Integrand size = 18, antiderivative size = 135 \[ \int x^2 (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {c^2 \left (b c^2+a d^2\right ) (c+d x)^{1+n}}{d^5 (1+n)}-\frac {2 c \left (2 b c^2+a d^2\right ) (c+d x)^{2+n}}{d^5 (2+n)}+\frac {\left (6 b c^2+a d^2\right ) (c+d x)^{3+n}}{d^5 (3+n)}-\frac {4 b c (c+d x)^{4+n}}{d^5 (4+n)}+\frac {b (c+d x)^{5+n}}{d^5 (5+n)} \] Output:
c^2*(a*d^2+b*c^2)*(d*x+c)^(1+n)/d^5/(1+n)-2*c*(a*d^2+2*b*c^2)*(d*x+c)^(2+n )/d^5/(2+n)+(a*d^2+6*b*c^2)*(d*x+c)^(3+n)/d^5/(3+n)-4*b*c*(d*x+c)^(4+n)/d^ 5/(4+n)+b*(d*x+c)^(5+n)/d^5/(5+n)
Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.84 \[ \int x^2 (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {(c+d x)^{1+n} \left (\frac {b c^4+a c^2 d^2}{1+n}-\frac {2 c \left (2 b c^2+a d^2\right ) (c+d x)}{2+n}+\frac {\left (6 b c^2+a d^2\right ) (c+d x)^2}{3+n}-\frac {4 b c (c+d x)^3}{4+n}+\frac {b (c+d x)^4}{5+n}\right )}{d^5} \] Input:
Integrate[x^2*(c + d*x)^n*(a + b*x^2),x]
Output:
((c + d*x)^(1 + n)*((b*c^4 + a*c^2*d^2)/(1 + n) - (2*c*(2*b*c^2 + a*d^2)*( c + d*x))/(2 + n) + ((6*b*c^2 + a*d^2)*(c + d*x)^2)/(3 + n) - (4*b*c*(c + d*x)^3)/(4 + n) + (b*(c + d*x)^4)/(5 + n)))/d^5
Time = 0.29 (sec) , antiderivative size = 135, 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.
| \(\displaystyle \int x^2 \left (a+b x^2\right ) (c+d x)^n \, dx\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (-\frac {2 \left (a c d^2+2 b c^3\right ) (c+d x)^{n+1}}{d^4}+\frac {\left (a d^2+6 b c^2\right ) (c+d x)^{n+2}}{d^4}+\frac {\left (a c^2 d^2+b c^4\right ) (c+d x)^n}{d^4}-\frac {4 b c (c+d x)^{n+3}}{d^4}+\frac {b (c+d x)^{n+4}}{d^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^2 \left (a d^2+b c^2\right ) (c+d x)^{n+1}}{d^5 (n+1)}-\frac {2 c \left (a d^2+2 b c^2\right ) (c+d x)^{n+2}}{d^5 (n+2)}+\frac {\left (a d^2+6 b c^2\right ) (c+d x)^{n+3}}{d^5 (n+3)}-\frac {4 b c (c+d x)^{n+4}}{d^5 (n+4)}+\frac {b (c+d x)^{n+5}}{d^5 (n+5)}\) |
Input:
Int[x^2*(c + d*x)^n*(a + b*x^2),x]
Output:
(c^2*(b*c^2 + a*d^2)*(c + d*x)^(1 + n))/(d^5*(1 + n)) - (2*c*(2*b*c^2 + a* d^2)*(c + d*x)^(2 + n))/(d^5*(2 + n)) + ((6*b*c^2 + a*d^2)*(c + d*x)^(3 + n))/(d^5*(3 + n)) - (4*b*c*(c + d*x)^(4 + n))/(d^5*(4 + n)) + (b*(c + d*x) ^(5 + n))/(d^5*(5 + 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. \(317\) vs. \(2(135)=270\).
Time = 0.30 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.36
| method | result | size |
| norman | \(\frac {b \,x^{5} {\mathrm e}^{n \ln \left (d x +c \right )}}{5+n}+\frac {\left (a \,d^{2} n^{2}+9 a \,d^{2} n -4 b \,c^{2} n +20 a \,d^{2}\right ) x^{3} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{2} \left (n^{3}+12 n^{2}+47 n +60\right )}+\frac {n b c \,x^{4} {\mathrm e}^{n \ln \left (d x +c \right )}}{d \left (n^{2}+9 n +20\right )}+\frac {\left (a \,d^{2} n^{2}+9 a \,d^{2} n +20 a \,d^{2}+12 b \,c^{2}\right ) c n \,x^{2} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{3} \left (n^{4}+14 n^{3}+71 n^{2}+154 n +120\right )}+\frac {2 c^{3} \left (a \,d^{2} n^{2}+9 a \,d^{2} n +20 a \,d^{2}+12 b \,c^{2}\right ) {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}-\frac {2 n \,c^{2} \left (a \,d^{2} n^{2}+9 a \,d^{2} n +20 a \,d^{2}+12 b \,c^{2}\right ) x \,{\mathrm e}^{n \ln \left (d x +c \right )}}{d^{4} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(318\) |
| gosper | \(\frac {\left (d x +c \right )^{1+n} \left (b \,d^{4} n^{4} x^{4}+10 b \,d^{4} n^{3} x^{4}+a \,d^{4} n^{4} x^{2}-4 b c \,d^{3} n^{3} x^{3}+35 b \,d^{4} n^{2} x^{4}+12 a \,d^{4} n^{3} x^{2}-24 b c \,d^{3} n^{2} x^{3}+50 b \,d^{4} n \,x^{4}-2 a c \,d^{3} n^{3} x +49 a \,d^{4} n^{2} x^{2}+12 b \,c^{2} d^{2} n^{2} x^{2}-44 b c \,d^{3} n \,x^{3}+24 b \,x^{4} d^{4}-20 a c \,d^{3} n^{2} x +78 a \,d^{4} n \,x^{2}+36 b \,c^{2} d^{2} n \,x^{2}-24 c b \,d^{3} x^{3}+2 a \,c^{2} d^{2} n^{2}-58 a c \,d^{3} n x +40 a \,d^{4} x^{2}-24 b \,c^{3} d n x +24 b \,c^{2} d^{2} x^{2}+18 a \,c^{2} d^{2} n -40 a c \,d^{3} x -24 b \,c^{3} d x +40 a \,c^{2} d^{2}+24 b \,c^{4}\right )}{d^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(328\) |
| orering | \(\frac {\left (d x +c \right ) \left (b \,d^{4} n^{4} x^{4}+10 b \,d^{4} n^{3} x^{4}+a \,d^{4} n^{4} x^{2}-4 b c \,d^{3} n^{3} x^{3}+35 b \,d^{4} n^{2} x^{4}+12 a \,d^{4} n^{3} x^{2}-24 b c \,d^{3} n^{2} x^{3}+50 b \,d^{4} n \,x^{4}-2 a c \,d^{3} n^{3} x +49 a \,d^{4} n^{2} x^{2}+12 b \,c^{2} d^{2} n^{2} x^{2}-44 b c \,d^{3} n \,x^{3}+24 b \,x^{4} d^{4}-20 a c \,d^{3} n^{2} x +78 a \,d^{4} n \,x^{2}+36 b \,c^{2} d^{2} n \,x^{2}-24 c b \,d^{3} x^{3}+2 a \,c^{2} d^{2} n^{2}-58 a c \,d^{3} n x +40 a \,d^{4} x^{2}-24 b \,c^{3} d n x +24 b \,c^{2} d^{2} x^{2}+18 a \,c^{2} d^{2} n -40 a c \,d^{3} x -24 b \,c^{3} d x +40 a \,c^{2} d^{2}+24 b \,c^{4}\right ) \left (d x +c \right )^{n}}{d^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(331\) |
| risch | \(\frac {\left (b \,d^{5} n^{4} x^{5}+b c \,d^{4} n^{4} x^{4}+10 b \,d^{5} n^{3} x^{5}+a \,d^{5} n^{4} x^{3}+6 b c \,d^{4} n^{3} x^{4}+35 b \,d^{5} n^{2} x^{5}+a c \,d^{4} n^{4} x^{2}+12 a \,d^{5} n^{3} x^{3}-4 b \,c^{2} d^{3} n^{3} x^{3}+11 b c \,d^{4} n^{2} x^{4}+50 b \,d^{5} n \,x^{5}+10 a c \,d^{4} n^{3} x^{2}+49 a \,d^{5} n^{2} x^{3}-12 b \,c^{2} d^{3} n^{2} x^{3}+6 b c \,d^{4} n \,x^{4}+24 b \,d^{5} x^{5}-2 a \,c^{2} d^{3} n^{3} x +29 a c \,d^{4} n^{2} x^{2}+78 a \,d^{5} n \,x^{3}+12 b \,c^{3} d^{2} n^{2} x^{2}-8 b \,c^{2} d^{3} n \,x^{3}-18 a \,c^{2} d^{3} n^{2} x +20 a c \,d^{4} n \,x^{2}+40 a \,d^{5} x^{3}+12 b \,c^{3} d^{2} n \,x^{2}+2 a \,c^{3} d^{2} n^{2}-40 a \,c^{2} d^{3} n x -24 b \,c^{4} d n x +18 a \,c^{3} d^{2} n +40 a \,c^{3} d^{2}+24 b \,c^{5}\right ) \left (d x +c \right )^{n}}{\left (4+n \right ) \left (5+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) d^{5}}\) | \(398\) |
| parallelrisch | \(\frac {6 x^{4} \left (d x +c \right )^{n} b c \,d^{4} n -12 x^{3} \left (d x +c \right )^{n} b \,c^{2} d^{3} n^{2}+x^{4} \left (d x +c \right )^{n} b c \,d^{4} n^{4}+6 x^{4} \left (d x +c \right )^{n} b c \,d^{4} n^{3}+11 x^{4} \left (d x +c \right )^{n} b c \,d^{4} n^{2}-4 x^{3} \left (d x +c \right )^{n} b \,c^{2} d^{3} n^{3}+x^{2} \left (d x +c \right )^{n} a c \,d^{4} n^{4}-8 x^{3} \left (d x +c \right )^{n} b \,c^{2} d^{3} n +29 x^{2} \left (d x +c \right )^{n} a c \,d^{4} n^{2}+12 x^{2} \left (d x +c \right )^{n} b \,c^{3} d^{2} n^{2}-2 x \left (d x +c \right )^{n} a \,c^{2} d^{3} n^{3}+20 x^{2} \left (d x +c \right )^{n} a c \,d^{4} n +12 x^{2} \left (d x +c \right )^{n} b \,c^{3} d^{2} n -18 x \left (d x +c \right )^{n} a \,c^{2} d^{3} n^{2}-40 x \left (d x +c \right )^{n} a \,c^{2} d^{3} n -24 x \left (d x +c \right )^{n} b \,c^{4} d n +18 \left (d x +c \right )^{n} a \,c^{3} d^{2} n +10 x^{2} \left (d x +c \right )^{n} a c \,d^{4} n^{3}+50 x^{5} \left (d x +c \right )^{n} b \,d^{5} n +12 x^{3} \left (d x +c \right )^{n} a \,d^{5} n^{3}+49 x^{3} \left (d x +c \right )^{n} a \,d^{5} n^{2}+78 x^{3} \left (d x +c \right )^{n} a \,d^{5} n +2 \left (d x +c \right )^{n} a \,c^{3} d^{2} n^{2}+x^{5} \left (d x +c \right )^{n} b \,d^{5} n^{4}+10 x^{5} \left (d x +c \right )^{n} b \,d^{5} n^{3}+35 x^{5} \left (d x +c \right )^{n} b \,d^{5} n^{2}+x^{3} \left (d x +c \right )^{n} a \,d^{5} n^{4}+24 x^{5} \left (d x +c \right )^{n} b \,d^{5}+40 x^{3} \left (d x +c \right )^{n} a \,d^{5}+40 \left (d x +c \right )^{n} a \,c^{3} d^{2}+24 \left (d x +c \right )^{n} b \,c^{5}}{d^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(608\) |
Input:
int(x^2*(d*x+c)^n*(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
b/(5+n)*x^5*exp(n*ln(d*x+c))+(a*d^2*n^2+9*a*d^2*n-4*b*c^2*n+20*a*d^2)/d^2/ (n^3+12*n^2+47*n+60)*x^3*exp(n*ln(d*x+c))+n*b*c/d/(n^2+9*n+20)*x^4*exp(n*l n(d*x+c))+(a*d^2*n^2+9*a*d^2*n+20*a*d^2+12*b*c^2)*c/d^3*n/(n^4+14*n^3+71*n ^2+154*n+120)*x^2*exp(n*ln(d*x+c))+2*c^3*(a*d^2*n^2+9*a*d^2*n+20*a*d^2+12* b*c^2)/d^5/(n^5+15*n^4+85*n^3+225*n^2+274*n+120)*exp(n*ln(d*x+c))-2/d^4*n* c^2*(a*d^2*n^2+9*a*d^2*n+20*a*d^2+12*b*c^2)/(n^5+15*n^4+85*n^3+225*n^2+274 *n+120)*x*exp(n*ln(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (135) = 270\).
Time = 0.09 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.73 \[ \int x^2 (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {{\left (2 \, a c^{3} d^{2} n^{2} + 18 \, a c^{3} d^{2} n + 24 \, b c^{5} + 40 \, a c^{3} d^{2} + {\left (b d^{5} n^{4} + 10 \, b d^{5} n^{3} + 35 \, b d^{5} n^{2} + 50 \, b d^{5} n + 24 \, b d^{5}\right )} x^{5} + {\left (b c d^{4} n^{4} + 6 \, b c d^{4} n^{3} + 11 \, b c d^{4} n^{2} + 6 \, b c d^{4} n\right )} x^{4} + {\left (a d^{5} n^{4} + 40 \, a d^{5} - 4 \, {\left (b c^{2} d^{3} - 3 \, a d^{5}\right )} n^{3} - {\left (12 \, b c^{2} d^{3} - 49 \, a d^{5}\right )} n^{2} - 2 \, {\left (4 \, b c^{2} d^{3} - 39 \, a d^{5}\right )} n\right )} x^{3} + {\left (a c d^{4} n^{4} + 10 \, a c d^{4} n^{3} + {\left (12 \, b c^{3} d^{2} + 29 \, a c d^{4}\right )} n^{2} + 4 \, {\left (3 \, b c^{3} d^{2} + 5 \, a c d^{4}\right )} n\right )} x^{2} - 2 \, {\left (a c^{2} d^{3} n^{3} + 9 \, a c^{2} d^{3} n^{2} + 4 \, {\left (3 \, b c^{4} d + 5 \, a c^{2} d^{3}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n}}{d^{5} n^{5} + 15 \, d^{5} n^{4} + 85 \, d^{5} n^{3} + 225 \, d^{5} n^{2} + 274 \, d^{5} n + 120 \, d^{5}} \] Input:
integrate(x^2*(d*x+c)^n*(b*x^2+a),x, algorithm="fricas")
Output:
(2*a*c^3*d^2*n^2 + 18*a*c^3*d^2*n + 24*b*c^5 + 40*a*c^3*d^2 + (b*d^5*n^4 + 10*b*d^5*n^3 + 35*b*d^5*n^2 + 50*b*d^5*n + 24*b*d^5)*x^5 + (b*c*d^4*n^4 + 6*b*c*d^4*n^3 + 11*b*c*d^4*n^2 + 6*b*c*d^4*n)*x^4 + (a*d^5*n^4 + 40*a*d^5 - 4*(b*c^2*d^3 - 3*a*d^5)*n^3 - (12*b*c^2*d^3 - 49*a*d^5)*n^2 - 2*(4*b*c^ 2*d^3 - 39*a*d^5)*n)*x^3 + (a*c*d^4*n^4 + 10*a*c*d^4*n^3 + (12*b*c^3*d^2 + 29*a*c*d^4)*n^2 + 4*(3*b*c^3*d^2 + 5*a*c*d^4)*n)*x^2 - 2*(a*c^2*d^3*n^3 + 9*a*c^2*d^3*n^2 + 4*(3*b*c^4*d + 5*a*c^2*d^3)*n)*x)*(d*x + c)^n/(d^5*n^5 + 15*d^5*n^4 + 85*d^5*n^3 + 225*d^5*n^2 + 274*d^5*n + 120*d^5)
Leaf count of result is larger than twice the leaf count of optimal. 4134 vs. \(2 (122) = 244\).
Time = 1.23 (sec) , antiderivative size = 4134, normalized size of antiderivative = 30.62 \[ \int x^2 (c+d x)^n \left (a+b x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate(x**2*(d*x+c)**n*(b*x**2+a),x)
Output:
Piecewise((c**n*(a*x**3/3 + b*x**5/5), Eq(d, 0)), (-a*c**2*d**2/(12*c**4*d **5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4) - 4*a*c*d**3*x/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d **8*x**3 + 12*d**9*x**4) - 6*a*d**4*x**2/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4) + 12*b*c**4*log(c/d + x )/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12 *d**9*x**4) + 25*b*c**4/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4) + 48*b*c**3*d*x*log(c/d + x)/(12*c**4*d* *5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4) + 88*b*c**3*d*x/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d **8*x**3 + 12*d**9*x**4) + 72*b*c**2*d**2*x**2*log(c/d + x)/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4) + 10 8*b*c**2*d**2*x**2/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48 *c*d**8*x**3 + 12*d**9*x**4) + 48*b*c*d**3*x**3*log(c/d + x)/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4) + 4 8*b*c*d**3*x**3/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c* d**8*x**3 + 12*d**9*x**4) + 12*b*d**4*x**4*log(c/d + x)/(12*c**4*d**5 + 48 *c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4), Eq(n, - 5)), (-a*c**2*d**2/(3*c**3*d**5 + 9*c**2*d**6*x + 9*c*d**7*x**2 + 3*d**8*x **3) - 3*a*c*d**3*x/(3*c**3*d**5 + 9*c**2*d**6*x + 9*c*d**7*x**2 + 3*d*...
Time = 0.05 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.56 \[ \int x^2 (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} x^{3} + {\left (n^{2} + n\right )} c d^{2} x^{2} - 2 \, c^{2} d n x + 2 \, c^{3}\right )} {\left (d x + c\right )}^{n} a}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} c d^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c^{2} d^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} c^{3} d^{2} x^{2} - 24 \, c^{4} d n x + 24 \, c^{5}\right )} {\left (d x + c\right )}^{n} b}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} d^{5}} \] Input:
integrate(x^2*(d*x+c)^n*(b*x^2+a),x, algorithm="maxima")
Output:
((n^2 + 3*n + 2)*d^3*x^3 + (n^2 + n)*c*d^2*x^2 - 2*c^2*d*n*x + 2*c^3)*(d*x + c)^n*a/((n^3 + 6*n^2 + 11*n + 6)*d^3) + ((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*c*d^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*c^2*d^3*x^3 + 12*(n^2 + n)*c^3*d^2*x^2 - 24*c^4*d*n*x + 24*c^5)*(d*x + c)^n*b/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*d^5)
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (135) = 270\).
Time = 0.13 (sec) , antiderivative size = 624, normalized size of antiderivative = 4.62 \[ \int x^2 (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {{\left (d x + c\right )}^{n} b d^{5} n^{4} x^{5} + {\left (d x + c\right )}^{n} b c d^{4} n^{4} x^{4} + 10 \, {\left (d x + c\right )}^{n} b d^{5} n^{3} x^{5} + {\left (d x + c\right )}^{n} a d^{5} n^{4} x^{3} + 6 \, {\left (d x + c\right )}^{n} b c d^{4} n^{3} x^{4} + 35 \, {\left (d x + c\right )}^{n} b d^{5} n^{2} x^{5} + {\left (d x + c\right )}^{n} a c d^{4} n^{4} x^{2} - 4 \, {\left (d x + c\right )}^{n} b c^{2} d^{3} n^{3} x^{3} + 12 \, {\left (d x + c\right )}^{n} a d^{5} n^{3} x^{3} + 11 \, {\left (d x + c\right )}^{n} b c d^{4} n^{2} x^{4} + 50 \, {\left (d x + c\right )}^{n} b d^{5} n x^{5} + 10 \, {\left (d x + c\right )}^{n} a c d^{4} n^{3} x^{2} - 12 \, {\left (d x + c\right )}^{n} b c^{2} d^{3} n^{2} x^{3} + 49 \, {\left (d x + c\right )}^{n} a d^{5} n^{2} x^{3} + 6 \, {\left (d x + c\right )}^{n} b c d^{4} n x^{4} + 24 \, {\left (d x + c\right )}^{n} b d^{5} x^{5} - 2 \, {\left (d x + c\right )}^{n} a c^{2} d^{3} n^{3} x + 12 \, {\left (d x + c\right )}^{n} b c^{3} d^{2} n^{2} x^{2} + 29 \, {\left (d x + c\right )}^{n} a c d^{4} n^{2} x^{2} - 8 \, {\left (d x + c\right )}^{n} b c^{2} d^{3} n x^{3} + 78 \, {\left (d x + c\right )}^{n} a d^{5} n x^{3} - 18 \, {\left (d x + c\right )}^{n} a c^{2} d^{3} n^{2} x + 12 \, {\left (d x + c\right )}^{n} b c^{3} d^{2} n x^{2} + 20 \, {\left (d x + c\right )}^{n} a c d^{4} n x^{2} + 40 \, {\left (d x + c\right )}^{n} a d^{5} x^{3} + 2 \, {\left (d x + c\right )}^{n} a c^{3} d^{2} n^{2} - 24 \, {\left (d x + c\right )}^{n} b c^{4} d n x - 40 \, {\left (d x + c\right )}^{n} a c^{2} d^{3} n x + 18 \, {\left (d x + c\right )}^{n} a c^{3} d^{2} n + 24 \, {\left (d x + c\right )}^{n} b c^{5} + 40 \, {\left (d x + c\right )}^{n} a c^{3} d^{2}}{d^{5} n^{5} + 15 \, d^{5} n^{4} + 85 \, d^{5} n^{3} + 225 \, d^{5} n^{2} + 274 \, d^{5} n + 120 \, d^{5}} \] Input:
integrate(x^2*(d*x+c)^n*(b*x^2+a),x, algorithm="giac")
Output:
((d*x + c)^n*b*d^5*n^4*x^5 + (d*x + c)^n*b*c*d^4*n^4*x^4 + 10*(d*x + c)^n* b*d^5*n^3*x^5 + (d*x + c)^n*a*d^5*n^4*x^3 + 6*(d*x + c)^n*b*c*d^4*n^3*x^4 + 35*(d*x + c)^n*b*d^5*n^2*x^5 + (d*x + c)^n*a*c*d^4*n^4*x^2 - 4*(d*x + c) ^n*b*c^2*d^3*n^3*x^3 + 12*(d*x + c)^n*a*d^5*n^3*x^3 + 11*(d*x + c)^n*b*c*d ^4*n^2*x^4 + 50*(d*x + c)^n*b*d^5*n*x^5 + 10*(d*x + c)^n*a*c*d^4*n^3*x^2 - 12*(d*x + c)^n*b*c^2*d^3*n^2*x^3 + 49*(d*x + c)^n*a*d^5*n^2*x^3 + 6*(d*x + c)^n*b*c*d^4*n*x^4 + 24*(d*x + c)^n*b*d^5*x^5 - 2*(d*x + c)^n*a*c^2*d^3* n^3*x + 12*(d*x + c)^n*b*c^3*d^2*n^2*x^2 + 29*(d*x + c)^n*a*c*d^4*n^2*x^2 - 8*(d*x + c)^n*b*c^2*d^3*n*x^3 + 78*(d*x + c)^n*a*d^5*n*x^3 - 18*(d*x + c )^n*a*c^2*d^3*n^2*x + 12*(d*x + c)^n*b*c^3*d^2*n*x^2 + 20*(d*x + c)^n*a*c* d^4*n*x^2 + 40*(d*x + c)^n*a*d^5*x^3 + 2*(d*x + c)^n*a*c^3*d^2*n^2 - 24*(d *x + c)^n*b*c^4*d*n*x - 40*(d*x + c)^n*a*c^2*d^3*n*x + 18*(d*x + c)^n*a*c^ 3*d^2*n + 24*(d*x + c)^n*b*c^5 + 40*(d*x + c)^n*a*c^3*d^2)/(d^5*n^5 + 15*d ^5*n^4 + 85*d^5*n^3 + 225*d^5*n^2 + 274*d^5*n + 120*d^5)
Time = 8.60 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.69 \[ \int x^2 (c+d x)^n \left (a+b x^2\right ) \, dx={\left (c+d\,x\right )}^n\,\left (\frac {2\,c^3\,\left (12\,b\,c^2+a\,d^2\,n^2+9\,a\,d^2\,n+20\,a\,d^2\right )}{d^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {b\,x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}+\frac {x^3\,\left (n^2+3\,n+2\right )\,\left (-4\,b\,c^2\,n+a\,d^2\,n^2+9\,a\,d^2\,n+20\,a\,d^2\right )}{d^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}-\frac {2\,c^2\,n\,x\,\left (12\,b\,c^2+a\,d^2\,n^2+9\,a\,d^2\,n+20\,a\,d^2\right )}{d^4\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {c\,n\,x^2\,\left (n+1\right )\,\left (12\,b\,c^2+a\,d^2\,n^2+9\,a\,d^2\,n+20\,a\,d^2\right )}{d^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {b\,c\,n\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{d\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}\right ) \] Input:
int(x^2*(a + b*x^2)*(c + d*x)^n,x)
Output:
(c + d*x)^n*((2*c^3*(20*a*d^2 + 12*b*c^2 + a*d^2*n^2 + 9*a*d^2*n))/(d^5*(2 74*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (b*x^5*(50*n + 35*n^2 + 1 0*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*a*d^2 + a*d^2*n^2 + 9*a*d^2*n - 4*b*c^2*n))/(d^2*(274* n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) - (2*c^2*n*x*(20*a*d^2 + 12*b* c^2 + a*d^2*n^2 + 9*a*d^2*n))/(d^4*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^ 5 + 120)) + (c*n*x^2*(n + 1)*(20*a*d^2 + 12*b*c^2 + a*d^2*n^2 + 9*a*d^2*n) )/(d^3*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (b*c*n*x^4*(11*n + 6*n^2 + n^3 + 6))/(d*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)))
Time = 0.22 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.94 \[ \int x^2 (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {\left (d x +c \right )^{n} \left (b \,d^{5} n^{4} x^{5}+b c \,d^{4} n^{4} x^{4}+10 b \,d^{5} n^{3} x^{5}+a \,d^{5} n^{4} x^{3}+6 b c \,d^{4} n^{3} x^{4}+35 b \,d^{5} n^{2} x^{5}+a c \,d^{4} n^{4} x^{2}+12 a \,d^{5} n^{3} x^{3}-4 b \,c^{2} d^{3} n^{3} x^{3}+11 b c \,d^{4} n^{2} x^{4}+50 b \,d^{5} n \,x^{5}+10 a c \,d^{4} n^{3} x^{2}+49 a \,d^{5} n^{2} x^{3}-12 b \,c^{2} d^{3} n^{2} x^{3}+6 b c \,d^{4} n \,x^{4}+24 b \,d^{5} x^{5}-2 a \,c^{2} d^{3} n^{3} x +29 a c \,d^{4} n^{2} x^{2}+78 a \,d^{5} n \,x^{3}+12 b \,c^{3} d^{2} n^{2} x^{2}-8 b \,c^{2} d^{3} n \,x^{3}-18 a \,c^{2} d^{3} n^{2} x +20 a c \,d^{4} n \,x^{2}+40 a \,d^{5} x^{3}+12 b \,c^{3} d^{2} n \,x^{2}+2 a \,c^{3} d^{2} n^{2}-40 a \,c^{2} d^{3} n x -24 b \,c^{4} d n x +18 a \,c^{3} d^{2} n +40 a \,c^{3} d^{2}+24 b \,c^{5}\right )}{d^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )} \] Input:
int(x^2*(d*x+c)^n*(b*x^2+a),x)
Output:
((c + d*x)**n*(2*a*c**3*d**2*n**2 + 18*a*c**3*d**2*n + 40*a*c**3*d**2 - 2* a*c**2*d**3*n**3*x - 18*a*c**2*d**3*n**2*x - 40*a*c**2*d**3*n*x + a*c*d**4 *n**4*x**2 + 10*a*c*d**4*n**3*x**2 + 29*a*c*d**4*n**2*x**2 + 20*a*c*d**4*n *x**2 + a*d**5*n**4*x**3 + 12*a*d**5*n**3*x**3 + 49*a*d**5*n**2*x**3 + 78* a*d**5*n*x**3 + 40*a*d**5*x**3 + 24*b*c**5 - 24*b*c**4*d*n*x + 12*b*c**3*d **2*n**2*x**2 + 12*b*c**3*d**2*n*x**2 - 4*b*c**2*d**3*n**3*x**3 - 12*b*c** 2*d**3*n**2*x**3 - 8*b*c**2*d**3*n*x**3 + b*c*d**4*n**4*x**4 + 6*b*c*d**4* n**3*x**4 + 11*b*c*d**4*n**2*x**4 + 6*b*c*d**4*n*x**4 + b*d**5*n**4*x**5 + 10*b*d**5*n**3*x**5 + 35*b*d**5*n**2*x**5 + 50*b*d**5*n*x**5 + 24*b*d**5* x**5))/(d**5*(n**5 + 15*n**4 + 85*n**3 + 225*n**2 + 274*n + 120))