Integrand size = 20, antiderivative size = 232 \[ \int x^2 (a+b x)^n \left (c+d x^2\right )^2 \, dx=\frac {a^2 \left (b^2 c+a^2 d\right )^2 (a+b x)^{1+n}}{b^7 (1+n)}-\frac {2 a \left (b^2 c+a^2 d\right ) \left (b^2 c+3 a^2 d\right ) (a+b x)^{2+n}}{b^7 (2+n)}+\frac {\left (b^4 c^2+12 a^2 b^2 c d+15 a^4 d^2\right ) (a+b x)^{3+n}}{b^7 (3+n)}-\frac {4 a d \left (2 b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^7 (4+n)}+\frac {d \left (2 b^2 c+15 a^2 d\right ) (a+b x)^{5+n}}{b^7 (5+n)}-\frac {6 a d^2 (a+b x)^{6+n}}{b^7 (6+n)}+\frac {d^2 (a+b x)^{7+n}}{b^7 (7+n)} \]
a^2*(a^2*d+b^2*c)^2*(b*x+a)^(1+n)/b^7/(1+n)-2*a*(a^2*d+b^2*c)*(3*a^2*d+b^2 *c)*(b*x+a)^(2+n)/b^7/(2+n)+(15*a^4*d^2+12*a^2*b^2*c*d+b^4*c^2)*(b*x+a)^(3 +n)/b^7/(3+n)-4*a*d*(5*a^2*d+2*b^2*c)*(b*x+a)^(4+n)/b^7/(4+n)+d*(15*a^2*d+ 2*b^2*c)*(b*x+a)^(5+n)/b^7/(5+n)-6*a*d^2*(b*x+a)^(6+n)/b^7/(6+n)+d^2*(b*x+ a)^(7+n)/b^7/(7+n)
Time = 0.17 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.86 \[ \int x^2 (a+b x)^n \left (c+d x^2\right )^2 \, dx=\frac {(a+b x)^{1+n} \left (\frac {\left (a b^2 c+a^3 d\right )^2}{1+n}-\frac {2 a \left (b^2 c+a^2 d\right ) \left (b^2 c+3 a^2 d\right ) (a+b x)}{2+n}+\frac {\left (b^4 c^2+12 a^2 b^2 c d+15 a^4 d^2\right ) (a+b x)^2}{3+n}-\frac {4 a d \left (2 b^2 c+5 a^2 d\right ) (a+b x)^3}{4+n}+\frac {d \left (2 b^2 c+15 a^2 d\right ) (a+b x)^4}{5+n}-\frac {6 a d^2 (a+b x)^5}{6+n}+\frac {d^2 (a+b x)^6}{7+n}\right )}{b^7} \]
((a + b*x)^(1 + n)*((a*b^2*c + a^3*d)^2/(1 + n) - (2*a*(b^2*c + a^2*d)*(b^ 2*c + 3*a^2*d)*(a + b*x))/(2 + n) + ((b^4*c^2 + 12*a^2*b^2*c*d + 15*a^4*d^ 2)*(a + b*x)^2)/(3 + n) - (4*a*d*(2*b^2*c + 5*a^2*d)*(a + b*x)^3)/(4 + n) + (d*(2*b^2*c + 15*a^2*d)*(a + b*x)^4)/(5 + n) - (6*a*d^2*(a + b*x)^5)/(6 + n) + (d^2*(a + b*x)^6)/(7 + n)))/b^7
Time = 0.41 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int x^2 \left (c+d x^2\right )^2 (a+b x)^n \, dx\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (\frac {\left (a^3 d+a b^2 c\right )^2 (a+b x)^n}{b^6}+\frac {2 a \left (-3 a^2 d-b^2 c\right ) \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^6}-\frac {4 a d \left (5 a^2 d+2 b^2 c\right ) (a+b x)^{n+3}}{b^6}+\frac {d \left (15 a^2 d+2 b^2 c\right ) (a+b x)^{n+4}}{b^6}+\frac {\left (15 a^4 d^2+12 a^2 b^2 c d+b^4 c^2\right ) (a+b x)^{n+2}}{b^6}-\frac {6 a d^2 (a+b x)^{n+5}}{b^6}+\frac {d^2 (a+b x)^{n+6}}{b^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+1}}{b^7 (n+1)}-\frac {2 a \left (a^2 d+b^2 c\right ) \left (3 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^7 (n+2)}-\frac {4 a d \left (5 a^2 d+2 b^2 c\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac {d \left (15 a^2 d+2 b^2 c\right ) (a+b x)^{n+5}}{b^7 (n+5)}+\frac {\left (15 a^4 d^2+12 a^2 b^2 c d+b^4 c^2\right ) (a+b x)^{n+3}}{b^7 (n+3)}-\frac {6 a d^2 (a+b x)^{n+6}}{b^7 (n+6)}+\frac {d^2 (a+b x)^{n+7}}{b^7 (n+7)}\) |
(a^2*(b^2*c + a^2*d)^2*(a + b*x)^(1 + n))/(b^7*(1 + n)) - (2*a*(b^2*c + a^ 2*d)*(b^2*c + 3*a^2*d)*(a + b*x)^(2 + n))/(b^7*(2 + n)) + ((b^4*c^2 + 12*a ^2*b^2*c*d + 15*a^4*d^2)*(a + b*x)^(3 + n))/(b^7*(3 + n)) - (4*a*d*(2*b^2* c + 5*a^2*d)*(a + b*x)^(4 + n))/(b^7*(4 + n)) + (d*(2*b^2*c + 15*a^2*d)*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (6*a*d^2*(a + b*x)^(6 + n))/(b^7*(6 + n)) + (d^2*(a + b*x)^(7 + n))/(b^7*(7 + n))
3.4.55.3.1 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. \(753\) vs. \(2(232)=464\).
Time = 0.48 (sec) , antiderivative size = 754, normalized size of antiderivative = 3.25
| method | result | size |
| norman | \(\frac {d^{2} x^{7} {\mathrm e}^{n \ln \left (b x +a \right )}}{7+n}+\frac {n \,d^{2} a \,x^{6} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+13 n +42\right )}+\frac {\left (b^{4} c^{2} n^{4}+22 b^{4} c^{2} n^{3}+24 a^{2} b^{2} c d \,n^{2}+179 b^{4} c^{2} n^{2}+312 a^{2} b^{2} c d n +638 b^{4} c^{2} n +360 d^{2} a^{4}+1008 a^{2} b^{2} c d +840 b^{4} c^{2}\right ) a n \,x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{6}+27 n^{5}+295 n^{4}+1665 n^{3}+5104 n^{2}+8028 n +5040\right )}+\frac {2 a^{3} \left (b^{4} c^{2} n^{4}+22 b^{4} c^{2} n^{3}+24 a^{2} b^{2} c d \,n^{2}+179 b^{4} c^{2} n^{2}+312 a^{2} b^{2} c d n +638 b^{4} c^{2} n +360 d^{2} a^{4}+1008 a^{2} b^{2} c d +840 b^{4} c^{2}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{7} \left (n^{7}+28 n^{6}+322 n^{5}+1960 n^{4}+6769 n^{3}+13132 n^{2}+13068 n +5040\right )}-\frac {\left (-b^{4} c^{2} n^{4}+8 a^{2} b^{2} c d \,n^{3}-22 b^{4} c^{2} n^{3}+104 a^{2} b^{2} c d \,n^{2}-179 b^{4} c^{2} n^{2}+120 a^{4} d^{2} n +336 a^{2} b^{2} c d n -638 b^{4} c^{2} n -840 b^{4} c^{2}\right ) x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+25 n^{4}+245 n^{3}+1175 n^{2}+2754 n +2520\right )}-\frac {2 d \left (-b^{2} c \,n^{2}+3 a^{2} d n -13 b^{2} c n -42 b^{2} c \right ) x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+18 n^{2}+107 n +210\right )}-\frac {2 n \,a^{2} \left (b^{4} c^{2} n^{4}+22 b^{4} c^{2} n^{3}+24 a^{2} b^{2} c d \,n^{2}+179 b^{4} c^{2} n^{2}+312 a^{2} b^{2} c d n +638 b^{4} c^{2} n +360 d^{2} a^{4}+1008 a^{2} b^{2} c d +840 b^{4} c^{2}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{6} \left (n^{7}+28 n^{6}+322 n^{5}+1960 n^{4}+6769 n^{3}+13132 n^{2}+13068 n +5040\right )}+\frac {2 \left (b^{2} c \,n^{2}+13 b^{2} c n +15 a^{2} d +42 b^{2} c \right ) a d n \,x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+22 n^{3}+179 n^{2}+638 n +840\right )}\) | \(754\) |
| gosper | \(\frac {\left (b x +a \right )^{1+n} \left (b^{6} d^{2} n^{6} x^{6}+21 b^{6} d^{2} n^{5} x^{6}-6 a \,b^{5} d^{2} n^{5} x^{5}+2 b^{6} c d \,n^{6} x^{4}+175 b^{6} d^{2} n^{4} x^{6}-90 a \,b^{5} d^{2} n^{4} x^{5}+46 b^{6} c d \,n^{5} x^{4}+735 b^{6} d^{2} n^{3} x^{6}+30 a^{2} b^{4} d^{2} n^{4} x^{4}-8 a \,b^{5} c d \,n^{5} x^{3}-510 a \,b^{5} d^{2} n^{3} x^{5}+b^{6} c^{2} n^{6} x^{2}+414 b^{6} c d \,n^{4} x^{4}+1624 b^{6} d^{2} n^{2} x^{6}+300 a^{2} b^{4} d^{2} n^{3} x^{4}-152 a \,b^{5} c d \,n^{4} x^{3}-1350 a \,b^{5} d^{2} n^{2} x^{5}+25 b^{6} c^{2} n^{5} x^{2}+1850 b^{6} c d \,n^{3} x^{4}+1764 b^{6} d^{2} n \,x^{6}-120 a^{3} b^{3} d^{2} n^{3} x^{3}+24 a^{2} b^{4} c d \,n^{4} x^{2}+1050 a^{2} b^{4} d^{2} n^{2} x^{4}-2 a \,b^{5} c^{2} n^{5} x -1048 a \,b^{5} c d \,n^{3} x^{3}-1644 a \,b^{5} d^{2} n \,x^{5}+247 b^{6} c^{2} n^{4} x^{2}+4288 b^{6} c d \,n^{2} x^{4}+720 x^{6} d^{2} b^{6}-720 a^{3} b^{3} d^{2} n^{2} x^{3}+384 a^{2} b^{4} c d \,n^{3} x^{2}+1500 a^{2} b^{4} d^{2} n \,x^{4}-46 a \,b^{5} c^{2} n^{4} x -3208 a \,b^{5} c d \,n^{2} x^{3}-720 x^{5} d^{2} b^{5} a +1219 b^{6} c^{2} n^{3} x^{2}+4824 b^{6} c d n \,x^{4}+360 a^{4} b^{2} d^{2} n^{2} x^{2}-48 a^{3} b^{3} c d \,n^{3} x -1320 a^{3} b^{3} d^{2} n \,x^{3}+2 a^{2} b^{4} c^{2} n^{4}+1992 a^{2} b^{4} c d \,n^{2} x^{2}+720 a^{2} b^{4} d^{2} x^{4}-402 a \,b^{5} c^{2} n^{3} x -4320 a \,b^{5} c d n \,x^{3}+3112 b^{6} c^{2} n^{2} x^{2}+2016 b^{6} c d \,x^{4}+1080 a^{4} b^{2} d^{2} n \,x^{2}-672 a^{3} b^{3} c d \,n^{2} x -720 x^{3} d^{2} a^{3} b^{3}+44 a^{2} b^{4} c^{2} n^{3}+3648 a^{2} b^{4} c d n \,x^{2}-1634 a \,b^{5} c^{2} n^{2} x -2016 a \,b^{5} c d \,x^{3}+3796 b^{6} c^{2} n \,x^{2}-720 a^{5} b \,d^{2} n x +48 a^{4} b^{2} c d \,n^{2}+720 x^{2} d^{2} a^{4} b^{2}-2640 a^{3} b^{3} c d n x +358 a^{2} b^{4} c^{2} n^{2}+2016 a^{2} b^{4} c d \,x^{2}-2956 a \,b^{5} c^{2} n x +1680 b^{6} c^{2} x^{2}-720 a^{5} b \,d^{2} x +624 a^{4} b^{2} c d n -2016 a^{3} b^{3} c d x +1276 a^{2} b^{4} c^{2} n -1680 a \,b^{5} c^{2} x +720 d^{2} a^{6}+2016 a^{4} b^{2} c d +1680 a^{2} b^{4} c^{2}\right )}{b^{7} \left (n^{7}+28 n^{6}+322 n^{5}+1960 n^{4}+6769 n^{3}+13132 n^{2}+13068 n +5040\right )}\) | \(1000\) |
| risch | \(\text {Expression too large to display}\) | \(1175\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1788\) |
d^2/(7+n)*x^7*exp(n*ln(b*x+a))+n*d^2/b*a/(n^2+13*n+42)*x^6*exp(n*ln(b*x+a) )+(b^4*c^2*n^4+22*b^4*c^2*n^3+24*a^2*b^2*c*d*n^2+179*b^4*c^2*n^2+312*a^2*b ^2*c*d*n+638*b^4*c^2*n+360*a^4*d^2+1008*a^2*b^2*c*d+840*b^4*c^2)*a/b^5*n/( n^6+27*n^5+295*n^4+1665*n^3+5104*n^2+8028*n+5040)*x^2*exp(n*ln(b*x+a))+2*a ^3*(b^4*c^2*n^4+22*b^4*c^2*n^3+24*a^2*b^2*c*d*n^2+179*b^4*c^2*n^2+312*a^2* b^2*c*d*n+638*b^4*c^2*n+360*a^4*d^2+1008*a^2*b^2*c*d+840*b^4*c^2)/b^7/(n^7 +28*n^6+322*n^5+1960*n^4+6769*n^3+13132*n^2+13068*n+5040)*exp(n*ln(b*x+a)) -(-b^4*c^2*n^4+8*a^2*b^2*c*d*n^3-22*b^4*c^2*n^3+104*a^2*b^2*c*d*n^2-179*b^ 4*c^2*n^2+120*a^4*d^2*n+336*a^2*b^2*c*d*n-638*b^4*c^2*n-840*b^4*c^2)/b^4/( n^5+25*n^4+245*n^3+1175*n^2+2754*n+2520)*x^3*exp(n*ln(b*x+a))-2*d*(-b^2*c* n^2+3*a^2*d*n-13*b^2*c*n-42*b^2*c)/b^2/(n^3+18*n^2+107*n+210)*x^5*exp(n*ln (b*x+a))-2/b^6*n*a^2*(b^4*c^2*n^4+22*b^4*c^2*n^3+24*a^2*b^2*c*d*n^2+179*b^ 4*c^2*n^2+312*a^2*b^2*c*d*n+638*b^4*c^2*n+360*a^4*d^2+1008*a^2*b^2*c*d+840 *b^4*c^2)/(n^7+28*n^6+322*n^5+1960*n^4+6769*n^3+13132*n^2+13068*n+5040)*x* exp(n*ln(b*x+a))+2*(b^2*c*n^2+13*b^2*c*n+15*a^2*d+42*b^2*c)*a/b^3*d*n/(n^4 +22*n^3+179*n^2+638*n+840)*x^4*exp(n*ln(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 1027 vs. \(2 (232) = 464\).
Time = 0.32 (sec) , antiderivative size = 1027, normalized size of antiderivative = 4.43 \[ \int x^2 (a+b x)^n \left (c+d x^2\right )^2 \, dx=\frac {{\left (2 \, a^{3} b^{4} c^{2} n^{4} + 44 \, a^{3} b^{4} c^{2} n^{3} + 1680 \, a^{3} b^{4} c^{2} + 2016 \, a^{5} b^{2} c d + 720 \, a^{7} d^{2} + {\left (b^{7} d^{2} n^{6} + 21 \, b^{7} d^{2} n^{5} + 175 \, b^{7} d^{2} n^{4} + 735 \, b^{7} d^{2} n^{3} + 1624 \, b^{7} d^{2} n^{2} + 1764 \, b^{7} d^{2} n + 720 \, b^{7} d^{2}\right )} x^{7} + {\left (a b^{6} d^{2} n^{6} + 15 \, a b^{6} d^{2} n^{5} + 85 \, a b^{6} d^{2} n^{4} + 225 \, a b^{6} d^{2} n^{3} + 274 \, a b^{6} d^{2} n^{2} + 120 \, a b^{6} d^{2} n\right )} x^{6} + 2 \, {\left (b^{7} c d n^{6} + 1008 \, b^{7} c d + {\left (23 \, b^{7} c d - 3 \, a^{2} b^{5} d^{2}\right )} n^{5} + 3 \, {\left (69 \, b^{7} c d - 10 \, a^{2} b^{5} d^{2}\right )} n^{4} + 5 \, {\left (185 \, b^{7} c d - 21 \, a^{2} b^{5} d^{2}\right )} n^{3} + 2 \, {\left (1072 \, b^{7} c d - 75 \, a^{2} b^{5} d^{2}\right )} n^{2} + 36 \, {\left (67 \, b^{7} c d - 2 \, a^{2} b^{5} d^{2}\right )} n\right )} x^{5} + 2 \, {\left (a b^{6} c d n^{6} + 19 \, a b^{6} c d n^{5} + {\left (131 \, a b^{6} c d + 15 \, a^{3} b^{4} d^{2}\right )} n^{4} + {\left (401 \, a b^{6} c d + 90 \, a^{3} b^{4} d^{2}\right )} n^{3} + 15 \, {\left (36 \, a b^{6} c d + 11 \, a^{3} b^{4} d^{2}\right )} n^{2} + 18 \, {\left (14 \, a b^{6} c d + 5 \, a^{3} b^{4} d^{2}\right )} n\right )} x^{4} + {\left (b^{7} c^{2} n^{6} + 1680 \, b^{7} c^{2} + {\left (25 \, b^{7} c^{2} - 8 \, a^{2} b^{5} c d\right )} n^{5} + {\left (247 \, b^{7} c^{2} - 128 \, a^{2} b^{5} c d\right )} n^{4} + {\left (1219 \, b^{7} c^{2} - 664 \, a^{2} b^{5} c d - 120 \, a^{4} b^{3} d^{2}\right )} n^{3} + 8 \, {\left (389 \, b^{7} c^{2} - 152 \, a^{2} b^{5} c d - 45 \, a^{4} b^{3} d^{2}\right )} n^{2} + 4 \, {\left (949 \, b^{7} c^{2} - 168 \, a^{2} b^{5} c d - 60 \, a^{4} b^{3} d^{2}\right )} n\right )} x^{3} + 2 \, {\left (179 \, a^{3} b^{4} c^{2} + 24 \, a^{5} b^{2} c d\right )} n^{2} + {\left (a b^{6} c^{2} n^{6} + 23 \, a b^{6} c^{2} n^{5} + 3 \, {\left (67 \, a b^{6} c^{2} + 8 \, a^{3} b^{4} c d\right )} n^{4} + {\left (817 \, a b^{6} c^{2} + 336 \, a^{3} b^{4} c d\right )} n^{3} + 2 \, {\left (739 \, a b^{6} c^{2} + 660 \, a^{3} b^{4} c d + 180 \, a^{5} b^{2} d^{2}\right )} n^{2} + 24 \, {\left (35 \, a b^{6} c^{2} + 42 \, a^{3} b^{4} c d + 15 \, a^{5} b^{2} d^{2}\right )} n\right )} x^{2} + 4 \, {\left (319 \, a^{3} b^{4} c^{2} + 156 \, a^{5} b^{2} c d\right )} n - 2 \, {\left (a^{2} b^{5} c^{2} n^{5} + 22 \, a^{2} b^{5} c^{2} n^{4} + {\left (179 \, a^{2} b^{5} c^{2} + 24 \, a^{4} b^{3} c d\right )} n^{3} + 2 \, {\left (319 \, a^{2} b^{5} c^{2} + 156 \, a^{4} b^{3} c d\right )} n^{2} + 24 \, {\left (35 \, a^{2} b^{5} c^{2} + 42 \, a^{4} b^{3} c d + 15 \, a^{6} b d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{7} n^{7} + 28 \, b^{7} n^{6} + 322 \, b^{7} n^{5} + 1960 \, b^{7} n^{4} + 6769 \, b^{7} n^{3} + 13132 \, b^{7} n^{2} + 13068 \, b^{7} n + 5040 \, b^{7}} \]
(2*a^3*b^4*c^2*n^4 + 44*a^3*b^4*c^2*n^3 + 1680*a^3*b^4*c^2 + 2016*a^5*b^2* c*d + 720*a^7*d^2 + (b^7*d^2*n^6 + 21*b^7*d^2*n^5 + 175*b^7*d^2*n^4 + 735* b^7*d^2*n^3 + 1624*b^7*d^2*n^2 + 1764*b^7*d^2*n + 720*b^7*d^2)*x^7 + (a*b^ 6*d^2*n^6 + 15*a*b^6*d^2*n^5 + 85*a*b^6*d^2*n^4 + 225*a*b^6*d^2*n^3 + 274* a*b^6*d^2*n^2 + 120*a*b^6*d^2*n)*x^6 + 2*(b^7*c*d*n^6 + 1008*b^7*c*d + (23 *b^7*c*d - 3*a^2*b^5*d^2)*n^5 + 3*(69*b^7*c*d - 10*a^2*b^5*d^2)*n^4 + 5*(1 85*b^7*c*d - 21*a^2*b^5*d^2)*n^3 + 2*(1072*b^7*c*d - 75*a^2*b^5*d^2)*n^2 + 36*(67*b^7*c*d - 2*a^2*b^5*d^2)*n)*x^5 + 2*(a*b^6*c*d*n^6 + 19*a*b^6*c*d* n^5 + (131*a*b^6*c*d + 15*a^3*b^4*d^2)*n^4 + (401*a*b^6*c*d + 90*a^3*b^4*d ^2)*n^3 + 15*(36*a*b^6*c*d + 11*a^3*b^4*d^2)*n^2 + 18*(14*a*b^6*c*d + 5*a^ 3*b^4*d^2)*n)*x^4 + (b^7*c^2*n^6 + 1680*b^7*c^2 + (25*b^7*c^2 - 8*a^2*b^5* c*d)*n^5 + (247*b^7*c^2 - 128*a^2*b^5*c*d)*n^4 + (1219*b^7*c^2 - 664*a^2*b ^5*c*d - 120*a^4*b^3*d^2)*n^3 + 8*(389*b^7*c^2 - 152*a^2*b^5*c*d - 45*a^4* b^3*d^2)*n^2 + 4*(949*b^7*c^2 - 168*a^2*b^5*c*d - 60*a^4*b^3*d^2)*n)*x^3 + 2*(179*a^3*b^4*c^2 + 24*a^5*b^2*c*d)*n^2 + (a*b^6*c^2*n^6 + 23*a*b^6*c^2* n^5 + 3*(67*a*b^6*c^2 + 8*a^3*b^4*c*d)*n^4 + (817*a*b^6*c^2 + 336*a^3*b^4* c*d)*n^3 + 2*(739*a*b^6*c^2 + 660*a^3*b^4*c*d + 180*a^5*b^2*d^2)*n^2 + 24* (35*a*b^6*c^2 + 42*a^3*b^4*c*d + 15*a^5*b^2*d^2)*n)*x^2 + 4*(319*a^3*b^4*c ^2 + 156*a^5*b^2*c*d)*n - 2*(a^2*b^5*c^2*n^5 + 22*a^2*b^5*c^2*n^4 + (179*a ^2*b^5*c^2 + 24*a^4*b^3*c*d)*n^3 + 2*(319*a^2*b^5*c^2 + 156*a^4*b^3*c*d...
Leaf count of result is larger than twice the leaf count of optimal. 14317 vs. \(2 (218) = 436\).
Time = 4.19 (sec) , antiderivative size = 14317, normalized size of antiderivative = 61.71 \[ \int x^2 (a+b x)^n \left (c+d x^2\right )^2 \, dx=\text {Too large to display} \]
Piecewise((a**n*(c**2*x**3/3 + 2*c*d*x**5/5 + d**2*x**7/7), Eq(b, 0)), (60 *a**6*d**2*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x* *2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b* *13*x**6) + 147*a**6*d**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9* x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60* b**13*x**6) + 360*a**5*b*d**2*x*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8 *x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360 *a*b**12*x**5 + 60*b**13*x**6) + 822*a**5*b*d**2*x/(60*a**6*b**7 + 360*a** 5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 4*a**4*b**2*c*d/(60*a**6*b**7 + 360 *a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11* x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 900*a**4*b**2*d**2*x**2*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b** 10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 1875*a **4*b**2*d**2*x**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x **6) - 24*a**3*b**3*c*d*x/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9* x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60* b**13*x**6) + 1200*a**3*b**3*d**2*x**3*log(a/b + x)/(60*a**6*b**7 + 360*a* *5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*...
Time = 0.21 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.93 \[ \int x^2 (a+b x)^n \left (c+d x^2\right )^2 \, dx=\frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {2 \, {\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} + \frac {{\left ({\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{7} x^{7} + {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a b^{6} x^{6} - 6 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{2} b^{5} x^{5} + 30 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{3} b^{4} x^{4} - 120 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{4} b^{3} x^{3} + 360 \, {\left (n^{2} + n\right )} a^{5} b^{2} x^{2} - 720 \, a^{6} b n x + 720 \, a^{7}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{7}} \]
((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^4 + 10*n^3 + 35*n^2 + 5 0*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*c*d/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^5) + ((n ^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*b^7*x^7 + (n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a*b^6*x^6 - 6*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^2*b^5*x^5 + 30*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a ^3*b^4*x^4 - 120*(n^3 + 3*n^2 + 2*n)*a^4*b^3*x^3 + 360*(n^2 + n)*a^5*b^2*x ^2 - 720*a^6*b*n*x + 720*a^7)*(b*x + a)^n*d^2/((n^7 + 28*n^6 + 322*n^5 + 1 960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*b^7)
Leaf count of result is larger than twice the leaf count of optimal. 1750 vs. \(2 (232) = 464\).
Time = 0.30 (sec) , antiderivative size = 1750, normalized size of antiderivative = 7.54 \[ \int x^2 (a+b x)^n \left (c+d x^2\right )^2 \, dx=\text {Too large to display} \]
((b*x + a)^n*b^7*d^2*n^6*x^7 + (b*x + a)^n*a*b^6*d^2*n^6*x^6 + 21*(b*x + a )^n*b^7*d^2*n^5*x^7 + 2*(b*x + a)^n*b^7*c*d*n^6*x^5 + 15*(b*x + a)^n*a*b^6 *d^2*n^5*x^6 + 175*(b*x + a)^n*b^7*d^2*n^4*x^7 + 2*(b*x + a)^n*a*b^6*c*d*n ^6*x^4 + 46*(b*x + a)^n*b^7*c*d*n^5*x^5 - 6*(b*x + a)^n*a^2*b^5*d^2*n^5*x^ 5 + 85*(b*x + a)^n*a*b^6*d^2*n^4*x^6 + 735*(b*x + a)^n*b^7*d^2*n^3*x^7 + ( b*x + a)^n*b^7*c^2*n^6*x^3 + 38*(b*x + a)^n*a*b^6*c*d*n^5*x^4 + 414*(b*x + a)^n*b^7*c*d*n^4*x^5 - 60*(b*x + a)^n*a^2*b^5*d^2*n^4*x^5 + 225*(b*x + a) ^n*a*b^6*d^2*n^3*x^6 + 1624*(b*x + a)^n*b^7*d^2*n^2*x^7 + (b*x + a)^n*a*b^ 6*c^2*n^6*x^2 + 25*(b*x + a)^n*b^7*c^2*n^5*x^3 - 8*(b*x + a)^n*a^2*b^5*c*d *n^5*x^3 + 262*(b*x + a)^n*a*b^6*c*d*n^4*x^4 + 30*(b*x + a)^n*a^3*b^4*d^2* n^4*x^4 + 1850*(b*x + a)^n*b^7*c*d*n^3*x^5 - 210*(b*x + a)^n*a^2*b^5*d^2*n ^3*x^5 + 274*(b*x + a)^n*a*b^6*d^2*n^2*x^6 + 1764*(b*x + a)^n*b^7*d^2*n*x^ 7 + 23*(b*x + a)^n*a*b^6*c^2*n^5*x^2 + 247*(b*x + a)^n*b^7*c^2*n^4*x^3 - 1 28*(b*x + a)^n*a^2*b^5*c*d*n^4*x^3 + 802*(b*x + a)^n*a*b^6*c*d*n^3*x^4 + 1 80*(b*x + a)^n*a^3*b^4*d^2*n^3*x^4 + 4288*(b*x + a)^n*b^7*c*d*n^2*x^5 - 30 0*(b*x + a)^n*a^2*b^5*d^2*n^2*x^5 + 120*(b*x + a)^n*a*b^6*d^2*n*x^6 + 720* (b*x + a)^n*b^7*d^2*x^7 - 2*(b*x + a)^n*a^2*b^5*c^2*n^5*x + 201*(b*x + a)^ n*a*b^6*c^2*n^4*x^2 + 24*(b*x + a)^n*a^3*b^4*c*d*n^4*x^2 + 1219*(b*x + a)^ n*b^7*c^2*n^3*x^3 - 664*(b*x + a)^n*a^2*b^5*c*d*n^3*x^3 - 120*(b*x + a)^n* a^4*b^3*d^2*n^3*x^3 + 1080*(b*x + a)^n*a*b^6*c*d*n^2*x^4 + 330*(b*x + a...
Time = 11.96 (sec) , antiderivative size = 932, normalized size of antiderivative = 4.02 \[ \int x^2 (a+b x)^n \left (c+d x^2\right )^2 \, dx=\frac {2\,a^3\,{\left (a+b\,x\right )}^n\,\left (360\,a^4\,d^2+24\,a^2\,b^2\,c\,d\,n^2+312\,a^2\,b^2\,c\,d\,n+1008\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+22\,b^4\,c^2\,n^3+179\,b^4\,c^2\,n^2+638\,b^4\,c^2\,n+840\,b^4\,c^2\right )}{b^7\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {d^2\,x^7\,{\left (a+b\,x\right )}^n\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}{n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040}+\frac {x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (-120\,a^4\,d^2\,n-8\,a^2\,b^2\,c\,d\,n^3-104\,a^2\,b^2\,c\,d\,n^2-336\,a^2\,b^2\,c\,d\,n+b^4\,c^2\,n^4+22\,b^4\,c^2\,n^3+179\,b^4\,c^2\,n^2+638\,b^4\,c^2\,n+840\,b^4\,c^2\right )}{b^4\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}-\frac {2\,a^2\,n\,x\,{\left (a+b\,x\right )}^n\,\left (360\,a^4\,d^2+24\,a^2\,b^2\,c\,d\,n^2+312\,a^2\,b^2\,c\,d\,n+1008\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+22\,b^4\,c^2\,n^3+179\,b^4\,c^2\,n^2+638\,b^4\,c^2\,n+840\,b^4\,c^2\right )}{b^6\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {2\,d\,x^5\,{\left (a+b\,x\right )}^n\,\left (-3\,d\,a^2\,n+c\,b^2\,n^2+13\,c\,b^2\,n+42\,c\,b^2\right )\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b^2\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {a\,d^2\,n\,x^6\,{\left (a+b\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{b\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {a\,n\,x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (360\,a^4\,d^2+24\,a^2\,b^2\,c\,d\,n^2+312\,a^2\,b^2\,c\,d\,n+1008\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+22\,b^4\,c^2\,n^3+179\,b^4\,c^2\,n^2+638\,b^4\,c^2\,n+840\,b^4\,c^2\right )}{b^5\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {2\,a\,d\,n\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )\,\left (15\,d\,a^2+c\,b^2\,n^2+13\,c\,b^2\,n+42\,c\,b^2\right )}{b^3\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )} \]
(2*a^3*(a + b*x)^n*(360*a^4*d^2 + 840*b^4*c^2 + 638*b^4*c^2*n + 179*b^4*c^ 2*n^2 + 22*b^4*c^2*n^3 + b^4*c^2*n^4 + 1008*a^2*b^2*c*d + 312*a^2*b^2*c*d* n + 24*a^2*b^2*c*d*n^2))/(b^7*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (d^2*x^7*(a + b*x)^n*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720))/(13068*n + 13132*n^2 + 6769*n^ 3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040) + (x^3*(a + b*x)^n*(3*n + n^ 2 + 2)*(840*b^4*c^2 - 120*a^4*d^2*n + 638*b^4*c^2*n + 179*b^4*c^2*n^2 + 22 *b^4*c^2*n^3 + b^4*c^2*n^4 - 336*a^2*b^2*c*d*n - 104*a^2*b^2*c*d*n^2 - 8*a ^2*b^2*c*d*n^3))/(b^4*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) - (2*a^2*n*x*(a + b*x)^n*(360*a^4*d^2 + 840*b^4*c ^2 + 638*b^4*c^2*n + 179*b^4*c^2*n^2 + 22*b^4*c^2*n^3 + b^4*c^2*n^4 + 1008 *a^2*b^2*c*d + 312*a^2*b^2*c*d*n + 24*a^2*b^2*c*d*n^2))/(b^6*(13068*n + 13 132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (2*d*x^5 *(a + b*x)^n*(42*b^2*c + b^2*c*n^2 - 3*a^2*d*n + 13*b^2*c*n)*(50*n + 35*n^ 2 + 10*n^3 + n^4 + 24))/(b^2*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (a*d^2*n*x^6*(a + b*x)^n*(274*n + 225*n^ 2 + 85*n^3 + 15*n^4 + n^5 + 120))/(b*(13068*n + 13132*n^2 + 6769*n^3 + 196 0*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (a*n*x^2*(n + 1)*(a + b*x)^n*(36 0*a^4*d^2 + 840*b^4*c^2 + 638*b^4*c^2*n + 179*b^4*c^2*n^2 + 22*b^4*c^2*n^3 + b^4*c^2*n^4 + 1008*a^2*b^2*c*d + 312*a^2*b^2*c*d*n + 24*a^2*b^2*c*d*...