Integrand size = 18, antiderivative size = 185 \[ \int x (c+d x)^n \left (a+b x^2\right )^2 \, dx=-\frac {c \left (b c^2+a d^2\right )^2 (c+d x)^{1+n}}{d^6 (1+n)}+\frac {\left (b c^2+a d^2\right ) \left (5 b c^2+a d^2\right ) (c+d x)^{2+n}}{d^6 (2+n)}-\frac {2 b c \left (5 b c^2+3 a d^2\right ) (c+d x)^{3+n}}{d^6 (3+n)}+\frac {2 b \left (5 b c^2+a d^2\right ) (c+d x)^{4+n}}{d^6 (4+n)}-\frac {5 b^2 c (c+d x)^{5+n}}{d^6 (5+n)}+\frac {b^2 (c+d x)^{6+n}}{d^6 (6+n)} \] Output:
-c*(a*d^2+b*c^2)^2*(d*x+c)^(1+n)/d^6/(1+n)+(a*d^2+b*c^2)*(a*d^2+5*b*c^2)*( d*x+c)^(2+n)/d^6/(2+n)-2*b*c*(3*a*d^2+5*b*c^2)*(d*x+c)^(3+n)/d^6/(3+n)+2*b *(a*d^2+5*b*c^2)*(d*x+c)^(4+n)/d^6/(4+n)-5*b^2*c*(d*x+c)^(5+n)/d^6/(5+n)+b ^2*(d*x+c)^(6+n)/d^6/(6+n)
Time = 0.37 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.92 \[ \int x (c+d x)^n \left (a+b x^2\right )^2 \, dx=\frac {(c+d x)^{1+n} \left (d^4 (1+n) (2+n) (3+n) (4+n) (5+n) (c+d x) \left (a+b x^2\right )^2-c (6+n) \left (d^4 (1+n) (2+n) (3+n) (4+n) \left (a+b x^2\right )^2+4 \left (b c^2+a d^2\right ) (4+n) \left (a d^2 \left (6+5 n+n^2\right )+b \left (2 c^2-2 c d (1+n) x+d^2 \left (2+3 n+n^2\right ) x^2\right )\right )-4 b c (1+n) (c+d x) \left (a d^2 \left (12+7 n+n^2\right )+b \left (2 c^2-2 c d (2+n) x+d^2 \left (6+5 n+n^2\right ) x^2\right )\right )\right )+4 (1+n) (c+d x) \left (\left (b c^2+a d^2\right ) (5+n) \left (a d^2 \left (12+7 n+n^2\right )+b \left (2 c^2-2 c d (2+n) x+d^2 \left (6+5 n+n^2\right ) x^2\right )\right )-b c (2+n) (c+d x) \left (a d^2 \left (20+9 n+n^2\right )+b \left (2 c^2-2 c d (3+n) x+d^2 \left (12+7 n+n^2\right ) x^2\right )\right )\right )\right )}{d^6 (1+n) (2+n) (3+n) (4+n) (5+n) (6+n)} \] Input:
Integrate[x*(c + d*x)^n*(a + b*x^2)^2,x]
Output:
((c + d*x)^(1 + n)*(d^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(c + d*x)* (a + b*x^2)^2 - c*(6 + n)*(d^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(a + b*x^2) ^2 + 4*(b*c^2 + a*d^2)*(4 + n)*(a*d^2*(6 + 5*n + n^2) + b*(2*c^2 - 2*c*d*( 1 + n)*x + d^2*(2 + 3*n + n^2)*x^2)) - 4*b*c*(1 + n)*(c + d*x)*(a*d^2*(12 + 7*n + n^2) + b*(2*c^2 - 2*c*d*(2 + n)*x + d^2*(6 + 5*n + n^2)*x^2))) + 4 *(1 + n)*(c + d*x)*((b*c^2 + a*d^2)*(5 + n)*(a*d^2*(12 + 7*n + n^2) + b*(2 *c^2 - 2*c*d*(2 + n)*x + d^2*(6 + 5*n + n^2)*x^2)) - b*c*(2 + n)*(c + d*x) *(a*d^2*(20 + 9*n + n^2) + b*(2*c^2 - 2*c*d*(3 + n)*x + d^2*(12 + 7*n + n^ 2)*x^2)))))/(d^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.
| \(\displaystyle \int x \left (a+b x^2\right )^2 (c+d x)^n \, dx\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (-\frac {c \left (a d^2+b c^2\right )^2 (c+d x)^n}{d^5}+\frac {\left (a d^2+b c^2\right ) \left (a d^2+5 b c^2\right ) (c+d x)^{n+1}}{d^5}-\frac {2 b c \left (3 a d^2+5 b c^2\right ) (c+d x)^{n+2}}{d^5}+\frac {2 b \left (a d^2+5 b c^2\right ) (c+d x)^{n+3}}{d^5}-\frac {5 b^2 c (c+d x)^{n+4}}{d^5}+\frac {b^2 (c+d x)^{n+5}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c \left (a d^2+b c^2\right )^2 (c+d x)^{n+1}}{d^6 (n+1)}+\frac {\left (a d^2+b c^2\right ) \left (a d^2+5 b c^2\right ) (c+d x)^{n+2}}{d^6 (n+2)}-\frac {2 b c \left (3 a d^2+5 b c^2\right ) (c+d x)^{n+3}}{d^6 (n+3)}+\frac {2 b \left (a d^2+5 b c^2\right ) (c+d x)^{n+4}}{d^6 (n+4)}-\frac {5 b^2 c (c+d x)^{n+5}}{d^6 (n+5)}+\frac {b^2 (c+d x)^{n+6}}{d^6 (n+6)}\) |
Input:
Int[x*(c + d*x)^n*(a + b*x^2)^2,x]
Output:
-((c*(b*c^2 + a*d^2)^2*(c + d*x)^(1 + n))/(d^6*(1 + n))) + ((b*c^2 + a*d^2 )*(5*b*c^2 + a*d^2)*(c + d*x)^(2 + n))/(d^6*(2 + n)) - (2*b*c*(5*b*c^2 + 3 *a*d^2)*(c + d*x)^(3 + n))/(d^6*(3 + n)) + (2*b*(5*b*c^2 + a*d^2)*(c + d*x )^(4 + n))/(d^6*(4 + n)) - (5*b^2*c*(c + d*x)^(5 + n))/(d^6*(5 + n)) + (b^ 2*(c + d*x)^(6 + n))/(d^6*(6 + n))
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. \(597\) vs. \(2(185)=370\).
Time = 0.34 (sec) , antiderivative size = 598, normalized size of antiderivative = 3.23
| method | result | size |
| norman | \(\frac {b^{2} x^{6} {\mathrm e}^{n \ln \left (d x +c \right )}}{6+n}+\frac {\left (a^{2} d^{4} n^{4}+18 a^{2} d^{4} n^{3}-6 a b \,c^{2} d^{2} n^{3}+119 a^{2} d^{4} n^{2}-66 a b \,c^{2} d^{2} n^{2}+342 a^{2} d^{4} n -180 a b \,c^{2} d^{2} n -60 b^{2} c^{4} n +360 a^{2} d^{4}\right ) x^{2} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{4} \left (n^{5}+20 n^{4}+155 n^{3}+580 n^{2}+1044 n +720\right )}+\frac {b \left (2 a \,d^{2} n^{2}+22 a \,d^{2} n -5 b \,c^{2} n +60 a \,d^{2}\right ) x^{4} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{2} \left (n^{3}+15 n^{2}+74 n +120\right )}+\frac {n c \left (a^{2} d^{4} n^{4}+18 a^{2} d^{4} n^{3}+119 a^{2} d^{4} n^{2}+12 a b \,c^{2} d^{2} n^{2}+342 a^{2} d^{4} n +132 a b \,c^{2} d^{2} n +360 a^{2} d^{4}+360 b \,c^{2} d^{2} a +120 b^{2} c^{4}\right ) x \,{\mathrm e}^{n \ln \left (d x +c \right )}}{d^{5} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}+\frac {n \,b^{2} c \,x^{5} {\mathrm e}^{n \ln \left (d x +c \right )}}{d \left (n^{2}+11 n +30\right )}-\frac {c^{2} \left (a^{2} d^{4} n^{4}+18 a^{2} d^{4} n^{3}+119 a^{2} d^{4} n^{2}+12 a b \,c^{2} d^{2} n^{2}+342 a^{2} d^{4} n +132 a b \,c^{2} d^{2} n +360 a^{2} d^{4}+360 b \,c^{2} d^{2} a +120 b^{2} c^{4}\right ) {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}+\frac {2 \left (a \,d^{2} n^{2}+11 a \,d^{2} n +30 a \,d^{2}+10 b \,c^{2}\right ) b c n \,x^{3} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{3} \left (n^{4}+18 n^{3}+119 n^{2}+342 n +360\right )}\) | \(598\) |
| gosper | \(-\frac {\left (d x +c \right )^{1+n} \left (-b^{2} d^{5} n^{5} x^{5}-15 b^{2} d^{5} n^{4} x^{5}-2 a b \,d^{5} n^{5} x^{3}+5 b^{2} c \,d^{4} n^{4} x^{4}-85 b^{2} d^{5} n^{3} x^{5}-34 a b \,d^{5} n^{4} x^{3}+50 b^{2} c \,d^{4} n^{3} x^{4}-225 b^{2} d^{5} n^{2} x^{5}-a^{2} d^{5} n^{5} x +6 a b c \,d^{4} n^{4} x^{2}-214 a b \,d^{5} n^{3} x^{3}-20 b^{2} c^{2} d^{3} n^{3} x^{3}+175 b^{2} c \,d^{4} n^{2} x^{4}-274 b^{2} d^{5} n \,x^{5}-19 a^{2} d^{5} n^{4} x +84 a b c \,d^{4} n^{3} x^{2}-614 a b \,d^{5} n^{2} x^{3}-120 b^{2} c^{2} d^{3} n^{2} x^{3}+250 b^{2} c \,d^{4} n \,x^{4}-120 b^{2} x^{5} d^{5}+a^{2} c \,d^{4} n^{4}-137 a^{2} d^{5} n^{3} x -12 a b \,c^{2} d^{3} n^{3} x +390 a b c \,d^{4} n^{2} x^{2}-792 a b \,d^{5} n \,x^{3}+60 b^{2} c^{3} d^{2} n^{2} x^{2}-220 b^{2} c^{2} d^{3} n \,x^{3}+120 b^{2} c \,x^{4} d^{4}+18 a^{2} c \,d^{4} n^{3}-461 a^{2} d^{5} n^{2} x -144 a b \,c^{2} d^{3} n^{2} x +672 a b c \,d^{4} n \,x^{2}-360 a b \,d^{5} x^{3}+180 b^{2} c^{3} d^{2} n \,x^{2}-120 c^{2} d^{3} x^{3} b^{2}+119 a^{2} c \,d^{4} n^{2}-702 a^{2} d^{5} n x +12 a b \,c^{3} d^{2} n^{2}-492 a b \,c^{2} d^{3} n x +360 a b c \,d^{4} x^{2}-120 b^{2} c^{4} d n x +120 b^{2} c^{3} d^{2} x^{2}+342 a^{2} c \,d^{4} n -360 a^{2} x \,d^{5}+132 a b \,c^{3} d^{2} n -360 a b \,c^{2} d^{3} x -120 b^{2} c^{4} d x +360 a^{2} c \,d^{4}+360 a \,c^{3} d^{2} b +120 c^{5} b^{2}\right )}{d^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}\) | \(677\) |
| orering | \(-\frac {\left (d x +c \right )^{n} \left (-b^{2} d^{5} n^{5} x^{5}-15 b^{2} d^{5} n^{4} x^{5}-2 a b \,d^{5} n^{5} x^{3}+5 b^{2} c \,d^{4} n^{4} x^{4}-85 b^{2} d^{5} n^{3} x^{5}-34 a b \,d^{5} n^{4} x^{3}+50 b^{2} c \,d^{4} n^{3} x^{4}-225 b^{2} d^{5} n^{2} x^{5}-a^{2} d^{5} n^{5} x +6 a b c \,d^{4} n^{4} x^{2}-214 a b \,d^{5} n^{3} x^{3}-20 b^{2} c^{2} d^{3} n^{3} x^{3}+175 b^{2} c \,d^{4} n^{2} x^{4}-274 b^{2} d^{5} n \,x^{5}-19 a^{2} d^{5} n^{4} x +84 a b c \,d^{4} n^{3} x^{2}-614 a b \,d^{5} n^{2} x^{3}-120 b^{2} c^{2} d^{3} n^{2} x^{3}+250 b^{2} c \,d^{4} n \,x^{4}-120 b^{2} x^{5} d^{5}+a^{2} c \,d^{4} n^{4}-137 a^{2} d^{5} n^{3} x -12 a b \,c^{2} d^{3} n^{3} x +390 a b c \,d^{4} n^{2} x^{2}-792 a b \,d^{5} n \,x^{3}+60 b^{2} c^{3} d^{2} n^{2} x^{2}-220 b^{2} c^{2} d^{3} n \,x^{3}+120 b^{2} c \,x^{4} d^{4}+18 a^{2} c \,d^{4} n^{3}-461 a^{2} d^{5} n^{2} x -144 a b \,c^{2} d^{3} n^{2} x +672 a b c \,d^{4} n \,x^{2}-360 a b \,d^{5} x^{3}+180 b^{2} c^{3} d^{2} n \,x^{2}-120 c^{2} d^{3} x^{3} b^{2}+119 a^{2} c \,d^{4} n^{2}-702 a^{2} d^{5} n x +12 a b \,c^{3} d^{2} n^{2}-492 a b \,c^{2} d^{3} n x +360 a b c \,d^{4} x^{2}-120 b^{2} c^{4} d n x +120 b^{2} c^{3} d^{2} x^{2}+342 a^{2} c \,d^{4} n -360 a^{2} x \,d^{5}+132 a b \,c^{3} d^{2} n -360 a b \,c^{2} d^{3} x -120 b^{2} c^{4} d x +360 a^{2} c \,d^{4}+360 a \,c^{3} d^{2} b +120 c^{5} b^{2}\right ) \left (d x +c \right )}{d^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}\) | \(680\) |
| risch | \(-\frac {\left (-b^{2} d^{6} n^{5} x^{6}-b^{2} c \,d^{5} n^{5} x^{5}-15 b^{2} d^{6} n^{4} x^{6}-2 a b \,d^{6} n^{5} x^{4}-10 b^{2} c \,d^{5} n^{4} x^{5}-85 b^{2} d^{6} n^{3} x^{6}-2 a b c \,d^{5} n^{5} x^{3}-34 a b \,d^{6} n^{4} x^{4}+5 b^{2} c^{2} d^{4} n^{4} x^{4}-35 b^{2} c \,d^{5} n^{3} x^{5}-225 b^{2} d^{6} n^{2} x^{6}-a^{2} d^{6} n^{5} x^{2}-28 a b c \,d^{5} n^{4} x^{3}-214 a b \,d^{6} n^{3} x^{4}+30 b^{2} c^{2} d^{4} n^{3} x^{4}-50 b^{2} c \,d^{5} n^{2} x^{5}-274 b^{2} d^{6} n \,x^{6}-a^{2} c \,d^{5} n^{5} x -19 a^{2} d^{6} n^{4} x^{2}+6 a b \,c^{2} d^{4} n^{4} x^{2}-130 a b c \,d^{5} n^{3} x^{3}-614 a b \,d^{6} n^{2} x^{4}-20 b^{2} c^{3} d^{3} n^{3} x^{3}+55 b^{2} c^{2} d^{4} n^{2} x^{4}-24 b^{2} c \,d^{5} n \,x^{5}-120 b^{2} d^{6} x^{6}-18 a^{2} c \,d^{5} n^{4} x -137 a^{2} d^{6} n^{3} x^{2}+72 a b \,c^{2} d^{4} n^{3} x^{2}-224 a b c \,d^{5} n^{2} x^{3}-792 a b \,d^{6} n \,x^{4}-60 b^{2} c^{3} d^{3} n^{2} x^{3}+30 b^{2} c^{2} d^{4} n \,x^{4}+a^{2} c^{2} d^{4} n^{4}-119 a^{2} c \,d^{5} n^{3} x -461 a^{2} d^{6} n^{2} x^{2}-12 a b \,c^{3} d^{3} n^{3} x +246 a b \,c^{2} d^{4} n^{2} x^{2}-120 a b c \,d^{5} n \,x^{3}-360 x^{4} a b \,d^{6}+60 b^{2} c^{4} d^{2} n^{2} x^{2}-40 b^{2} c^{3} d^{3} n \,x^{3}+18 a^{2} c^{2} d^{4} n^{3}-342 a^{2} c \,d^{5} n^{2} x -702 a^{2} d^{6} n \,x^{2}-132 a b \,c^{3} d^{3} n^{2} x +180 a b \,c^{2} d^{4} n \,x^{2}+60 b^{2} c^{4} d^{2} n \,x^{2}+119 a^{2} c^{2} d^{4} n^{2}-360 a^{2} c \,d^{5} n x -360 x^{2} a^{2} d^{6}+12 a b \,c^{4} d^{2} n^{2}-360 a b \,c^{3} d^{3} n x -120 b^{2} c^{5} d n x +342 a^{2} c^{2} d^{4} n +132 a b \,c^{4} d^{2} n +360 a^{2} c^{2} d^{4}+360 a b \,c^{4} d^{2}+120 c^{6} b^{2}\right ) \left (d x +c \right )^{n}}{\left (5+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) d^{6}}\) | \(845\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1247\) |
Input:
int(x*(d*x+c)^n*(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
b^2/(6+n)*x^6*exp(n*ln(d*x+c))+(a^2*d^4*n^4+18*a^2*d^4*n^3-6*a*b*c^2*d^2*n ^3+119*a^2*d^4*n^2-66*a*b*c^2*d^2*n^2+342*a^2*d^4*n-180*a*b*c^2*d^2*n-60*b ^2*c^4*n+360*a^2*d^4)/d^4/(n^5+20*n^4+155*n^3+580*n^2+1044*n+720)*x^2*exp( n*ln(d*x+c))+b*(2*a*d^2*n^2+22*a*d^2*n-5*b*c^2*n+60*a*d^2)/d^2/(n^3+15*n^2 +74*n+120)*x^4*exp(n*ln(d*x+c))+1/d^5*n*c*(a^2*d^4*n^4+18*a^2*d^4*n^3+119* a^2*d^4*n^2+12*a*b*c^2*d^2*n^2+342*a^2*d^4*n+132*a*b*c^2*d^2*n+360*a^2*d^4 +360*a*b*c^2*d^2+120*b^2*c^4)/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1764*n+ 720)*x*exp(n*ln(d*x+c))+n*b^2*c/d/(n^2+11*n+30)*x^5*exp(n*ln(d*x+c))-c^2*( a^2*d^4*n^4+18*a^2*d^4*n^3+119*a^2*d^4*n^2+12*a*b*c^2*d^2*n^2+342*a^2*d^4* n+132*a*b*c^2*d^2*n+360*a^2*d^4+360*a*b*c^2*d^2+120*b^2*c^4)/d^6/(n^6+21*n ^5+175*n^4+735*n^3+1624*n^2+1764*n+720)*exp(n*ln(d*x+c))+2*(a*d^2*n^2+11*a *d^2*n+30*a*d^2+10*b*c^2)*b*c/d^3*n/(n^4+18*n^3+119*n^2+342*n+360)*x^3*exp (n*ln(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (185) = 370\).
Time = 0.09 (sec) , antiderivative size = 762, normalized size of antiderivative = 4.12 \[ \int x (c+d x)^n \left (a+b x^2\right )^2 \, dx =\text {Too large to display} \] Input:
integrate(x*(d*x+c)^n*(b*x^2+a)^2,x, algorithm="fricas")
Output:
-(a^2*c^2*d^4*n^4 + 18*a^2*c^2*d^4*n^3 + 120*b^2*c^6 + 360*a*b*c^4*d^2 + 3 60*a^2*c^2*d^4 - (b^2*d^6*n^5 + 15*b^2*d^6*n^4 + 85*b^2*d^6*n^3 + 225*b^2* d^6*n^2 + 274*b^2*d^6*n + 120*b^2*d^6)*x^6 - (b^2*c*d^5*n^5 + 10*b^2*c*d^5 *n^4 + 35*b^2*c*d^5*n^3 + 50*b^2*c*d^5*n^2 + 24*b^2*c*d^5*n)*x^5 - (2*a*b* d^6*n^5 + 360*a*b*d^6 - (5*b^2*c^2*d^4 - 34*a*b*d^6)*n^4 - 2*(15*b^2*c^2*d ^4 - 107*a*b*d^6)*n^3 - (55*b^2*c^2*d^4 - 614*a*b*d^6)*n^2 - 6*(5*b^2*c^2* d^4 - 132*a*b*d^6)*n)*x^4 - 2*(a*b*c*d^5*n^5 + 14*a*b*c*d^5*n^4 + 5*(2*b^2 *c^3*d^3 + 13*a*b*c*d^5)*n^3 + 2*(15*b^2*c^3*d^3 + 56*a*b*c*d^5)*n^2 + 20* (b^2*c^3*d^3 + 3*a*b*c*d^5)*n)*x^3 + (12*a*b*c^4*d^2 + 119*a^2*c^2*d^4)*n^ 2 - (a^2*d^6*n^5 + 360*a^2*d^6 - (6*a*b*c^2*d^4 - 19*a^2*d^6)*n^4 - (72*a* b*c^2*d^4 - 137*a^2*d^6)*n^3 - (60*b^2*c^4*d^2 + 246*a*b*c^2*d^4 - 461*a^2 *d^6)*n^2 - 6*(10*b^2*c^4*d^2 + 30*a*b*c^2*d^4 - 117*a^2*d^6)*n)*x^2 + 6*( 22*a*b*c^4*d^2 + 57*a^2*c^2*d^4)*n - (a^2*c*d^5*n^5 + 18*a^2*c*d^5*n^4 + ( 12*a*b*c^3*d^3 + 119*a^2*c*d^5)*n^3 + 6*(22*a*b*c^3*d^3 + 57*a^2*c*d^5)*n^ 2 + 120*(b^2*c^5*d + 3*a*b*c^3*d^3 + 3*a^2*c*d^5)*n)*x)*(d*x + c)^n/(d^6*n ^6 + 21*d^6*n^5 + 175*d^6*n^4 + 735*d^6*n^3 + 1624*d^6*n^2 + 1764*d^6*n + 720*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 8940 vs. \(2 (170) = 340\).
Time = 2.11 (sec) , antiderivative size = 8940, normalized size of antiderivative = 48.32 \[ \int x (c+d x)^n \left (a+b x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate(x*(d*x+c)**n*(b*x**2+a)**2,x)
Output:
Piecewise((c**n*(a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6), Eq(d, 0)), (-3*a **2*c*d**4/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2 *d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) - 15*a**2*d**5*x/(60*c**5*d **6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d* *10*x**4 + 60*d**11*x**5) - 6*a*b*c**3*d**2/(60*c**5*d**6 + 300*c**4*d**7* x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11* x**5) - 30*a*b*c**2*d**3*x/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8 *x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) - 60*a*b*c* d**4*x**2/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2* d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) - 60*a*b*d**5*x**3/(60*c**5* d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d **10*x**4 + 60*d**11*x**5) + 60*b**2*c**5*log(c/d + x)/(60*c**5*d**6 + 300 *c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) + 137*b**2*c**5/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c** 3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) + 300 *b**2*c**4*d*x*log(c/d + x)/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d** 8*x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) + 625*b**2 *c**4*d*x/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2* d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) + 600*b**2*c**3*d**2*x**2*lo g(c/d + x)/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c...
Time = 0.04 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.81 \[ \int x (c+d x)^n \left (a+b x^2\right )^2 \, dx=\frac {{\left (d^{2} {\left (n + 1\right )} x^{2} + c d n x - c^{2}\right )} {\left (d x + c\right )}^{n} a^{2}}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {2 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c d^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} c^{2} d^{2} x^{2} + 6 \, c^{3} d n x - 6 \, c^{4}\right )} {\left (d x + c\right )}^{n} a b}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{4}} + \frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} d^{6} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} c d^{5} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} c^{2} d^{4} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c^{3} d^{3} x^{3} - 60 \, {\left (n^{2} + n\right )} c^{4} d^{2} x^{2} + 120 \, c^{5} d n x - 120 \, c^{6}\right )} {\left (d x + c\right )}^{n} b^{2}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} d^{6}} \] Input:
integrate(x*(d*x+c)^n*(b*x^2+a)^2,x, algorithm="maxima")
Output:
(d^2*(n + 1)*x^2 + c*d*n*x - c^2)*(d*x + c)^n*a^2/((n^2 + 3*n + 2)*d^2) + 2*((n^3 + 6*n^2 + 11*n + 6)*d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3 - 3*(n ^2 + n)*c^2*d^2*x^2 + 6*c^3*d*n*x - 6*c^4)*(d*x + c)^n*a*b/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^4) + ((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 1 20)*d^6*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*c*d^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*c^2*d^4*x^4 + 20*(n^3 + 3*n^2 + 2*n)*c^3*d^3*x^3 - 60*(n^2 + n)*c^4*d^2*x^2 + 120*c^5*d*n*x - 120*c^6)*(d*x + c)^n*b^2/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 1266 vs. \(2 (185) = 370\).
Time = 0.13 (sec) , antiderivative size = 1266, normalized size of antiderivative = 6.84 \[ \int x (c+d x)^n \left (a+b x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate(x*(d*x+c)^n*(b*x^2+a)^2,x, algorithm="giac")
Output:
((d*x + c)^n*b^2*d^6*n^5*x^6 + (d*x + c)^n*b^2*c*d^5*n^5*x^5 + 15*(d*x + c )^n*b^2*d^6*n^4*x^6 + 2*(d*x + c)^n*a*b*d^6*n^5*x^4 + 10*(d*x + c)^n*b^2*c *d^5*n^4*x^5 + 85*(d*x + c)^n*b^2*d^6*n^3*x^6 + 2*(d*x + c)^n*a*b*c*d^5*n^ 5*x^3 - 5*(d*x + c)^n*b^2*c^2*d^4*n^4*x^4 + 34*(d*x + c)^n*a*b*d^6*n^4*x^4 + 35*(d*x + c)^n*b^2*c*d^5*n^3*x^5 + 225*(d*x + c)^n*b^2*d^6*n^2*x^6 + (d *x + c)^n*a^2*d^6*n^5*x^2 + 28*(d*x + c)^n*a*b*c*d^5*n^4*x^3 - 30*(d*x + c )^n*b^2*c^2*d^4*n^3*x^4 + 214*(d*x + c)^n*a*b*d^6*n^3*x^4 + 50*(d*x + c)^n *b^2*c*d^5*n^2*x^5 + 274*(d*x + c)^n*b^2*d^6*n*x^6 + (d*x + c)^n*a^2*c*d^5 *n^5*x - 6*(d*x + c)^n*a*b*c^2*d^4*n^4*x^2 + 19*(d*x + c)^n*a^2*d^6*n^4*x^ 2 + 20*(d*x + c)^n*b^2*c^3*d^3*n^3*x^3 + 130*(d*x + c)^n*a*b*c*d^5*n^3*x^3 - 55*(d*x + c)^n*b^2*c^2*d^4*n^2*x^4 + 614*(d*x + c)^n*a*b*d^6*n^2*x^4 + 24*(d*x + c)^n*b^2*c*d^5*n*x^5 + 120*(d*x + c)^n*b^2*d^6*x^6 + 18*(d*x + c )^n*a^2*c*d^5*n^4*x - 72*(d*x + c)^n*a*b*c^2*d^4*n^3*x^2 + 137*(d*x + c)^n *a^2*d^6*n^3*x^2 + 60*(d*x + c)^n*b^2*c^3*d^3*n^2*x^3 + 224*(d*x + c)^n*a* b*c*d^5*n^2*x^3 - 30*(d*x + c)^n*b^2*c^2*d^4*n*x^4 + 792*(d*x + c)^n*a*b*d ^6*n*x^4 - (d*x + c)^n*a^2*c^2*d^4*n^4 + 12*(d*x + c)^n*a*b*c^3*d^3*n^3*x + 119*(d*x + c)^n*a^2*c*d^5*n^3*x - 60*(d*x + c)^n*b^2*c^4*d^2*n^2*x^2 - 2 46*(d*x + c)^n*a*b*c^2*d^4*n^2*x^2 + 461*(d*x + c)^n*a^2*d^6*n^2*x^2 + 40* (d*x + c)^n*b^2*c^3*d^3*n*x^3 + 120*(d*x + c)^n*a*b*c*d^5*n*x^3 + 360*(d*x + c)^n*a*b*d^6*x^4 - 18*(d*x + c)^n*a^2*c^2*d^4*n^3 + 132*(d*x + c)^n*...
Time = 9.02 (sec) , antiderivative size = 723, normalized size of antiderivative = 3.91 \[ \int x (c+d x)^n \left (a+b x^2\right )^2 \, dx=\frac {b^2\,x^6\,{\left (c+d\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720}-\frac {c^2\,{\left (c+d\,x\right )}^n\,\left (a^2\,d^4\,n^4+18\,a^2\,d^4\,n^3+119\,a^2\,d^4\,n^2+342\,a^2\,d^4\,n+360\,a^2\,d^4+12\,a\,b\,c^2\,d^2\,n^2+132\,a\,b\,c^2\,d^2\,n+360\,a\,b\,c^2\,d^2+120\,b^2\,c^4\right )}{d^6\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {x^2\,\left (n+1\right )\,{\left (c+d\,x\right )}^n\,\left (a^2\,d^4\,n^4+18\,a^2\,d^4\,n^3+119\,a^2\,d^4\,n^2+342\,a^2\,d^4\,n+360\,a^2\,d^4-6\,a\,b\,c^2\,d^2\,n^3-66\,a\,b\,c^2\,d^2\,n^2-180\,a\,b\,c^2\,d^2\,n-60\,b^2\,c^4\,n\right )}{d^4\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {c\,n\,x\,{\left (c+d\,x\right )}^n\,\left (a^2\,d^4\,n^4+18\,a^2\,d^4\,n^3+119\,a^2\,d^4\,n^2+342\,a^2\,d^4\,n+360\,a^2\,d^4+12\,a\,b\,c^2\,d^2\,n^2+132\,a\,b\,c^2\,d^2\,n+360\,a\,b\,c^2\,d^2+120\,b^2\,c^4\right )}{d^5\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {b\,x^4\,{\left (c+d\,x\right )}^n\,\left (-5\,b\,c^2\,n+2\,a\,d^2\,n^2+22\,a\,d^2\,n+60\,a\,d^2\right )\,\left (n^3+6\,n^2+11\,n+6\right )}{d^2\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {b^2\,c\,n\,x^5\,{\left (c+d\,x\right )}^n\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {2\,b\,c\,n\,x^3\,{\left (c+d\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (10\,b\,c^2+a\,d^2\,n^2+11\,a\,d^2\,n+30\,a\,d^2\right )}{d^3\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )} \] Input:
int(x*(a + b*x^2)^2*(c + d*x)^n,x)
Output:
(b^2*x^6*(c + d*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) - (c^2*(c + d*x)^ n*(360*a^2*d^4 + 120*b^2*c^4 + 342*a^2*d^4*n + 119*a^2*d^4*n^2 + 18*a^2*d^ 4*n^3 + a^2*d^4*n^4 + 360*a*b*c^2*d^2 + 132*a*b*c^2*d^2*n + 12*a*b*c^2*d^2 *n^2))/(d^6*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (x^2*(n + 1)*(c + d*x)^n*(360*a^2*d^4 + 342*a^2*d^4*n - 60*b^2*c^4*n + 1 19*a^2*d^4*n^2 + 18*a^2*d^4*n^3 + a^2*d^4*n^4 - 180*a*b*c^2*d^2*n - 66*a*b *c^2*d^2*n^2 - 6*a*b*c^2*d^2*n^3))/(d^4*(1764*n + 1624*n^2 + 735*n^3 + 175 *n^4 + 21*n^5 + n^6 + 720)) + (c*n*x*(c + d*x)^n*(360*a^2*d^4 + 120*b^2*c^ 4 + 342*a^2*d^4*n + 119*a^2*d^4*n^2 + 18*a^2*d^4*n^3 + a^2*d^4*n^4 + 360*a *b*c^2*d^2 + 132*a*b*c^2*d^2*n + 12*a*b*c^2*d^2*n^2))/(d^5*(1764*n + 1624* n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (b*x^4*(c + d*x)^n*(60*a* d^2 + 2*a*d^2*n^2 + 22*a*d^2*n - 5*b*c^2*n)*(11*n + 6*n^2 + n^3 + 6))/(d^2 *(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (b^2*c*n* x^5*(c + d*x)^n*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (2*b*c*n*x^3*(c + d*x)^n*(3* n + n^2 + 2)*(30*a*d^2 + 10*b*c^2 + a*d^2*n^2 + 11*a*d^2*n))/(d^3*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720))
Time = 0.26 (sec) , antiderivative size = 840, normalized size of antiderivative = 4.54 \[ \int x (c+d x)^n \left (a+b x^2\right )^2 \, dx=\frac {\left (d x +c \right )^{n} \left (b^{2} d^{6} n^{5} x^{6}+b^{2} c \,d^{5} n^{5} x^{5}+15 b^{2} d^{6} n^{4} x^{6}+2 a b \,d^{6} n^{5} x^{4}+10 b^{2} c \,d^{5} n^{4} x^{5}+85 b^{2} d^{6} n^{3} x^{6}+2 a b c \,d^{5} n^{5} x^{3}+34 a b \,d^{6} n^{4} x^{4}-5 b^{2} c^{2} d^{4} n^{4} x^{4}+35 b^{2} c \,d^{5} n^{3} x^{5}+225 b^{2} d^{6} n^{2} x^{6}+a^{2} d^{6} n^{5} x^{2}+28 a b c \,d^{5} n^{4} x^{3}+214 a b \,d^{6} n^{3} x^{4}-30 b^{2} c^{2} d^{4} n^{3} x^{4}+50 b^{2} c \,d^{5} n^{2} x^{5}+274 b^{2} d^{6} n \,x^{6}+a^{2} c \,d^{5} n^{5} x +19 a^{2} d^{6} n^{4} x^{2}-6 a b \,c^{2} d^{4} n^{4} x^{2}+130 a b c \,d^{5} n^{3} x^{3}+614 a b \,d^{6} n^{2} x^{4}+20 b^{2} c^{3} d^{3} n^{3} x^{3}-55 b^{2} c^{2} d^{4} n^{2} x^{4}+24 b^{2} c \,d^{5} n \,x^{5}+120 b^{2} d^{6} x^{6}+18 a^{2} c \,d^{5} n^{4} x +137 a^{2} d^{6} n^{3} x^{2}-72 a b \,c^{2} d^{4} n^{3} x^{2}+224 a b c \,d^{5} n^{2} x^{3}+792 a b \,d^{6} n \,x^{4}+60 b^{2} c^{3} d^{3} n^{2} x^{3}-30 b^{2} c^{2} d^{4} n \,x^{4}-a^{2} c^{2} d^{4} n^{4}+119 a^{2} c \,d^{5} n^{3} x +461 a^{2} d^{6} n^{2} x^{2}+12 a b \,c^{3} d^{3} n^{3} x -246 a b \,c^{2} d^{4} n^{2} x^{2}+120 a b c \,d^{5} n \,x^{3}+360 a b \,d^{6} x^{4}-60 b^{2} c^{4} d^{2} n^{2} x^{2}+40 b^{2} c^{3} d^{3} n \,x^{3}-18 a^{2} c^{2} d^{4} n^{3}+342 a^{2} c \,d^{5} n^{2} x +702 a^{2} d^{6} n \,x^{2}+132 a b \,c^{3} d^{3} n^{2} x -180 a b \,c^{2} d^{4} n \,x^{2}-60 b^{2} c^{4} d^{2} n \,x^{2}-119 a^{2} c^{2} d^{4} n^{2}+360 a^{2} c \,d^{5} n x +360 a^{2} d^{6} x^{2}-12 a b \,c^{4} d^{2} n^{2}+360 a b \,c^{3} d^{3} n x +120 b^{2} c^{5} d n x -342 a^{2} c^{2} d^{4} n -132 a b \,c^{4} d^{2} n -360 a^{2} c^{2} d^{4}-360 a b \,c^{4} d^{2}-120 b^{2} c^{6}\right )}{d^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )} \] Input:
int(x*(d*x+c)^n*(b*x^2+a)^2,x)
Output:
((c + d*x)**n*( - a**2*c**2*d**4*n**4 - 18*a**2*c**2*d**4*n**3 - 119*a**2* c**2*d**4*n**2 - 342*a**2*c**2*d**4*n - 360*a**2*c**2*d**4 + a**2*c*d**5*n **5*x + 18*a**2*c*d**5*n**4*x + 119*a**2*c*d**5*n**3*x + 342*a**2*c*d**5*n **2*x + 360*a**2*c*d**5*n*x + a**2*d**6*n**5*x**2 + 19*a**2*d**6*n**4*x**2 + 137*a**2*d**6*n**3*x**2 + 461*a**2*d**6*n**2*x**2 + 702*a**2*d**6*n*x** 2 + 360*a**2*d**6*x**2 - 12*a*b*c**4*d**2*n**2 - 132*a*b*c**4*d**2*n - 360 *a*b*c**4*d**2 + 12*a*b*c**3*d**3*n**3*x + 132*a*b*c**3*d**3*n**2*x + 360* a*b*c**3*d**3*n*x - 6*a*b*c**2*d**4*n**4*x**2 - 72*a*b*c**2*d**4*n**3*x**2 - 246*a*b*c**2*d**4*n**2*x**2 - 180*a*b*c**2*d**4*n*x**2 + 2*a*b*c*d**5*n **5*x**3 + 28*a*b*c*d**5*n**4*x**3 + 130*a*b*c*d**5*n**3*x**3 + 224*a*b*c* d**5*n**2*x**3 + 120*a*b*c*d**5*n*x**3 + 2*a*b*d**6*n**5*x**4 + 34*a*b*d** 6*n**4*x**4 + 214*a*b*d**6*n**3*x**4 + 614*a*b*d**6*n**2*x**4 + 792*a*b*d* *6*n*x**4 + 360*a*b*d**6*x**4 - 120*b**2*c**6 + 120*b**2*c**5*d*n*x - 60*b **2*c**4*d**2*n**2*x**2 - 60*b**2*c**4*d**2*n*x**2 + 20*b**2*c**3*d**3*n** 3*x**3 + 60*b**2*c**3*d**3*n**2*x**3 + 40*b**2*c**3*d**3*n*x**3 - 5*b**2*c **2*d**4*n**4*x**4 - 30*b**2*c**2*d**4*n**3*x**4 - 55*b**2*c**2*d**4*n**2* x**4 - 30*b**2*c**2*d**4*n*x**4 + b**2*c*d**5*n**5*x**5 + 10*b**2*c*d**5*n **4*x**5 + 35*b**2*c*d**5*n**3*x**5 + 50*b**2*c*d**5*n**2*x**5 + 24*b**2*c *d**5*n*x**5 + b**2*d**6*n**5*x**6 + 15*b**2*d**6*n**4*x**6 + 85*b**2*d**6 *n**3*x**6 + 225*b**2*d**6*n**2*x**6 + 274*b**2*d**6*n*x**6 + 120*b**2*...