Integrand size = 18, antiderivative size = 157 \[ \int x^2 (a+b x)^n (c+d x)^2 \, dx=\frac {a^2 (b c-a d)^2 (a+b x)^{1+n}}{b^5 (1+n)}-\frac {2 a (b c-2 a d) (b c-a d) (a+b x)^{2+n}}{b^5 (2+n)}+\frac {\left (b^2 c^2-6 a b c d+6 a^2 d^2\right ) (a+b x)^{3+n}}{b^5 (3+n)}+\frac {2 d (b c-2 a d) (a+b x)^{4+n}}{b^5 (4+n)}+\frac {d^2 (a+b x)^{5+n}}{b^5 (5+n)} \]
a^2*(-a*d+b*c)^2*(b*x+a)^(1+n)/b^5/(1+n)-2*a*(-2*a*d+b*c)*(-a*d+b*c)*(b*x+ a)^(2+n)/b^5/(2+n)+(6*a^2*d^2-6*a*b*c*d+b^2*c^2)*(b*x+a)^(3+n)/b^5/(3+n)+2 *d*(-2*a*d+b*c)*(b*x+a)^(4+n)/b^5/(4+n)+d^2*(b*x+a)^(5+n)/b^5/(5+n)
Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.87 \[ \int x^2 (a+b x)^n (c+d x)^2 \, dx=\frac {(a+b x)^{1+n} \left (\frac {a^2 (b c-a d)^2}{1+n}-\frac {2 a (b c-2 a d) (b c-a d) (a+b x)}{2+n}+\frac {\left (b^2 c^2-6 a b c d+6 a^2 d^2\right ) (a+b x)^2}{3+n}+\frac {2 d (b c-2 a d) (a+b x)^3}{4+n}+\frac {d^2 (a+b x)^4}{5+n}\right )}{b^5} \]
((a + b*x)^(1 + n)*((a^2*(b*c - a*d)^2)/(1 + n) - (2*a*(b*c - 2*a*d)*(b*c - a*d)*(a + b*x))/(2 + n) + ((b^2*c^2 - 6*a*b*c*d + 6*a^2*d^2)*(a + b*x)^2 )/(3 + n) + (2*d*(b*c - 2*a*d)*(a + b*x)^3)/(4 + n) + (d^2*(a + b*x)^4)/(5 + n)))/b^5
Time = 0.31 (sec) , antiderivative size = 157, 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 = {99, 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.
| \(\displaystyle \int x^2 (c+d x)^2 (a+b x)^n \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {a^2 (a d-b c)^2 (a+b x)^n}{b^4}+\frac {\left (6 a^2 d^2-6 a b c d+b^2 c^2\right ) (a+b x)^{n+2}}{b^4}+\frac {2 a (b c-2 a d) (a d-b c) (a+b x)^{n+1}}{b^4}+\frac {2 d (b c-2 a d) (a+b x)^{n+3}}{b^4}+\frac {d^2 (a+b x)^{n+4}}{b^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 (b c-a d)^2 (a+b x)^{n+1}}{b^5 (n+1)}+\frac {\left (6 a^2 d^2-6 a b c d+b^2 c^2\right ) (a+b x)^{n+3}}{b^5 (n+3)}-\frac {2 a (b c-2 a d) (b c-a d) (a+b x)^{n+2}}{b^5 (n+2)}+\frac {2 d (b c-2 a d) (a+b x)^{n+4}}{b^5 (n+4)}+\frac {d^2 (a+b x)^{n+5}}{b^5 (n+5)}\) |
(a^2*(b*c - a*d)^2*(a + b*x)^(1 + n))/(b^5*(1 + n)) - (2*a*(b*c - 2*a*d)*( b*c - a*d)*(a + b*x)^(2 + n))/(b^5*(2 + n)) + ((b^2*c^2 - 6*a*b*c*d + 6*a^ 2*d^2)*(a + b*x)^(3 + n))/(b^5*(3 + n)) + (2*d*(b*c - 2*a*d)*(a + b*x)^(4 + n))/(b^5*(4 + n)) + (d^2*(a + b*x)^(5 + n))/(b^5*(5 + n))
3.10.23.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Leaf count of result is larger than twice the leaf count of optimal. \(420\) vs. \(2(157)=314\).
Time = 0.66 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.68
| method | result | size |
| norman | \(\frac {d^{2} x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{5+n}+\frac {\left (a d n +2 b c n +10 b c \right ) d \,x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+9 n +20\right )}+\frac {\left (b^{2} c^{2} n^{2}-6 a b c d n +9 b^{2} c^{2} n +12 a^{2} d^{2}-30 a b c d +20 b^{2} c^{2}\right ) a n \,x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+14 n^{3}+71 n^{2}+154 n +120\right )}+\frac {2 a^{3} \left (b^{2} c^{2} n^{2}-6 a b c d n +9 b^{2} c^{2} n +12 a^{2} d^{2}-30 a b c d +20 b^{2} c^{2}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}-\frac {\left (-2 a b c d \,n^{2}-b^{2} c^{2} n^{2}+4 a^{2} d^{2} n -10 a b c d n -9 b^{2} c^{2} n -20 b^{2} c^{2}\right ) x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+12 n^{2}+47 n +60\right )}-\frac {2 n \,a^{2} \left (b^{2} c^{2} n^{2}-6 a b c d n +9 b^{2} c^{2} n +12 a^{2} d^{2}-30 a b c d +20 b^{2} c^{2}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(421\) |
| gosper | \(\frac {\left (b x +a \right )^{1+n} \left (b^{4} d^{2} n^{4} x^{4}+2 b^{4} c d \,n^{4} x^{3}+10 b^{4} d^{2} n^{3} x^{4}-4 a \,b^{3} d^{2} n^{3} x^{3}+b^{4} c^{2} n^{4} x^{2}+22 b^{4} c d \,n^{3} x^{3}+35 b^{4} d^{2} n^{2} x^{4}-6 a \,b^{3} c d \,n^{3} x^{2}-24 a \,b^{3} d^{2} n^{2} x^{3}+12 b^{4} c^{2} n^{3} x^{2}+82 b^{4} c d \,n^{2} x^{3}+50 b^{4} d^{2} n \,x^{4}+12 a^{2} b^{2} d^{2} n^{2} x^{2}-2 a \,b^{3} c^{2} n^{3} x -48 a \,b^{3} c d \,n^{2} x^{2}-44 a \,b^{3} d^{2} n \,x^{3}+49 b^{4} c^{2} n^{2} x^{2}+122 b^{4} c d n \,x^{3}+24 d^{2} x^{4} b^{4}+12 a^{2} b^{2} c d \,n^{2} x +36 a^{2} b^{2} d^{2} n \,x^{2}-20 a \,b^{3} c^{2} n^{2} x -102 a \,b^{3} c d n \,x^{2}-24 x^{3} a \,b^{3} d^{2}+78 b^{4} c^{2} n \,x^{2}+60 x^{3} b^{4} c d -24 a^{3} b \,d^{2} n x +2 a^{2} b^{2} c^{2} n^{2}+72 a^{2} b^{2} c d n x +24 x^{2} a^{2} b^{2} d^{2}-58 a \,b^{3} c^{2} n x -60 x^{2} a \,b^{3} c d +40 x^{2} b^{4} c^{2}-12 a^{3} b c d n -24 a^{3} b \,d^{2} x +18 a^{2} b^{2} c^{2} n +60 a^{2} b^{2} c d x -40 a \,b^{3} c^{2} x +24 a^{4} d^{2}-60 a^{3} b c d +40 a^{2} b^{2} c^{2}\right )}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(547\) |
| risch | \(\frac {\left (b^{5} d^{2} n^{4} x^{5}+a \,b^{4} d^{2} n^{4} x^{4}+2 b^{5} c d \,n^{4} x^{4}+10 b^{5} d^{2} n^{3} x^{5}+2 a \,b^{4} c d \,n^{4} x^{3}+6 a \,b^{4} d^{2} n^{3} x^{4}+b^{5} c^{2} n^{4} x^{3}+22 b^{5} c d \,n^{3} x^{4}+35 b^{5} d^{2} n^{2} x^{5}-4 a^{2} b^{3} d^{2} n^{3} x^{3}+a \,b^{4} c^{2} n^{4} x^{2}+16 a \,b^{4} c d \,n^{3} x^{3}+11 a \,b^{4} d^{2} n^{2} x^{4}+12 b^{5} c^{2} n^{3} x^{3}+82 b^{5} c d \,n^{2} x^{4}+50 b^{5} d^{2} n \,x^{5}-6 a^{2} b^{3} c d \,n^{3} x^{2}-12 a^{2} b^{3} d^{2} n^{2} x^{3}+10 a \,b^{4} c^{2} n^{3} x^{2}+34 a \,b^{4} c d \,n^{2} x^{3}+6 a \,b^{4} d^{2} n \,x^{4}+49 b^{5} c^{2} n^{2} x^{3}+122 b^{5} c d n \,x^{4}+24 d^{2} x^{5} b^{5}+12 a^{3} b^{2} d^{2} n^{2} x^{2}-2 a^{2} b^{3} c^{2} n^{3} x -36 a^{2} b^{3} c d \,n^{2} x^{2}-8 a^{2} b^{3} d^{2} n \,x^{3}+29 a \,b^{4} c^{2} n^{2} x^{2}+20 a \,b^{4} c d n \,x^{3}+78 b^{5} c^{2} n \,x^{3}+60 x^{4} b^{5} c d +12 a^{3} b^{2} c d \,n^{2} x +12 a^{3} b^{2} d^{2} n \,x^{2}-18 a^{2} b^{3} c^{2} n^{2} x -30 a^{2} b^{3} c d n \,x^{2}+20 a \,b^{4} c^{2} n \,x^{2}+40 x^{3} b^{5} c^{2}-24 a^{4} b \,d^{2} n x +2 a^{3} b^{2} c^{2} n^{2}+60 a^{3} b^{2} c d n x -40 a^{2} b^{3} c^{2} n x -12 a^{4} b c d n +18 a^{3} b^{2} c^{2} n +24 a^{5} d^{2}-60 a^{4} b c d +40 a^{3} b^{2} c^{2}\right ) \left (b x +a \right )^{n}}{\left (4+n \right ) \left (5+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{5}}\) | \(663\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1027\) |
d^2/(5+n)*x^5*exp(n*ln(b*x+a))+(a*d*n+2*b*c*n+10*b*c)*d/b/(n^2+9*n+20)*x^4 *exp(n*ln(b*x+a))+(b^2*c^2*n^2-6*a*b*c*d*n+9*b^2*c^2*n+12*a^2*d^2-30*a*b*c *d+20*b^2*c^2)*a/b^3*n/(n^4+14*n^3+71*n^2+154*n+120)*x^2*exp(n*ln(b*x+a))+ 2*a^3*(b^2*c^2*n^2-6*a*b*c*d*n+9*b^2*c^2*n+12*a^2*d^2-30*a*b*c*d+20*b^2*c^ 2)/b^5/(n^5+15*n^4+85*n^3+225*n^2+274*n+120)*exp(n*ln(b*x+a))-(-2*a*b*c*d* n^2-b^2*c^2*n^2+4*a^2*d^2*n-10*a*b*c*d*n-9*b^2*c^2*n-20*b^2*c^2)/b^2/(n^3+ 12*n^2+47*n+60)*x^3*exp(n*ln(b*x+a))-2/b^4*n*a^2*(b^2*c^2*n^2-6*a*b*c*d*n+ 9*b^2*c^2*n+12*a^2*d^2-30*a*b*c*d+20*b^2*c^2)/(n^5+15*n^4+85*n^3+225*n^2+2 74*n+120)*x*exp(n*ln(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (157) = 314\).
Time = 0.23 (sec) , antiderivative size = 584, normalized size of antiderivative = 3.72 \[ \int x^2 (a+b x)^n (c+d x)^2 \, dx=\frac {{\left (2 \, a^{3} b^{2} c^{2} n^{2} + 40 \, a^{3} b^{2} c^{2} - 60 \, a^{4} b c d + 24 \, a^{5} d^{2} + {\left (b^{5} d^{2} n^{4} + 10 \, b^{5} d^{2} n^{3} + 35 \, b^{5} d^{2} n^{2} + 50 \, b^{5} d^{2} n + 24 \, b^{5} d^{2}\right )} x^{5} + {\left (60 \, b^{5} c d + {\left (2 \, b^{5} c d + a b^{4} d^{2}\right )} n^{4} + 2 \, {\left (11 \, b^{5} c d + 3 \, a b^{4} d^{2}\right )} n^{3} + {\left (82 \, b^{5} c d + 11 \, a b^{4} d^{2}\right )} n^{2} + 2 \, {\left (61 \, b^{5} c d + 3 \, a b^{4} d^{2}\right )} n\right )} x^{4} + {\left (40 \, b^{5} c^{2} + {\left (b^{5} c^{2} + 2 \, a b^{4} c d\right )} n^{4} + 4 \, {\left (3 \, b^{5} c^{2} + 4 \, a b^{4} c d - a^{2} b^{3} d^{2}\right )} n^{3} + {\left (49 \, b^{5} c^{2} + 34 \, a b^{4} c d - 12 \, a^{2} b^{3} d^{2}\right )} n^{2} + 2 \, {\left (39 \, b^{5} c^{2} + 10 \, a b^{4} c d - 4 \, a^{2} b^{3} d^{2}\right )} n\right )} x^{3} + {\left (a b^{4} c^{2} n^{4} + 2 \, {\left (5 \, a b^{4} c^{2} - 3 \, a^{2} b^{3} c d\right )} n^{3} + {\left (29 \, a b^{4} c^{2} - 36 \, a^{2} b^{3} c d + 12 \, a^{3} b^{2} d^{2}\right )} n^{2} + 2 \, {\left (10 \, a b^{4} c^{2} - 15 \, a^{2} b^{3} c d + 6 \, a^{3} b^{2} d^{2}\right )} n\right )} x^{2} + 6 \, {\left (3 \, a^{3} b^{2} c^{2} - 2 \, a^{4} b c d\right )} n - 2 \, {\left (a^{2} b^{3} c^{2} n^{3} + 3 \, {\left (3 \, a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d\right )} n^{2} + 2 \, {\left (10 \, a^{2} b^{3} c^{2} - 15 \, a^{3} b^{2} c d + 6 \, a^{4} b d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \]
(2*a^3*b^2*c^2*n^2 + 40*a^3*b^2*c^2 - 60*a^4*b*c*d + 24*a^5*d^2 + (b^5*d^2 *n^4 + 10*b^5*d^2*n^3 + 35*b^5*d^2*n^2 + 50*b^5*d^2*n + 24*b^5*d^2)*x^5 + (60*b^5*c*d + (2*b^5*c*d + a*b^4*d^2)*n^4 + 2*(11*b^5*c*d + 3*a*b^4*d^2)*n ^3 + (82*b^5*c*d + 11*a*b^4*d^2)*n^2 + 2*(61*b^5*c*d + 3*a*b^4*d^2)*n)*x^4 + (40*b^5*c^2 + (b^5*c^2 + 2*a*b^4*c*d)*n^4 + 4*(3*b^5*c^2 + 4*a*b^4*c*d - a^2*b^3*d^2)*n^3 + (49*b^5*c^2 + 34*a*b^4*c*d - 12*a^2*b^3*d^2)*n^2 + 2* (39*b^5*c^2 + 10*a*b^4*c*d - 4*a^2*b^3*d^2)*n)*x^3 + (a*b^4*c^2*n^4 + 2*(5 *a*b^4*c^2 - 3*a^2*b^3*c*d)*n^3 + (29*a*b^4*c^2 - 36*a^2*b^3*c*d + 12*a^3* b^2*d^2)*n^2 + 2*(10*a*b^4*c^2 - 15*a^2*b^3*c*d + 6*a^3*b^2*d^2)*n)*x^2 + 6*(3*a^3*b^2*c^2 - 2*a^4*b*c*d)*n - 2*(a^2*b^3*c^2*n^3 + 3*(3*a^2*b^3*c^2 - 2*a^3*b^2*c*d)*n^2 + 2*(10*a^2*b^3*c^2 - 15*a^3*b^2*c*d + 6*a^4*b*d^2)*n )*x)*(b*x + a)^n/(b^5*n^5 + 15*b^5*n^4 + 85*b^5*n^3 + 225*b^5*n^2 + 274*b^ 5*n + 120*b^5)
Leaf count of result is larger than twice the leaf count of optimal. 6418 vs. \(2 (144) = 288\).
Time = 1.56 (sec) , antiderivative size = 6418, normalized size of antiderivative = 40.88 \[ \int x^2 (a+b x)^n (c+d x)^2 \, dx=\text {Too large to display} \]
Piecewise((a**n*(c**2*x**3/3 + c*d*x**4/2 + d**2*x**5/5), Eq(b, 0)), (12*a **4*d**2*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 25*a**4*d**2/(12*a**4*b**5 + 48*a**3*b** 6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 6*a**3*b*c*d/(1 2*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b** 9*x**4) + 48*a**3*b*d**2*x*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 7 2*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 88*a**3*b*d**2*x/(12*a **4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x **4) - a**2*b**2*c**2/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 24*a**2*b**2*c*d*x/(12*a**4*b**5 + 48*a* *3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 72*a**2*b **2*d**2*x**2*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x **2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 108*a**2*b**2*d**2*x**2/(12*a**4*b* *5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 4*a*b**3*c**2*x/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a *b**8*x**3 + 12*b**9*x**4) - 36*a*b**3*c*d*x**2/(12*a**4*b**5 + 48*a**3*b* *6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 48*a*b**3*d**2 *x**3*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48 *a*b**8*x**3 + 12*b**9*x**4) + 48*a*b**3*d**2*x**3/(12*a**4*b**5 + 48*a**3 *b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 6*b**4*c...
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (157) = 314\).
Time = 0.21 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.03 \[ \int x^2 (a+b x)^n (c+d x)^2 \, dx=\frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {2 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \]
((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^2/((n^3 + 6*n^2 + 11*n + 6)*b^3) + 2*((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*c*d/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4) + ((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n )*a*b^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*a^2*b^3*x^3 + 12*(n^2 + n)*a^3*b^2*x^2 - 24*a^4*b*n*x + 24*a^5)*(b*x + a)^n*d^2/((n^5 + 15*n^4 + 85*n^3 + 225*n^ 2 + 274*n + 120)*b^5)
Leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (157) = 314\).
Time = 0.28 (sec) , antiderivative size = 1001, normalized size of antiderivative = 6.38 \[ \int x^2 (a+b x)^n (c+d x)^2 \, dx=\frac {{\left (b x + a\right )}^{n} b^{5} d^{2} n^{4} x^{5} + 2 \, {\left (b x + a\right )}^{n} b^{5} c d n^{4} x^{4} + {\left (b x + a\right )}^{n} a b^{4} d^{2} n^{4} x^{4} + 10 \, {\left (b x + a\right )}^{n} b^{5} d^{2} n^{3} x^{5} + {\left (b x + a\right )}^{n} b^{5} c^{2} n^{4} x^{3} + 2 \, {\left (b x + a\right )}^{n} a b^{4} c d n^{4} x^{3} + 22 \, {\left (b x + a\right )}^{n} b^{5} c d n^{3} x^{4} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d^{2} n^{3} x^{4} + 35 \, {\left (b x + a\right )}^{n} b^{5} d^{2} n^{2} x^{5} + {\left (b x + a\right )}^{n} a b^{4} c^{2} n^{4} x^{2} + 12 \, {\left (b x + a\right )}^{n} b^{5} c^{2} n^{3} x^{3} + 16 \, {\left (b x + a\right )}^{n} a b^{4} c d n^{3} x^{3} - 4 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d^{2} n^{3} x^{3} + 82 \, {\left (b x + a\right )}^{n} b^{5} c d n^{2} x^{4} + 11 \, {\left (b x + a\right )}^{n} a b^{4} d^{2} n^{2} x^{4} + 50 \, {\left (b x + a\right )}^{n} b^{5} d^{2} n x^{5} + 10 \, {\left (b x + a\right )}^{n} a b^{4} c^{2} n^{3} x^{2} - 6 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c d n^{3} x^{2} + 49 \, {\left (b x + a\right )}^{n} b^{5} c^{2} n^{2} x^{3} + 34 \, {\left (b x + a\right )}^{n} a b^{4} c d n^{2} x^{3} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d^{2} n^{2} x^{3} + 122 \, {\left (b x + a\right )}^{n} b^{5} c d n x^{4} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d^{2} n x^{4} + 24 \, {\left (b x + a\right )}^{n} b^{5} d^{2} x^{5} - 2 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c^{2} n^{3} x + 29 \, {\left (b x + a\right )}^{n} a b^{4} c^{2} n^{2} x^{2} - 36 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c d n^{2} x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d^{2} n^{2} x^{2} + 78 \, {\left (b x + a\right )}^{n} b^{5} c^{2} n x^{3} + 20 \, {\left (b x + a\right )}^{n} a b^{4} c d n x^{3} - 8 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d^{2} n x^{3} + 60 \, {\left (b x + a\right )}^{n} b^{5} c d x^{4} - 18 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c^{2} n^{2} x + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c d n^{2} x + 20 \, {\left (b x + a\right )}^{n} a b^{4} c^{2} n x^{2} - 30 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c d n x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d^{2} n x^{2} + 40 \, {\left (b x + a\right )}^{n} b^{5} c^{2} x^{3} + 2 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c^{2} n^{2} - 40 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c^{2} n x + 60 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c d n x - 24 \, {\left (b x + a\right )}^{n} a^{4} b d^{2} n x + 18 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c^{2} n - 12 \, {\left (b x + a\right )}^{n} a^{4} b c d n + 40 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c^{2} - 60 \, {\left (b x + a\right )}^{n} a^{4} b c d + 24 \, {\left (b x + a\right )}^{n} a^{5} d^{2}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \]
((b*x + a)^n*b^5*d^2*n^4*x^5 + 2*(b*x + a)^n*b^5*c*d*n^4*x^4 + (b*x + a)^n *a*b^4*d^2*n^4*x^4 + 10*(b*x + a)^n*b^5*d^2*n^3*x^5 + (b*x + a)^n*b^5*c^2* n^4*x^3 + 2*(b*x + a)^n*a*b^4*c*d*n^4*x^3 + 22*(b*x + a)^n*b^5*c*d*n^3*x^4 + 6*(b*x + a)^n*a*b^4*d^2*n^3*x^4 + 35*(b*x + a)^n*b^5*d^2*n^2*x^5 + (b*x + a)^n*a*b^4*c^2*n^4*x^2 + 12*(b*x + a)^n*b^5*c^2*n^3*x^3 + 16*(b*x + a)^ n*a*b^4*c*d*n^3*x^3 - 4*(b*x + a)^n*a^2*b^3*d^2*n^3*x^3 + 82*(b*x + a)^n*b ^5*c*d*n^2*x^4 + 11*(b*x + a)^n*a*b^4*d^2*n^2*x^4 + 50*(b*x + a)^n*b^5*d^2 *n*x^5 + 10*(b*x + a)^n*a*b^4*c^2*n^3*x^2 - 6*(b*x + a)^n*a^2*b^3*c*d*n^3* x^2 + 49*(b*x + a)^n*b^5*c^2*n^2*x^3 + 34*(b*x + a)^n*a*b^4*c*d*n^2*x^3 - 12*(b*x + a)^n*a^2*b^3*d^2*n^2*x^3 + 122*(b*x + a)^n*b^5*c*d*n*x^4 + 6*(b* x + a)^n*a*b^4*d^2*n*x^4 + 24*(b*x + a)^n*b^5*d^2*x^5 - 2*(b*x + a)^n*a^2* b^3*c^2*n^3*x + 29*(b*x + a)^n*a*b^4*c^2*n^2*x^2 - 36*(b*x + a)^n*a^2*b^3* c*d*n^2*x^2 + 12*(b*x + a)^n*a^3*b^2*d^2*n^2*x^2 + 78*(b*x + a)^n*b^5*c^2* n*x^3 + 20*(b*x + a)^n*a*b^4*c*d*n*x^3 - 8*(b*x + a)^n*a^2*b^3*d^2*n*x^3 + 60*(b*x + a)^n*b^5*c*d*x^4 - 18*(b*x + a)^n*a^2*b^3*c^2*n^2*x + 12*(b*x + a)^n*a^3*b^2*c*d*n^2*x + 20*(b*x + a)^n*a*b^4*c^2*n*x^2 - 30*(b*x + a)^n* a^2*b^3*c*d*n*x^2 + 12*(b*x + a)^n*a^3*b^2*d^2*n*x^2 + 40*(b*x + a)^n*b^5* c^2*x^3 + 2*(b*x + a)^n*a^3*b^2*c^2*n^2 - 40*(b*x + a)^n*a^2*b^3*c^2*n*x + 60*(b*x + a)^n*a^3*b^2*c*d*n*x - 24*(b*x + a)^n*a^4*b*d^2*n*x + 18*(b*x + a)^n*a^3*b^2*c^2*n - 12*(b*x + a)^n*a^4*b*c*d*n + 40*(b*x + a)^n*a^3*b...
Time = 1.55 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.96 \[ \int x^2 (a+b x)^n (c+d x)^2 \, dx={\left (a+b\,x\right )}^n\,\left (\frac {2\,a^3\,\left (12\,a^2\,d^2-6\,a\,b\,c\,d\,n-30\,a\,b\,c\,d+b^2\,c^2\,n^2+9\,b^2\,c^2\,n+20\,b^2\,c^2\right )}{b^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {d^2\,x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}+\frac {x^3\,\left (n^2+3\,n+2\right )\,\left (-4\,a^2\,d^2\,n+2\,a\,b\,c\,d\,n^2+10\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+9\,b^2\,c^2\,n+20\,b^2\,c^2\right )}{b^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {d\,x^4\,\left (10\,b\,c+a\,d\,n+2\,b\,c\,n\right )\,\left (n^3+6\,n^2+11\,n+6\right )}{b\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}-\frac {2\,a^2\,n\,x\,\left (12\,a^2\,d^2-6\,a\,b\,c\,d\,n-30\,a\,b\,c\,d+b^2\,c^2\,n^2+9\,b^2\,c^2\,n+20\,b^2\,c^2\right )}{b^4\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,n\,x^2\,\left (n+1\right )\,\left (12\,a^2\,d^2-6\,a\,b\,c\,d\,n-30\,a\,b\,c\,d+b^2\,c^2\,n^2+9\,b^2\,c^2\,n+20\,b^2\,c^2\right )}{b^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}\right ) \]
(a + b*x)^n*((2*a^3*(12*a^2*d^2 + 20*b^2*c^2 + 9*b^2*c^2*n + b^2*c^2*n^2 - 30*a*b*c*d - 6*a*b*c*d*n))/(b^5*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (d^2*x^5*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120) + (x^3*(3*n + n^2 + 2)*(20*b^2*c^2 - 4*a^2*d ^2*n + 9*b^2*c^2*n + b^2*c^2*n^2 + 10*a*b*c*d*n + 2*a*b*c*d*n^2))/(b^2*(27 4*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (d*x^4*(10*b*c + a*d*n + 2 *b*c*n)*(11*n + 6*n^2 + n^3 + 6))/(b*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) - (2*a^2*n*x*(12*a^2*d^2 + 20*b^2*c^2 + 9*b^2*c^2*n + b^2*c^2* n^2 - 30*a*b*c*d - 6*a*b*c*d*n))/(b^4*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (a*n*x^2*(n + 1)*(12*a^2*d^2 + 20*b^2*c^2 + 9*b^2*c^2*n + b ^2*c^2*n^2 - 30*a*b*c*d - 6*a*b*c*d*n))/(b^3*(274*n + 225*n^2 + 85*n^3 + 1 5*n^4 + n^5 + 120)))