\,x^6\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720}+\frac {2\,a^3\,\left (-60\,d\,a^3+c\,b^3\,n^3+15\,c\,b^3\,n^2+74\,c\,b^3\,n+120\,c\,b^3\right )}{b^6\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {x^3\,\left (n^2+3\,n+2\right )\,\left (20\,d\,a^3\,n+c\,b^3\,n^3+15\,c\,b^3\,n^2+74\,c\,b^3\,n+120\,c\,b^3\right )}{b^3\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {2\,a^2\,n\,x\,\left (-60\,d\,a^3+c\,b^3\,n^3+15\,c\,b^3\,n^2+74\,c\,b^3\,n+120\,c\,b^3\right )}{b^5\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {a\,d\,n\,x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {a\,n\,x^2\,\left (n+1\right )\,\left (-60\,d\,a^3+c\,b^3\,n^3+15\,c\,b^3\,n^2+74\,c\,b^3\,n+120\,c\,b^3\right )}{b^4\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {5\,a^2\,d\,n\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{b^2\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}\right ) \]
(a + b*x)^n*((d*x^6*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(1764 *n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720) + (2*a^3*(120*b^3* c - 60*a^3*d + 15*b^3*c*n^2 + b^3*c*n^3 + 74*b^3*c*n))/(b^6*(1764*n + 1624 *n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (x^3*(3*n + n^2 + 2)*(12 0*b^3*c + 15*b^3*c*n^2 + b^3*c*n^3 + 20*a^3*d*n + 74*b^3*c*n))/(b^3*(1764* n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - (2*a^2*n*x*(120* b^3*c - 60*a^3*d + 15*b^3*c*n^2 + b^3*c*n^3 + 74*b^3*c*n))/(b^5*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (a*d*n*x^5*(50*n + 3 5*n^2 + 10*n^3 + n^4 + 24))/(b*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21 *n^5 + n^6 + 720)) + (a*n*x^2*(n + 1)*(120*b^3*c - 60*a^3*d + 15*b^3*c*n^2 + b^3*c*n^3 + 74*b^3*c*n))/(b^4*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - (5*a^2*d*n*x^4*(11*n + 6*n^2 + n^3 + 6))/(b^2*(1764 *n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)))