Integrand size = 16, antiderivative size = 114 \[ \int x (a+b x)^n (c+d x)^2 \, dx=-\frac {a (b c-a d)^2 (a+b x)^{1+n}}{b^4 (1+n)}+\frac {(b c-3 a d) (b c-a d) (a+b x)^{2+n}}{b^4 (2+n)}+\frac {d (2 b c-3 a d) (a+b x)^{3+n}}{b^4 (3+n)}+\frac {d^2 (a+b x)^{4+n}}{b^4 (4+n)} \]
-a*(-a*d+b*c)^2*(b*x+a)^(1+n)/b^4/(1+n)+(-3*a*d+b*c)*(-a*d+b*c)*(b*x+a)^(2 +n)/b^4/(2+n)+d*(-3*a*d+2*b*c)*(b*x+a)^(3+n)/b^4/(3+n)+d^2*(b*x+a)^(4+n)/b ^4/(4+n)
Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.86 \[ \int x (a+b x)^n (c+d x)^2 \, dx=\frac {(a+b x)^{1+n} \left (-\frac {a (b c-a d)^2}{1+n}+\frac {(b c-3 a d) (b c-a d) (a+b x)}{2+n}+\frac {d (2 b c-3 a d) (a+b x)^2}{3+n}+\frac {d^2 (a+b x)^3}{4+n}\right )}{b^4} \]
((a + b*x)^(1 + n)*(-((a*(b*c - a*d)^2)/(1 + n)) + ((b*c - 3*a*d)*(b*c - a *d)*(a + b*x))/(2 + n) + (d*(2*b*c - 3*a*d)*(a + b*x)^2)/(3 + n) + (d^2*(a + b*x)^3)/(4 + n)))/b^4
Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {86, 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.
| \(\displaystyle \int x (c+d x)^2 (a+b x)^n \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (-\frac {a (a d-b c)^2 (a+b x)^n}{b^3}+\frac {(b c-3 a d) (b c-a d) (a+b x)^{n+1}}{b^3}+\frac {d (2 b c-3 a d) (a+b x)^{n+2}}{b^3}+\frac {d^2 (a+b x)^{n+3}}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a (b c-a d)^2 (a+b x)^{n+1}}{b^4 (n+1)}+\frac {(b c-3 a d) (b c-a d) (a+b x)^{n+2}}{b^4 (n+2)}+\frac {d (2 b c-3 a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac {d^2 (a+b x)^{n+4}}{b^4 (n+4)}\) |
-((a*(b*c - a*d)^2*(a + b*x)^(1 + n))/(b^4*(1 + n))) + ((b*c - 3*a*d)*(b*c - a*d)*(a + b*x)^(2 + n))/(b^4*(2 + n)) + (d*(2*b*c - 3*a*d)*(a + b*x)^(3 + n))/(b^4*(3 + n)) + (d^2*(a + b*x)^(4 + n))/(b^4*(4 + n))
3.10.24.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Leaf count of result is larger than twice the leaf count of optimal. \(320\) vs. \(2(114)=228\).
Time = 0.67 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.82
| method | result | size |
| norman | \(\frac {d^{2} x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{4+n}+\frac {d \left (a d n +2 b c n +8 b c \right ) x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+7 n +12\right )}+\frac {n a \left (b^{2} c^{2} n^{2}-4 a b c d n +7 b^{2} c^{2} n +6 a^{2} d^{2}-16 a b c d +12 b^{2} c^{2}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}-\frac {a^{2} \left (b^{2} c^{2} n^{2}-4 a b c d n +7 b^{2} c^{2} n +6 a^{2} d^{2}-16 a b c d +12 b^{2} c^{2}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}-\frac {\left (-2 a b c d \,n^{2}-b^{2} c^{2} n^{2}+3 a^{2} d^{2} n -8 a b c d n -7 b^{2} c^{2} n -12 b^{2} c^{2}\right ) x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+9 n^{2}+26 n +24\right )}\) | \(321\) |
| gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (-b^{3} d^{2} n^{3} x^{3}-2 b^{3} c d \,n^{3} x^{2}-6 b^{3} d^{2} n^{2} x^{3}+3 a \,b^{2} d^{2} n^{2} x^{2}-b^{3} c^{2} n^{3} x -14 b^{3} c d \,n^{2} x^{2}-11 b^{3} d^{2} n \,x^{3}+4 a \,b^{2} c d \,n^{2} x +9 a \,b^{2} d^{2} n \,x^{2}-8 b^{3} c^{2} n^{2} x -28 b^{3} c d n \,x^{2}-6 d^{2} x^{3} b^{3}-6 a^{2} b \,d^{2} n x +a \,b^{2} c^{2} n^{2}+20 a \,b^{2} c d n x +6 a \,b^{2} d^{2} x^{2}-19 b^{3} c^{2} n x -16 b^{3} c d \,x^{2}-4 a^{2} b c d n -6 a^{2} b \,d^{2} x +7 a \,b^{2} c^{2} n +16 a \,b^{2} c d x -12 b^{3} c^{2} x +6 a^{3} d^{2}-16 a^{2} b c d +12 a \,b^{2} c^{2}\right )}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\) | \(324\) |
| risch | \(-\frac {\left (-b^{4} d^{2} n^{3} x^{4}-a \,b^{3} d^{2} n^{3} x^{3}-2 b^{4} c d \,n^{3} x^{3}-6 b^{4} d^{2} n^{2} x^{4}-2 a \,b^{3} c d \,n^{3} x^{2}-3 a \,b^{3} d^{2} n^{2} x^{3}-b^{4} c^{2} n^{3} x^{2}-14 b^{4} c d \,n^{2} x^{3}-11 b^{4} d^{2} n \,x^{4}+3 a^{2} b^{2} d^{2} n^{2} x^{2}-a \,b^{3} c^{2} n^{3} x -10 a \,b^{3} c d \,n^{2} x^{2}-2 a \,b^{3} d^{2} n \,x^{3}-8 b^{4} c^{2} n^{2} x^{2}-28 b^{4} c d n \,x^{3}-6 d^{2} x^{4} b^{4}+4 a^{2} b^{2} c d \,n^{2} x +3 a^{2} b^{2} d^{2} n \,x^{2}-7 a \,b^{3} c^{2} n^{2} x -8 a \,b^{3} c d n \,x^{2}-19 b^{4} c^{2} n \,x^{2}-16 x^{3} b^{4} c d -6 a^{3} b \,d^{2} n x +a^{2} b^{2} c^{2} n^{2}+16 a^{2} b^{2} c d n x -12 a \,b^{3} c^{2} n x -12 x^{2} b^{4} c^{2}-4 a^{3} b c d n +7 a^{2} b^{2} c^{2} n +6 a^{4} d^{2}-16 a^{3} b c d +12 a^{2} b^{2} c^{2}\right ) \left (b x +a \right )^{n}}{\left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) b^{4}}\) | \(433\) |
| parallelrisch | \(\frac {8 x^{2} \left (b x +a \right )^{n} a \,b^{3} c d n -4 x \left (b x +a \right )^{n} a^{2} b^{2} c d \,n^{2}-16 x \left (b x +a \right )^{n} a^{2} b^{2} c d n +16 \left (b x +a \right )^{n} a^{3} b c d +6 x^{4} \left (b x +a \right )^{n} b^{4} d^{2}+12 x^{2} \left (b x +a \right )^{n} b^{4} c^{2}-12 \left (b x +a \right )^{n} a^{2} b^{2} c^{2}+x^{4} \left (b x +a \right )^{n} b^{4} d^{2} n^{3}+6 x^{4} \left (b x +a \right )^{n} b^{4} d^{2} n^{2}+11 x^{4} \left (b x +a \right )^{n} b^{4} d^{2} n +x^{2} \left (b x +a \right )^{n} b^{4} c^{2} n^{3}+8 x^{2} \left (b x +a \right )^{n} b^{4} c^{2} n^{2}+16 x^{3} \left (b x +a \right )^{n} b^{4} c d +19 x^{2} \left (b x +a \right )^{n} b^{4} c^{2} n -\left (b x +a \right )^{n} a^{2} b^{2} c^{2} n^{2}+x^{3} \left (b x +a \right )^{n} a \,b^{3} d^{2} n^{3}+x \left (b x +a \right )^{n} a \,b^{3} c^{2} n^{3}-3 x^{2} \left (b x +a \right )^{n} a^{2} b^{2} d^{2} n +7 x \left (b x +a \right )^{n} a \,b^{3} c^{2} n^{2}+12 x \left (b x +a \right )^{n} a \,b^{3} c^{2} n +4 \left (b x +a \right )^{n} a^{3} b c d n +6 x \left (b x +a \right )^{n} a^{3} b \,d^{2} n -7 \left (b x +a \right )^{n} a^{2} b^{2} c^{2} n +2 x^{3} \left (b x +a \right )^{n} b^{4} c d \,n^{3}+3 x^{3} \left (b x +a \right )^{n} a \,b^{3} d^{2} n^{2}+14 x^{3} \left (b x +a \right )^{n} b^{4} c d \,n^{2}+2 x^{3} \left (b x +a \right )^{n} a \,b^{3} d^{2} n +28 x^{3} \left (b x +a \right )^{n} b^{4} c d n -3 x^{2} \left (b x +a \right )^{n} a^{2} b^{2} d^{2} n^{2}+2 x^{2} \left (b x +a \right )^{n} a \,b^{3} c d \,n^{3}+10 x^{2} \left (b x +a \right )^{n} a \,b^{3} c d \,n^{2}-6 \left (b x +a \right )^{n} a^{4} d^{2}}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\) | \(646\) |
d^2/(4+n)*x^4*exp(n*ln(b*x+a))+d*(a*d*n+2*b*c*n+8*b*c)/b/(n^2+7*n+12)*x^3* exp(n*ln(b*x+a))+1/b^3*n*a*(b^2*c^2*n^2-4*a*b*c*d*n+7*b^2*c^2*n+6*a^2*d^2- 16*a*b*c*d+12*b^2*c^2)/(n^4+10*n^3+35*n^2+50*n+24)*x*exp(n*ln(b*x+a))-a^2* (b^2*c^2*n^2-4*a*b*c*d*n+7*b^2*c^2*n+6*a^2*d^2-16*a*b*c*d+12*b^2*c^2)/b^4/ (n^4+10*n^3+35*n^2+50*n+24)*exp(n*ln(b*x+a))-(-2*a*b*c*d*n^2-b^2*c^2*n^2+3 *a^2*d^2*n-8*a*b*c*d*n-7*b^2*c^2*n-12*b^2*c^2)/b^2/(n^3+9*n^2+26*n+24)*x^2 *exp(n*ln(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (114) = 228\).
Time = 0.22 (sec) , antiderivative size = 393, normalized size of antiderivative = 3.45 \[ \int x (a+b x)^n (c+d x)^2 \, dx=-\frac {{\left (a^{2} b^{2} c^{2} n^{2} + 12 \, a^{2} b^{2} c^{2} - 16 \, a^{3} b c d + 6 \, a^{4} d^{2} - {\left (b^{4} d^{2} n^{3} + 6 \, b^{4} d^{2} n^{2} + 11 \, b^{4} d^{2} n + 6 \, b^{4} d^{2}\right )} x^{4} - {\left (16 \, b^{4} c d + {\left (2 \, b^{4} c d + a b^{3} d^{2}\right )} n^{3} + {\left (14 \, b^{4} c d + 3 \, a b^{3} d^{2}\right )} n^{2} + 2 \, {\left (14 \, b^{4} c d + a b^{3} d^{2}\right )} n\right )} x^{3} - {\left (12 \, b^{4} c^{2} + {\left (b^{4} c^{2} + 2 \, a b^{3} c d\right )} n^{3} + {\left (8 \, b^{4} c^{2} + 10 \, a b^{3} c d - 3 \, a^{2} b^{2} d^{2}\right )} n^{2} + {\left (19 \, b^{4} c^{2} + 8 \, a b^{3} c d - 3 \, a^{2} b^{2} d^{2}\right )} n\right )} x^{2} + {\left (7 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d\right )} n - {\left (a b^{3} c^{2} n^{3} + {\left (7 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d\right )} n^{2} + 2 \, {\left (6 \, a b^{3} c^{2} - 8 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
-(a^2*b^2*c^2*n^2 + 12*a^2*b^2*c^2 - 16*a^3*b*c*d + 6*a^4*d^2 - (b^4*d^2*n ^3 + 6*b^4*d^2*n^2 + 11*b^4*d^2*n + 6*b^4*d^2)*x^4 - (16*b^4*c*d + (2*b^4* c*d + a*b^3*d^2)*n^3 + (14*b^4*c*d + 3*a*b^3*d^2)*n^2 + 2*(14*b^4*c*d + a* b^3*d^2)*n)*x^3 - (12*b^4*c^2 + (b^4*c^2 + 2*a*b^3*c*d)*n^3 + (8*b^4*c^2 + 10*a*b^3*c*d - 3*a^2*b^2*d^2)*n^2 + (19*b^4*c^2 + 8*a*b^3*c*d - 3*a^2*b^2 *d^2)*n)*x^2 + (7*a^2*b^2*c^2 - 4*a^3*b*c*d)*n - (a*b^3*c^2*n^3 + (7*a*b^3 *c^2 - 4*a^2*b^2*c*d)*n^2 + 2*(6*a*b^3*c^2 - 8*a^2*b^2*c*d + 3*a^3*b*d^2)* n)*x)*(b*x + a)^n/(b^4*n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b^4*n + 24*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 3412 vs. \(2 (100) = 200\).
Time = 0.99 (sec) , antiderivative size = 3412, normalized size of antiderivative = 29.93 \[ \int x (a+b x)^n (c+d x)^2 \, dx=\text {Too large to display} \]
Piecewise((a**n*(c**2*x**2/2 + 2*c*d*x**3/3 + d**2*x**4/4), Eq(b, 0)), (6* a**3*d**2*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6* b**7*x**3) + 11*a**3*d**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 4*a**2*b*c*d/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x** 2 + 6*b**7*x**3) + 18*a**2*b*d**2*x*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b* *5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 27*a**2*b*d**2*x/(6*a**3*b**4 + 18* a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - a*b**2*c**2/(6*a**3*b**4 + 1 8*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 12*a*b**2*c*d*x/(6*a**3*b* *4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*b**2*d**2*x**2* log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*b**2*d**2*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6* b**7*x**3) - 3*b**3*c**2*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 12*b**3*c*d*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b** 6*x**2 + 6*b**7*x**3) + 6*b**3*d**2*x**3*log(a/b + x)/(6*a**3*b**4 + 18*a* *2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3), Eq(n, -4)), (-6*a**3*d**2*log(a /b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 9*a**3*d**2/(2*a**2*b** 4 + 4*a*b**5*x + 2*b**6*x**2) + 4*a**2*b*c*d*log(a/b + x)/(2*a**2*b**4 + 4 *a*b**5*x + 2*b**6*x**2) + 6*a**2*b*c*d/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6 *x**2) - 12*a**2*b*d**2*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6* x**2) - 12*a**2*b*d**2*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - a*b...
Time = 0.21 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.94 \[ \int x (a+b x)^n (c+d x)^2 \, dx=\frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c^{2}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {2 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \]
(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^2/((n^2 + 3*n + 2)*b^2) + 2*((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*d/((n^3 + 6*n^2 + 11*n + 6)*b^3) + ((n^3 + 6*n^2 + 11*n + 6)*b ^4*x^4 + (n^3 + 3*n^2 + 2*n)*a*b^3*x^3 - 3*(n^2 + n)*a^2*b^2*x^2 + 6*a^3*b *n*x - 6*a^4)*(b*x + a)^n*d^2/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (114) = 228\).
Time = 0.28 (sec) , antiderivative size = 659, normalized size of antiderivative = 5.78 \[ \int x (a+b x)^n (c+d x)^2 \, dx=\frac {{\left (b x + a\right )}^{n} b^{4} d^{2} n^{3} x^{4} + 2 \, {\left (b x + a\right )}^{n} b^{4} c d n^{3} x^{3} + {\left (b x + a\right )}^{n} a b^{3} d^{2} n^{3} x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d^{2} n^{2} x^{4} + {\left (b x + a\right )}^{n} b^{4} c^{2} n^{3} x^{2} + 2 \, {\left (b x + a\right )}^{n} a b^{3} c d n^{3} x^{2} + 14 \, {\left (b x + a\right )}^{n} b^{4} c d n^{2} x^{3} + 3 \, {\left (b x + a\right )}^{n} a b^{3} d^{2} n^{2} x^{3} + 11 \, {\left (b x + a\right )}^{n} b^{4} d^{2} n x^{4} + {\left (b x + a\right )}^{n} a b^{3} c^{2} n^{3} x + 8 \, {\left (b x + a\right )}^{n} b^{4} c^{2} n^{2} x^{2} + 10 \, {\left (b x + a\right )}^{n} a b^{3} c d n^{2} x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d^{2} n^{2} x^{2} + 28 \, {\left (b x + a\right )}^{n} b^{4} c d n x^{3} + 2 \, {\left (b x + a\right )}^{n} a b^{3} d^{2} n x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d^{2} x^{4} + 7 \, {\left (b x + a\right )}^{n} a b^{3} c^{2} n^{2} x - 4 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c d n^{2} x + 19 \, {\left (b x + a\right )}^{n} b^{4} c^{2} n x^{2} + 8 \, {\left (b x + a\right )}^{n} a b^{3} c d n x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d^{2} n x^{2} + 16 \, {\left (b x + a\right )}^{n} b^{4} c d x^{3} - {\left (b x + a\right )}^{n} a^{2} b^{2} c^{2} n^{2} + 12 \, {\left (b x + a\right )}^{n} a b^{3} c^{2} n x - 16 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c d n x + 6 \, {\left (b x + a\right )}^{n} a^{3} b d^{2} n x + 12 \, {\left (b x + a\right )}^{n} b^{4} c^{2} x^{2} - 7 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c^{2} n + 4 \, {\left (b x + a\right )}^{n} a^{3} b c d n - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c^{2} + 16 \, {\left (b x + a\right )}^{n} a^{3} b c d - 6 \, {\left (b x + a\right )}^{n} a^{4} d^{2}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
((b*x + a)^n*b^4*d^2*n^3*x^4 + 2*(b*x + a)^n*b^4*c*d*n^3*x^3 + (b*x + a)^n *a*b^3*d^2*n^3*x^3 + 6*(b*x + a)^n*b^4*d^2*n^2*x^4 + (b*x + a)^n*b^4*c^2*n ^3*x^2 + 2*(b*x + a)^n*a*b^3*c*d*n^3*x^2 + 14*(b*x + a)^n*b^4*c*d*n^2*x^3 + 3*(b*x + a)^n*a*b^3*d^2*n^2*x^3 + 11*(b*x + a)^n*b^4*d^2*n*x^4 + (b*x + a)^n*a*b^3*c^2*n^3*x + 8*(b*x + a)^n*b^4*c^2*n^2*x^2 + 10*(b*x + a)^n*a*b^ 3*c*d*n^2*x^2 - 3*(b*x + a)^n*a^2*b^2*d^2*n^2*x^2 + 28*(b*x + a)^n*b^4*c*d *n*x^3 + 2*(b*x + a)^n*a*b^3*d^2*n*x^3 + 6*(b*x + a)^n*b^4*d^2*x^4 + 7*(b* x + a)^n*a*b^3*c^2*n^2*x - 4*(b*x + a)^n*a^2*b^2*c*d*n^2*x + 19*(b*x + a)^ n*b^4*c^2*n*x^2 + 8*(b*x + a)^n*a*b^3*c*d*n*x^2 - 3*(b*x + a)^n*a^2*b^2*d^ 2*n*x^2 + 16*(b*x + a)^n*b^4*c*d*x^3 - (b*x + a)^n*a^2*b^2*c^2*n^2 + 12*(b *x + a)^n*a*b^3*c^2*n*x - 16*(b*x + a)^n*a^2*b^2*c*d*n*x + 6*(b*x + a)^n*a ^3*b*d^2*n*x + 12*(b*x + a)^n*b^4*c^2*x^2 - 7*(b*x + a)^n*a^2*b^2*c^2*n + 4*(b*x + a)^n*a^3*b*c*d*n - 12*(b*x + a)^n*a^2*b^2*c^2 + 16*(b*x + a)^n*a^ 3*b*c*d - 6*(b*x + a)^n*a^4*d^2)/(b^4*n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b ^4*n + 24*b^4)
Time = 1.27 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.94 \[ \int x (a+b x)^n (c+d x)^2 \, dx={\left (a+b\,x\right )}^n\,\left (\frac {d^2\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {a^2\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d\,n-16\,a\,b\,c\,d+b^2\,c^2\,n^2+7\,b^2\,c^2\,n+12\,b^2\,c^2\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {x^2\,\left (n+1\right )\,\left (-3\,a^2\,d^2\,n+2\,a\,b\,c\,d\,n^2+8\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+7\,b^2\,c^2\,n+12\,b^2\,c^2\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,n\,x\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d\,n-16\,a\,b\,c\,d+b^2\,c^2\,n^2+7\,b^2\,c^2\,n+12\,b^2\,c^2\right )}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {d\,x^3\,\left (8\,b\,c+a\,d\,n+2\,b\,c\,n\right )\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \]
(a + b*x)^n*((d^2*x^4*(11*n + 6*n^2 + n^3 + 6))/(50*n + 35*n^2 + 10*n^3 + n^4 + 24) - (a^2*(6*a^2*d^2 + 12*b^2*c^2 + 7*b^2*c^2*n + b^2*c^2*n^2 - 16* a*b*c*d - 4*a*b*c*d*n))/(b^4*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (x^2*( n + 1)*(12*b^2*c^2 - 3*a^2*d^2*n + 7*b^2*c^2*n + b^2*c^2*n^2 + 8*a*b*c*d*n + 2*a*b*c*d*n^2))/(b^2*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (a*n*x*(6*a ^2*d^2 + 12*b^2*c^2 + 7*b^2*c^2*n + b^2*c^2*n^2 - 16*a*b*c*d - 4*a*b*c*d*n ))/(b^3*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (d*x^3*(8*b*c + a*d*n + 2*b *c*n)*(3*n + n^2 + 2))/(b*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)))