Integrand size = 18, antiderivative size = 185 \[ \int x (a+b x)^n \left (c+d x^2\right )^2 \, dx=-\frac {a \left (b^2 c+a^2 d\right )^2 (a+b x)^{1+n}}{b^6 (1+n)}+\frac {\left (b^2 c+a^2 d\right ) \left (b^2 c+5 a^2 d\right ) (a+b x)^{2+n}}{b^6 (2+n)}-\frac {2 a d \left (3 b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {2 d \left (b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^6 (4+n)}-\frac {5 a d^2 (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d^2 (a+b x)^{6+n}}{b^6 (6+n)} \]
-a*(a^2*d+b^2*c)^2*(b*x+a)^(1+n)/b^6/(1+n)+(a^2*d+b^2*c)*(5*a^2*d+b^2*c)*( b*x+a)^(2+n)/b^6/(2+n)-2*a*d*(5*a^2*d+3*b^2*c)*(b*x+a)^(3+n)/b^6/(3+n)+2*d *(5*a^2*d+b^2*c)*(b*x+a)^(4+n)/b^6/(4+n)-5*a*d^2*(b*x+a)^(5+n)/b^6/(5+n)+d ^2*(b*x+a)^(6+n)/b^6/(6+n)
Time = 0.34 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.75 \[ \int x (a+b x)^n \left (c+d x^2\right )^2 \, dx=\frac {(a+b x)^{1+n} \left (b^4 (1+n) (2+n) (3+n) (4+n) (5+n) (a+b x) \left (c+d x^2\right )^2-a (6+n) \left (b^4 (1+n) (2+n) (3+n) (4+n) \left (c+d x^2\right )^2+4 \left (b^2 c+a^2 d\right ) (4+n) \left (2 a^2 d-2 a b d (1+n) x+b^2 (2+n) \left (c (3+n)+d (1+n) x^2\right )\right )-4 a d (1+n) (a+b x) \left (2 a^2 d-2 a b d (2+n) x+b^2 (3+n) \left (c (4+n)+d (2+n) x^2\right )\right )\right )+4 (1+n) (a+b x) \left (\left (b^2 c+a^2 d\right ) (5+n) \left (2 a^2 d-2 a b d (2+n) x+b^2 (3+n) \left (c (4+n)+d (2+n) x^2\right )\right )-a d (2+n) (a+b x) \left (2 a^2 d-2 a b d (3+n) x+b^2 (4+n) \left (c (5+n)+d (3+n) x^2\right )\right )\right )\right )}{b^6 (1+n) (2+n) (3+n) (4+n) (5+n) (6+n)} \]
((a + b*x)^(1 + n)*(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(a + b*x)* (c + d*x^2)^2 - a*(6 + n)*(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(c + d*x^2) ^2 + 4*(b^2*c + a^2*d)*(4 + n)*(2*a^2*d - 2*a*b*d*(1 + n)*x + b^2*(2 + n)* (c*(3 + n) + d*(1 + n)*x^2)) - 4*a*d*(1 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d* (2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2))) + 4*(1 + n)*(a + b*x )*((b^2*c + a^2*d)*(5 + n)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*( 4 + n) + d*(2 + n)*x^2)) - a*d*(2 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(3 + n )*x + b^2*(4 + n)*(c*(5 + n) + d*(3 + n)*x^2)))))/(b^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n))
Time = 0.35 (sec) , antiderivative size = 185, 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 = {522, 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.
| \(\displaystyle \int x \left (c+d x^2\right )^2 (a+b x)^n \, dx\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (-\frac {a \left (a^2 d+b^2 c\right )^2 (a+b x)^n}{b^5}+\frac {\left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^5}-\frac {2 a d \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+2}}{b^5}+\frac {2 d \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^5}-\frac {5 a d^2 (a+b x)^{n+4}}{b^5}+\frac {d^2 (a+b x)^{n+5}}{b^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+1}}{b^6 (n+1)}+\frac {\left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^6 (n+2)}-\frac {2 a d \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+3}}{b^6 (n+3)}+\frac {2 d \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+4}}{b^6 (n+4)}-\frac {5 a d^2 (a+b x)^{n+5}}{b^6 (n+5)}+\frac {d^2 (a+b x)^{n+6}}{b^6 (n+6)}\) |
-((a*(b^2*c + a^2*d)^2*(a + b*x)^(1 + n))/(b^6*(1 + n))) + ((b^2*c + a^2*d )*(b^2*c + 5*a^2*d)*(a + b*x)^(2 + n))/(b^6*(2 + n)) - (2*a*d*(3*b^2*c + 5 *a^2*d)*(a + b*x)^(3 + n))/(b^6*(3 + n)) + (2*d*(b^2*c + 5*a^2*d)*(a + b*x )^(4 + n))/(b^6*(4 + n)) - (5*a*d^2*(a + b*x)^(5 + n))/(b^6*(5 + n)) + (d^ 2*(a + b*x)^(6 + n))/(b^6*(6 + n))
3.4.56.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(600\) vs. \(2(185)=370\).
Time = 0.43 (sec) , antiderivative size = 601, normalized size of antiderivative = 3.25
| method | result | size |
| norman | \(\frac {d^{2} x^{6} {\mathrm e}^{n \ln \left (b x +a \right )}}{6+n}+\frac {n a \left (b^{4} c^{2} n^{4}+18 b^{4} c^{2} n^{3}+12 a^{2} b^{2} c d \,n^{2}+119 b^{4} c^{2} n^{2}+132 a^{2} b^{2} c d n +342 b^{4} c^{2} n +120 d^{2} a^{4}+360 a^{2} b^{2} c d +360 b^{4} c^{2}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}+\frac {n \,d^{2} a \,x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+11 n +30\right )}-\frac {a^{2} \left (b^{4} c^{2} n^{4}+18 b^{4} c^{2} n^{3}+12 a^{2} b^{2} c d \,n^{2}+119 b^{4} c^{2} n^{2}+132 a^{2} b^{2} c d n +342 b^{4} c^{2} n +120 d^{2} a^{4}+360 a^{2} b^{2} c d +360 b^{4} c^{2}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}-\frac {\left (-b^{4} c^{2} n^{4}+6 a^{2} b^{2} c d \,n^{3}-18 b^{4} c^{2} n^{3}+66 a^{2} b^{2} c d \,n^{2}-119 b^{4} c^{2} n^{2}+60 a^{4} d^{2} n +180 a^{2} b^{2} c d n -342 b^{4} c^{2} n -360 b^{4} c^{2}\right ) x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+20 n^{4}+155 n^{3}+580 n^{2}+1044 n +720\right )}-\frac {d \left (-2 b^{2} c \,n^{2}+5 a^{2} d n -22 b^{2} c n -60 b^{2} c \right ) x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+15 n^{2}+74 n +120\right )}+\frac {2 \left (b^{2} c \,n^{2}+11 b^{2} c n +10 a^{2} d +30 b^{2} c \right ) a d n \,x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+18 n^{3}+119 n^{2}+342 n +360\right )}\) | \(601\) |
| gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (-b^{5} d^{2} n^{5} x^{5}-15 b^{5} d^{2} n^{4} x^{5}+5 a \,b^{4} d^{2} n^{4} x^{4}-2 b^{5} c d \,n^{5} x^{3}-85 b^{5} d^{2} n^{3} x^{5}+50 a \,b^{4} d^{2} n^{3} x^{4}-34 b^{5} c d \,n^{4} x^{3}-225 b^{5} d^{2} n^{2} x^{5}-20 a^{2} b^{3} d^{2} n^{3} x^{3}+6 a \,b^{4} c d \,n^{4} x^{2}+175 a \,b^{4} d^{2} n^{2} x^{4}-b^{5} c^{2} n^{5} x -214 b^{5} c d \,n^{3} x^{3}-274 b^{5} d^{2} n \,x^{5}-120 a^{2} b^{3} d^{2} n^{2} x^{3}+84 a \,b^{4} c d \,n^{3} x^{2}+250 a \,b^{4} d^{2} n \,x^{4}-19 b^{5} c^{2} n^{4} x -614 b^{5} c d \,n^{2} x^{3}-120 d^{2} x^{5} b^{5}+60 a^{3} b^{2} d^{2} n^{2} x^{2}-12 a^{2} b^{3} c d \,n^{3} x -220 a^{2} b^{3} d^{2} n \,x^{3}+a \,b^{4} c^{2} n^{4}+390 a \,b^{4} c d \,n^{2} x^{2}+120 a \,d^{2} x^{4} b^{4}-137 b^{5} c^{2} n^{3} x -792 b^{5} c d n \,x^{3}+180 a^{3} b^{2} d^{2} n \,x^{2}-144 a^{2} b^{3} c d \,n^{2} x -120 a^{2} b^{3} d^{2} x^{3}+18 a \,b^{4} c^{2} n^{3}+672 a \,b^{4} c d n \,x^{2}-461 b^{5} c^{2} n^{2} x -360 b^{5} c d \,x^{3}-120 a^{4} b \,d^{2} n x +12 a^{3} b^{2} c d \,n^{2}+120 a^{3} b^{2} d^{2} x^{2}-492 a^{2} b^{3} c d n x +119 a \,b^{4} c^{2} n^{2}+360 a \,b^{4} c d \,x^{2}-702 b^{5} c^{2} n x -120 a^{4} b \,d^{2} x +132 a^{3} b^{2} c d n -360 a^{2} b^{3} c d x +342 a \,b^{4} c^{2} n -360 b^{5} c^{2} x +120 d^{2} a^{5}+360 a^{3} b^{2} c d +360 a \,b^{4} c^{2}\right )}{b^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}\) | \(677\) |
| risch | \(-\frac {\left (-b^{6} d^{2} n^{5} x^{6}-a \,b^{5} d^{2} n^{5} x^{5}-15 b^{6} d^{2} n^{4} x^{6}-10 a \,b^{5} d^{2} n^{4} x^{5}-2 b^{6} c d \,n^{5} x^{4}-85 b^{6} d^{2} n^{3} x^{6}+5 a^{2} b^{4} d^{2} n^{4} x^{4}-2 a \,b^{5} c d \,n^{5} x^{3}-35 a \,b^{5} d^{2} n^{3} x^{5}-34 b^{6} c d \,n^{4} x^{4}-225 b^{6} d^{2} n^{2} x^{6}+30 a^{2} b^{4} d^{2} n^{3} x^{4}-28 a \,b^{5} c d \,n^{4} x^{3}-50 a \,b^{5} d^{2} n^{2} x^{5}-b^{6} c^{2} n^{5} x^{2}-214 b^{6} c d \,n^{3} x^{4}-274 b^{6} d^{2} n \,x^{6}-20 a^{3} b^{3} d^{2} n^{3} x^{3}+6 a^{2} b^{4} c d \,n^{4} x^{2}+55 a^{2} b^{4} d^{2} n^{2} x^{4}-a \,b^{5} c^{2} n^{5} x -130 a \,b^{5} c d \,n^{3} x^{3}-24 a \,b^{5} d^{2} n \,x^{5}-19 b^{6} c^{2} n^{4} x^{2}-614 b^{6} c d \,n^{2} x^{4}-120 x^{6} d^{2} b^{6}-60 a^{3} b^{3} d^{2} n^{2} x^{3}+72 a^{2} b^{4} c d \,n^{3} x^{2}+30 a^{2} b^{4} d^{2} n \,x^{4}-18 a \,b^{5} c^{2} n^{4} x -224 a \,b^{5} c d \,n^{2} x^{3}-137 b^{6} c^{2} n^{3} x^{2}-792 b^{6} c d n \,x^{4}+60 a^{4} b^{2} d^{2} n^{2} x^{2}-12 a^{3} b^{3} c d \,n^{3} x -40 a^{3} b^{3} d^{2} n \,x^{3}+a^{2} b^{4} c^{2} n^{4}+246 a^{2} b^{4} c d \,n^{2} x^{2}-119 a \,b^{5} c^{2} n^{3} x -120 a \,b^{5} c d n \,x^{3}-461 b^{6} c^{2} n^{2} x^{2}-360 b^{6} c d \,x^{4}+60 a^{4} b^{2} d^{2} n \,x^{2}-132 a^{3} b^{3} c d \,n^{2} x +18 a^{2} b^{4} c^{2} n^{3}+180 a^{2} b^{4} c d n \,x^{2}-342 a \,b^{5} c^{2} n^{2} x -702 b^{6} c^{2} n \,x^{2}-120 a^{5} b \,d^{2} n x +12 a^{4} b^{2} c d \,n^{2}-360 a^{3} b^{3} c d n x +119 a^{2} b^{4} c^{2} n^{2}-360 a \,b^{5} c^{2} n x -360 b^{6} c^{2} x^{2}+132 a^{4} b^{2} c d n +342 a^{2} b^{4} c^{2} n +120 d^{2} a^{6}+360 a^{4} b^{2} c d +360 a^{2} b^{4} c^{2}\right ) \left (b x +a \right )^{n}}{\left (5+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{6}}\) | \(845\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1247\) |
d^2/(6+n)*x^6*exp(n*ln(b*x+a))+1/b^5*n*a*(b^4*c^2*n^4+18*b^4*c^2*n^3+12*a^ 2*b^2*c*d*n^2+119*b^4*c^2*n^2+132*a^2*b^2*c*d*n+342*b^4*c^2*n+120*a^4*d^2+ 360*a^2*b^2*c*d+360*b^4*c^2)/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1764*n+7 20)*x*exp(n*ln(b*x+a))+n*d^2/b*a/(n^2+11*n+30)*x^5*exp(n*ln(b*x+a))-a^2*(b ^4*c^2*n^4+18*b^4*c^2*n^3+12*a^2*b^2*c*d*n^2+119*b^4*c^2*n^2+132*a^2*b^2*c *d*n+342*b^4*c^2*n+120*a^4*d^2+360*a^2*b^2*c*d+360*b^4*c^2)/b^6/(n^6+21*n^ 5+175*n^4+735*n^3+1624*n^2+1764*n+720)*exp(n*ln(b*x+a))-(-b^4*c^2*n^4+6*a^ 2*b^2*c*d*n^3-18*b^4*c^2*n^3+66*a^2*b^2*c*d*n^2-119*b^4*c^2*n^2+60*a^4*d^2 *n+180*a^2*b^2*c*d*n-342*b^4*c^2*n-360*b^4*c^2)/b^4/(n^5+20*n^4+155*n^3+58 0*n^2+1044*n+720)*x^2*exp(n*ln(b*x+a))-d*(-2*b^2*c*n^2+5*a^2*d*n-22*b^2*c* n-60*b^2*c)/b^2/(n^3+15*n^2+74*n+120)*x^4*exp(n*ln(b*x+a))+2*(b^2*c*n^2+11 *b^2*c*n+10*a^2*d+30*b^2*c)*a/b^3*d*n/(n^4+18*n^3+119*n^2+342*n+360)*x^3*e xp(n*ln(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 757 vs. \(2 (185) = 370\).
Time = 0.30 (sec) , antiderivative size = 757, normalized size of antiderivative = 4.09 \[ \int x (a+b x)^n \left (c+d x^2\right )^2 \, dx=-\frac {{\left (a^{2} b^{4} c^{2} n^{4} + 18 \, a^{2} b^{4} c^{2} n^{3} + 360 \, a^{2} b^{4} c^{2} + 360 \, a^{4} b^{2} c d + 120 \, a^{6} d^{2} - {\left (b^{6} d^{2} n^{5} + 15 \, b^{6} d^{2} n^{4} + 85 \, b^{6} d^{2} n^{3} + 225 \, b^{6} d^{2} n^{2} + 274 \, b^{6} d^{2} n + 120 \, b^{6} d^{2}\right )} x^{6} - {\left (a b^{5} d^{2} n^{5} + 10 \, a b^{5} d^{2} n^{4} + 35 \, a b^{5} d^{2} n^{3} + 50 \, a b^{5} d^{2} n^{2} + 24 \, a b^{5} d^{2} n\right )} x^{5} - {\left (2 \, b^{6} c d n^{5} + 360 \, b^{6} c d + {\left (34 \, b^{6} c d - 5 \, a^{2} b^{4} d^{2}\right )} n^{4} + 2 \, {\left (107 \, b^{6} c d - 15 \, a^{2} b^{4} d^{2}\right )} n^{3} + {\left (614 \, b^{6} c d - 55 \, a^{2} b^{4} d^{2}\right )} n^{2} + 6 \, {\left (132 \, b^{6} c d - 5 \, a^{2} b^{4} d^{2}\right )} n\right )} x^{4} - 2 \, {\left (a b^{5} c d n^{5} + 14 \, a b^{5} c d n^{4} + 5 \, {\left (13 \, a b^{5} c d + 2 \, a^{3} b^{3} d^{2}\right )} n^{3} + 2 \, {\left (56 \, a b^{5} c d + 15 \, a^{3} b^{3} d^{2}\right )} n^{2} + 20 \, {\left (3 \, a b^{5} c d + a^{3} b^{3} d^{2}\right )} n\right )} x^{3} + {\left (119 \, a^{2} b^{4} c^{2} + 12 \, a^{4} b^{2} c d\right )} n^{2} - {\left (b^{6} c^{2} n^{5} + 360 \, b^{6} c^{2} + {\left (19 \, b^{6} c^{2} - 6 \, a^{2} b^{4} c d\right )} n^{4} + {\left (137 \, b^{6} c^{2} - 72 \, a^{2} b^{4} c d\right )} n^{3} + {\left (461 \, b^{6} c^{2} - 246 \, a^{2} b^{4} c d - 60 \, a^{4} b^{2} d^{2}\right )} n^{2} + 6 \, {\left (117 \, b^{6} c^{2} - 30 \, a^{2} b^{4} c d - 10 \, a^{4} b^{2} d^{2}\right )} n\right )} x^{2} + 6 \, {\left (57 \, a^{2} b^{4} c^{2} + 22 \, a^{4} b^{2} c d\right )} n - {\left (a b^{5} c^{2} n^{5} + 18 \, a b^{5} c^{2} n^{4} + {\left (119 \, a b^{5} c^{2} + 12 \, a^{3} b^{3} c d\right )} n^{3} + 6 \, {\left (57 \, a b^{5} c^{2} + 22 \, a^{3} b^{3} c d\right )} n^{2} + 120 \, {\left (3 \, a b^{5} c^{2} + 3 \, a^{3} b^{3} c d + a^{5} b d^{2}\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}} \]
-(a^2*b^4*c^2*n^4 + 18*a^2*b^4*c^2*n^3 + 360*a^2*b^4*c^2 + 360*a^4*b^2*c*d + 120*a^6*d^2 - (b^6*d^2*n^5 + 15*b^6*d^2*n^4 + 85*b^6*d^2*n^3 + 225*b^6* d^2*n^2 + 274*b^6*d^2*n + 120*b^6*d^2)*x^6 - (a*b^5*d^2*n^5 + 10*a*b^5*d^2 *n^4 + 35*a*b^5*d^2*n^3 + 50*a*b^5*d^2*n^2 + 24*a*b^5*d^2*n)*x^5 - (2*b^6* c*d*n^5 + 360*b^6*c*d + (34*b^6*c*d - 5*a^2*b^4*d^2)*n^4 + 2*(107*b^6*c*d - 15*a^2*b^4*d^2)*n^3 + (614*b^6*c*d - 55*a^2*b^4*d^2)*n^2 + 6*(132*b^6*c* d - 5*a^2*b^4*d^2)*n)*x^4 - 2*(a*b^5*c*d*n^5 + 14*a*b^5*c*d*n^4 + 5*(13*a* b^5*c*d + 2*a^3*b^3*d^2)*n^3 + 2*(56*a*b^5*c*d + 15*a^3*b^3*d^2)*n^2 + 20* (3*a*b^5*c*d + a^3*b^3*d^2)*n)*x^3 + (119*a^2*b^4*c^2 + 12*a^4*b^2*c*d)*n^ 2 - (b^6*c^2*n^5 + 360*b^6*c^2 + (19*b^6*c^2 - 6*a^2*b^4*c*d)*n^4 + (137*b ^6*c^2 - 72*a^2*b^4*c*d)*n^3 + (461*b^6*c^2 - 246*a^2*b^4*c*d - 60*a^4*b^2 *d^2)*n^2 + 6*(117*b^6*c^2 - 30*a^2*b^4*c*d - 10*a^4*b^2*d^2)*n)*x^2 + 6*( 57*a^2*b^4*c^2 + 22*a^4*b^2*c*d)*n - (a*b^5*c^2*n^5 + 18*a*b^5*c^2*n^4 + ( 119*a*b^5*c^2 + 12*a^3*b^3*c*d)*n^3 + 6*(57*a*b^5*c^2 + 22*a^3*b^3*c*d)*n^ 2 + 120*(3*a*b^5*c^2 + 3*a^3*b^3*c*d + a^5*b*d^2)*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. 8940 vs. \(2 (170) = 340\).
Time = 2.45 (sec) , antiderivative size = 8940, normalized size of antiderivative = 48.32 \[ \int x (a+b x)^n \left (c+d x^2\right )^2 \, dx=\text {Too large to display} \]
Piecewise((a**n*(c**2*x**2/2 + c*d*x**4/2 + d**2*x**6/6), Eq(b, 0)), (60*a **5*d**2*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**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) + 300*a**4*b*d**2*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**2*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) - 6*a**3*b**2*c*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) + 600*a **3*b**2*d**2*x**2*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) + 1100 *a**3*b**2*d**2*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) - 30*a**2*b**3*c* 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**2*b**3*d**2*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**2*x**3/(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) - 3*a*b**4*c**2/(60*a**5*b**6 + 300*a**4*b...
Time = 0.21 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.81 \[ \int x (a+b x)^n \left (c+d x^2\right )^2 \, dx=\frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c^{2}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {2 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \, {\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} \]
(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^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^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 1 20)*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^2/((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. 1266 vs. \(2 (185) = 370\).
Time = 0.30 (sec) , antiderivative size = 1266, normalized size of antiderivative = 6.84 \[ \int x (a+b x)^n \left (c+d x^2\right )^2 \, dx=\text {Too large to display} \]
((b*x + a)^n*b^6*d^2*n^5*x^6 + (b*x + a)^n*a*b^5*d^2*n^5*x^5 + 15*(b*x + a )^n*b^6*d^2*n^4*x^6 + 2*(b*x + a)^n*b^6*c*d*n^5*x^4 + 10*(b*x + a)^n*a*b^5 *d^2*n^4*x^5 + 85*(b*x + a)^n*b^6*d^2*n^3*x^6 + 2*(b*x + a)^n*a*b^5*c*d*n^ 5*x^3 + 34*(b*x + a)^n*b^6*c*d*n^4*x^4 - 5*(b*x + a)^n*a^2*b^4*d^2*n^4*x^4 + 35*(b*x + a)^n*a*b^5*d^2*n^3*x^5 + 225*(b*x + a)^n*b^6*d^2*n^2*x^6 + (b *x + a)^n*b^6*c^2*n^5*x^2 + 28*(b*x + a)^n*a*b^5*c*d*n^4*x^3 + 214*(b*x + a)^n*b^6*c*d*n^3*x^4 - 30*(b*x + a)^n*a^2*b^4*d^2*n^3*x^4 + 50*(b*x + a)^n *a*b^5*d^2*n^2*x^5 + 274*(b*x + a)^n*b^6*d^2*n*x^6 + (b*x + a)^n*a*b^5*c^2 *n^5*x + 19*(b*x + a)^n*b^6*c^2*n^4*x^2 - 6*(b*x + a)^n*a^2*b^4*c*d*n^4*x^ 2 + 130*(b*x + a)^n*a*b^5*c*d*n^3*x^3 + 20*(b*x + a)^n*a^3*b^3*d^2*n^3*x^3 + 614*(b*x + a)^n*b^6*c*d*n^2*x^4 - 55*(b*x + a)^n*a^2*b^4*d^2*n^2*x^4 + 24*(b*x + a)^n*a*b^5*d^2*n*x^5 + 120*(b*x + a)^n*b^6*d^2*x^6 + 18*(b*x + a )^n*a*b^5*c^2*n^4*x + 137*(b*x + a)^n*b^6*c^2*n^3*x^2 - 72*(b*x + a)^n*a^2 *b^4*c*d*n^3*x^2 + 224*(b*x + a)^n*a*b^5*c*d*n^2*x^3 + 60*(b*x + a)^n*a^3* b^3*d^2*n^2*x^3 + 792*(b*x + a)^n*b^6*c*d*n*x^4 - 30*(b*x + a)^n*a^2*b^4*d ^2*n*x^4 - (b*x + a)^n*a^2*b^4*c^2*n^4 + 119*(b*x + a)^n*a*b^5*c^2*n^3*x + 12*(b*x + a)^n*a^3*b^3*c*d*n^3*x + 461*(b*x + a)^n*b^6*c^2*n^2*x^2 - 246* (b*x + a)^n*a^2*b^4*c*d*n^2*x^2 - 60*(b*x + a)^n*a^4*b^2*d^2*n^2*x^2 + 120 *(b*x + a)^n*a*b^5*c*d*n*x^3 + 40*(b*x + a)^n*a^3*b^3*d^2*n*x^3 + 360*(b*x + a)^n*b^6*c*d*x^4 - 18*(b*x + a)^n*a^2*b^4*c^2*n^3 + 342*(b*x + a)^n*...
Time = 11.74 (sec) , antiderivative size = 723, normalized size of antiderivative = 3.91 \[ \int x (a+b x)^n \left (c+d x^2\right )^2 \, dx=\frac {d^2\,x^6\,{\left (a+b\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720}-\frac {a^2\,{\left (a+b\,x\right )}^n\,\left (120\,a^4\,d^2+12\,a^2\,b^2\,c\,d\,n^2+132\,a^2\,b^2\,c\,d\,n+360\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+18\,b^4\,c^2\,n^3+119\,b^4\,c^2\,n^2+342\,b^4\,c^2\,n+360\,b^4\,c^2\right )}{b^6\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (-60\,a^4\,d^2\,n-6\,a^2\,b^2\,c\,d\,n^3-66\,a^2\,b^2\,c\,d\,n^2-180\,a^2\,b^2\,c\,d\,n+b^4\,c^2\,n^4+18\,b^4\,c^2\,n^3+119\,b^4\,c^2\,n^2+342\,b^4\,c^2\,n+360\,b^4\,c^2\right )}{b^4\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {d\,x^4\,{\left (a+b\,x\right )}^n\,\left (-5\,d\,a^2\,n+2\,c\,b^2\,n^2+22\,c\,b^2\,n+60\,c\,b^2\right )\,\left (n^3+6\,n^2+11\,n+6\right )}{b^2\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {a\,n\,x\,{\left (a+b\,x\right )}^n\,\left (120\,a^4\,d^2+12\,a^2\,b^2\,c\,d\,n^2+132\,a^2\,b^2\,c\,d\,n+360\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+18\,b^4\,c^2\,n^3+119\,b^4\,c^2\,n^2+342\,b^4\,c^2\,n+360\,b^4\,c^2\right )}{b^5\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {a\,d^2\,n\,x^5\,{\left (a+b\,x\right )}^n\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {2\,a\,d\,n\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (10\,d\,a^2+c\,b^2\,n^2+11\,c\,b^2\,n+30\,c\,b^2\right )}{b^3\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )} \]
(d^2*x^6*(a + b*x)^n*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(176 4*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720) - (a^2*(a + b*x)^ n*(120*a^4*d^2 + 360*b^4*c^2 + 342*b^4*c^2*n + 119*b^4*c^2*n^2 + 18*b^4*c^ 2*n^3 + b^4*c^2*n^4 + 360*a^2*b^2*c*d + 132*a^2*b^2*c*d*n + 12*a^2*b^2*c*d *n^2))/(b^6*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (x^2*(n + 1)*(a + b*x)^n*(360*b^4*c^2 - 60*a^4*d^2*n + 342*b^4*c^2*n + 1 19*b^4*c^2*n^2 + 18*b^4*c^2*n^3 + b^4*c^2*n^4 - 180*a^2*b^2*c*d*n - 66*a^2 *b^2*c*d*n^2 - 6*a^2*b^2*c*d*n^3))/(b^4*(1764*n + 1624*n^2 + 735*n^3 + 175 *n^4 + 21*n^5 + n^6 + 720)) + (d*x^4*(a + b*x)^n*(60*b^2*c + 2*b^2*c*n^2 - 5*a^2*d*n + 22*b^2*c*n)*(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)) + (a*n*x*(a + b*x)^n*(120*a^4* d^2 + 360*b^4*c^2 + 342*b^4*c^2*n + 119*b^4*c^2*n^2 + 18*b^4*c^2*n^3 + b^4 *c^2*n^4 + 360*a^2*b^2*c*d + 132*a^2*b^2*c*d*n + 12*a^2*b^2*c*d*n^2))/(b^5 *(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (a*d^2*n* x^5*(a + b*x)^n*(50*n + 35*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)) + (2*a*d*n*x^3*(a + b*x)^n*(3* n + n^2 + 2)*(10*a^2*d + 30*b^2*c + b^2*c*n^2 + 11*b^2*c*n))/(b^3*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720))