Integrand size = 20, antiderivative size = 138 \[ \int x^2 (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {1}{3} a^3 c^2 x^3+\frac {1}{2} a^3 c d x^4+\frac {1}{5} a^2 \left (3 b c^2+a d^2\right ) x^5+a^2 b c d x^6+\frac {3}{7} a b \left (b c^2+a d^2\right ) x^7+\frac {3}{4} a b^2 c d x^8+\frac {1}{9} b^2 \left (b c^2+3 a d^2\right ) x^9+\frac {1}{5} b^3 c d x^{10}+\frac {1}{11} b^3 d^2 x^{11} \] Output:
1/3*a^3*c^2*x^3+1/2*a^3*c*d*x^4+1/5*a^2*(a*d^2+3*b*c^2)*x^5+a^2*b*c*d*x^6+ 3/7*a*b*(a*d^2+b*c^2)*x^7+3/4*a*b^2*c*d*x^8+1/9*b^2*(3*a*d^2+b*c^2)*x^9+1/ 5*b^3*c*d*x^10+1/11*b^3*d^2*x^11
Time = 0.02 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.88 \[ \int x^2 (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {1}{30} a^3 x^3 \left (10 c^2+15 c d x+6 d^2 x^2\right )+\frac {1}{84} a b^2 x^7 \left (36 c^2+63 c d x+28 d^2 x^2\right )+\frac {1}{495} b^3 x^9 \left (55 c^2+99 c d x+45 d^2 x^2\right )+a^2 b \left (\frac {3 c^2 x^5}{5}+c d x^6+\frac {3 d^2 x^7}{7}\right ) \] Input:
Integrate[x^2*(c + d*x)^2*(a + b*x^2)^3,x]
Output:
(a^3*x^3*(10*c^2 + 15*c*d*x + 6*d^2*x^2))/30 + (a*b^2*x^7*(36*c^2 + 63*c*d *x + 28*d^2*x^2))/84 + (b^3*x^9*(55*c^2 + 99*c*d*x + 45*d^2*x^2))/495 + a^ 2*b*((3*c^2*x^5)/5 + c*d*x^6 + (3*d^2*x^7)/7)
Time = 0.52 (sec) , antiderivative size = 138, 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.
\(\displaystyle \int x^2 \left (a+b x^2\right )^3 (c+d x)^2 \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (a^3 c^2 x^2+2 a^3 c d x^3+a^2 x^4 \left (a d^2+3 b c^2\right )+6 a^2 b c d x^5+b^2 x^8 \left (3 a d^2+b c^2\right )+6 a b^2 c d x^7+3 a b x^6 \left (a d^2+b c^2\right )+2 b^3 c d x^9+b^3 d^2 x^{10}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} a^3 c^2 x^3+\frac {1}{2} a^3 c d x^4+\frac {1}{5} a^2 x^5 \left (a d^2+3 b c^2\right )+a^2 b c d x^6+\frac {1}{9} b^2 x^9 \left (3 a d^2+b c^2\right )+\frac {3}{4} a b^2 c d x^8+\frac {3}{7} a b x^7 \left (a d^2+b c^2\right )+\frac {1}{5} b^3 c d x^{10}+\frac {1}{11} b^3 d^2 x^{11}\) |
Input:
Int[x^2*(c + d*x)^2*(a + b*x^2)^3,x]
Output:
(a^3*c^2*x^3)/3 + (a^3*c*d*x^4)/2 + (a^2*(3*b*c^2 + a*d^2)*x^5)/5 + a^2*b* c*d*x^6 + (3*a*b*(b*c^2 + a*d^2)*x^7)/7 + (3*a*b^2*c*d*x^8)/4 + (b^2*(b*c^ 2 + 3*a*d^2)*x^9)/9 + (b^3*c*d*x^10)/5 + (b^3*d^2*x^11)/11
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]
Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {b^{3} d^{2} x^{11}}{11}+\frac {b^{3} c d \,x^{10}}{5}+\left (\frac {1}{3} a \,b^{2} d^{2}+\frac {1}{9} b^{3} c^{2}\right ) x^{9}+\frac {3 a \,b^{2} c d \,x^{8}}{4}+\left (\frac {3}{7} a^{2} b \,d^{2}+\frac {3}{7} a \,c^{2} b^{2}\right ) x^{7}+a^{2} b c d \,x^{6}+\left (\frac {1}{5} a^{3} d^{2}+\frac {3}{5} a^{2} b \,c^{2}\right ) x^{5}+\frac {a^{3} c d \,x^{4}}{2}+\frac {a^{3} c^{2} x^{3}}{3}\) | \(132\) |
default | \(\frac {b^{3} d^{2} x^{11}}{11}+\frac {b^{3} c d \,x^{10}}{5}+\frac {\left (3 a \,b^{2} d^{2}+b^{3} c^{2}\right ) x^{9}}{9}+\frac {3 a \,b^{2} c d \,x^{8}}{4}+\frac {\left (3 a^{2} b \,d^{2}+3 a \,c^{2} b^{2}\right ) x^{7}}{7}+a^{2} b c d \,x^{6}+\frac {\left (a^{3} d^{2}+3 a^{2} b \,c^{2}\right ) x^{5}}{5}+\frac {a^{3} c d \,x^{4}}{2}+\frac {a^{3} c^{2} x^{3}}{3}\) | \(133\) |
gosper | \(\frac {1}{11} b^{3} d^{2} x^{11}+\frac {1}{5} b^{3} c d \,x^{10}+\frac {1}{3} x^{9} a \,b^{2} d^{2}+\frac {1}{9} x^{9} b^{3} c^{2}+\frac {3}{4} a \,b^{2} c d \,x^{8}+\frac {3}{7} x^{7} a^{2} b \,d^{2}+\frac {3}{7} x^{7} a \,c^{2} b^{2}+a^{2} b c d \,x^{6}+\frac {1}{5} x^{5} a^{3} d^{2}+\frac {3}{5} x^{5} a^{2} b \,c^{2}+\frac {1}{2} a^{3} c d \,x^{4}+\frac {1}{3} a^{3} c^{2} x^{3}\) | \(135\) |
risch | \(\frac {1}{11} b^{3} d^{2} x^{11}+\frac {1}{5} b^{3} c d \,x^{10}+\frac {1}{3} x^{9} a \,b^{2} d^{2}+\frac {1}{9} x^{9} b^{3} c^{2}+\frac {3}{4} a \,b^{2} c d \,x^{8}+\frac {3}{7} x^{7} a^{2} b \,d^{2}+\frac {3}{7} x^{7} a \,c^{2} b^{2}+a^{2} b c d \,x^{6}+\frac {1}{5} x^{5} a^{3} d^{2}+\frac {3}{5} x^{5} a^{2} b \,c^{2}+\frac {1}{2} a^{3} c d \,x^{4}+\frac {1}{3} a^{3} c^{2} x^{3}\) | \(135\) |
parallelrisch | \(\frac {1}{11} b^{3} d^{2} x^{11}+\frac {1}{5} b^{3} c d \,x^{10}+\frac {1}{3} x^{9} a \,b^{2} d^{2}+\frac {1}{9} x^{9} b^{3} c^{2}+\frac {3}{4} a \,b^{2} c d \,x^{8}+\frac {3}{7} x^{7} a^{2} b \,d^{2}+\frac {3}{7} x^{7} a \,c^{2} b^{2}+a^{2} b c d \,x^{6}+\frac {1}{5} x^{5} a^{3} d^{2}+\frac {3}{5} x^{5} a^{2} b \,c^{2}+\frac {1}{2} a^{3} c d \,x^{4}+\frac {1}{3} a^{3} c^{2} x^{3}\) | \(135\) |
orering | \(\frac {x^{3} \left (1260 b^{3} d^{2} x^{8}+2772 d \,x^{7} c \,b^{3}+4620 a \,b^{2} d^{2} x^{6}+1540 b^{3} c^{2} x^{6}+10395 a \,b^{2} c d \,x^{5}+5940 a^{2} b \,d^{2} x^{4}+5940 a \,b^{2} c^{2} x^{4}+13860 a^{2} b c d \,x^{3}+2772 a^{3} d^{2} x^{2}+8316 a^{2} b \,c^{2} x^{2}+6930 a^{3} c d x +4620 c^{2} a^{3}\right )}{13860}\) | \(136\) |
Input:
int(x^2*(d*x+c)^2*(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
1/11*b^3*d^2*x^11+1/5*b^3*c*d*x^10+(1/3*a*b^2*d^2+1/9*b^3*c^2)*x^9+3/4*a*b ^2*c*d*x^8+(3/7*a^2*b*d^2+3/7*a*c^2*b^2)*x^7+a^2*b*c*d*x^6+(1/5*a^3*d^2+3/ 5*a^2*b*c^2)*x^5+1/2*a^3*c*d*x^4+1/3*a^3*c^2*x^3
Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.94 \[ \int x^2 (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {1}{11} \, b^{3} d^{2} x^{11} + \frac {1}{5} \, b^{3} c d x^{10} + \frac {3}{4} \, a b^{2} c d x^{8} + a^{2} b c d x^{6} + \frac {1}{9} \, {\left (b^{3} c^{2} + 3 \, a b^{2} d^{2}\right )} x^{9} + \frac {1}{2} \, a^{3} c d x^{4} + \frac {1}{3} \, a^{3} c^{2} x^{3} + \frac {3}{7} \, {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )} x^{7} + \frac {1}{5} \, {\left (3 \, a^{2} b c^{2} + a^{3} d^{2}\right )} x^{5} \] Input:
integrate(x^2*(d*x+c)^2*(b*x^2+a)^3,x, algorithm="fricas")
Output:
1/11*b^3*d^2*x^11 + 1/5*b^3*c*d*x^10 + 3/4*a*b^2*c*d*x^8 + a^2*b*c*d*x^6 + 1/9*(b^3*c^2 + 3*a*b^2*d^2)*x^9 + 1/2*a^3*c*d*x^4 + 1/3*a^3*c^2*x^3 + 3/7 *(a*b^2*c^2 + a^2*b*d^2)*x^7 + 1/5*(3*a^2*b*c^2 + a^3*d^2)*x^5
Time = 0.03 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.06 \[ \int x^2 (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {a^{3} c^{2} x^{3}}{3} + \frac {a^{3} c d x^{4}}{2} + a^{2} b c d x^{6} + \frac {3 a b^{2} c d x^{8}}{4} + \frac {b^{3} c d x^{10}}{5} + \frac {b^{3} d^{2} x^{11}}{11} + x^{9} \left (\frac {a b^{2} d^{2}}{3} + \frac {b^{3} c^{2}}{9}\right ) + x^{7} \cdot \left (\frac {3 a^{2} b d^{2}}{7} + \frac {3 a b^{2} c^{2}}{7}\right ) + x^{5} \left (\frac {a^{3} d^{2}}{5} + \frac {3 a^{2} b c^{2}}{5}\right ) \] Input:
integrate(x**2*(d*x+c)**2*(b*x**2+a)**3,x)
Output:
a**3*c**2*x**3/3 + a**3*c*d*x**4/2 + a**2*b*c*d*x**6 + 3*a*b**2*c*d*x**8/4 + b**3*c*d*x**10/5 + b**3*d**2*x**11/11 + x**9*(a*b**2*d**2/3 + b**3*c**2 /9) + x**7*(3*a**2*b*d**2/7 + 3*a*b**2*c**2/7) + x**5*(a**3*d**2/5 + 3*a** 2*b*c**2/5)
Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.94 \[ \int x^2 (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {1}{11} \, b^{3} d^{2} x^{11} + \frac {1}{5} \, b^{3} c d x^{10} + \frac {3}{4} \, a b^{2} c d x^{8} + a^{2} b c d x^{6} + \frac {1}{9} \, {\left (b^{3} c^{2} + 3 \, a b^{2} d^{2}\right )} x^{9} + \frac {1}{2} \, a^{3} c d x^{4} + \frac {1}{3} \, a^{3} c^{2} x^{3} + \frac {3}{7} \, {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )} x^{7} + \frac {1}{5} \, {\left (3 \, a^{2} b c^{2} + a^{3} d^{2}\right )} x^{5} \] Input:
integrate(x^2*(d*x+c)^2*(b*x^2+a)^3,x, algorithm="maxima")
Output:
1/11*b^3*d^2*x^11 + 1/5*b^3*c*d*x^10 + 3/4*a*b^2*c*d*x^8 + a^2*b*c*d*x^6 + 1/9*(b^3*c^2 + 3*a*b^2*d^2)*x^9 + 1/2*a^3*c*d*x^4 + 1/3*a^3*c^2*x^3 + 3/7 *(a*b^2*c^2 + a^2*b*d^2)*x^7 + 1/5*(3*a^2*b*c^2 + a^3*d^2)*x^5
Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.97 \[ \int x^2 (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {1}{11} \, b^{3} d^{2} x^{11} + \frac {1}{5} \, b^{3} c d x^{10} + \frac {1}{9} \, b^{3} c^{2} x^{9} + \frac {1}{3} \, a b^{2} d^{2} x^{9} + \frac {3}{4} \, a b^{2} c d x^{8} + \frac {3}{7} \, a b^{2} c^{2} x^{7} + \frac {3}{7} \, a^{2} b d^{2} x^{7} + a^{2} b c d x^{6} + \frac {3}{5} \, a^{2} b c^{2} x^{5} + \frac {1}{5} \, a^{3} d^{2} x^{5} + \frac {1}{2} \, a^{3} c d x^{4} + \frac {1}{3} \, a^{3} c^{2} x^{3} \] Input:
integrate(x^2*(d*x+c)^2*(b*x^2+a)^3,x, algorithm="giac")
Output:
1/11*b^3*d^2*x^11 + 1/5*b^3*c*d*x^10 + 1/9*b^3*c^2*x^9 + 1/3*a*b^2*d^2*x^9 + 3/4*a*b^2*c*d*x^8 + 3/7*a*b^2*c^2*x^7 + 3/7*a^2*b*d^2*x^7 + a^2*b*c*d*x ^6 + 3/5*a^2*b*c^2*x^5 + 1/5*a^3*d^2*x^5 + 1/2*a^3*c*d*x^4 + 1/3*a^3*c^2*x ^3
Time = 0.05 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91 \[ \int x^2 (c+d x)^2 \left (a+b x^2\right )^3 \, dx=x^5\,\left (\frac {a^3\,d^2}{5}+\frac {3\,b\,a^2\,c^2}{5}\right )+x^9\,\left (\frac {b^3\,c^2}{9}+\frac {a\,b^2\,d^2}{3}\right )+\frac {a^3\,c^2\,x^3}{3}+\frac {b^3\,d^2\,x^{11}}{11}+\frac {3\,a\,b\,x^7\,\left (b\,c^2+a\,d^2\right )}{7}+\frac {a^3\,c\,d\,x^4}{2}+\frac {b^3\,c\,d\,x^{10}}{5}+a^2\,b\,c\,d\,x^6+\frac {3\,a\,b^2\,c\,d\,x^8}{4} \] Input:
int(x^2*(a + b*x^2)^3*(c + d*x)^2,x)
Output:
x^5*((a^3*d^2)/5 + (3*a^2*b*c^2)/5) + x^9*((b^3*c^2)/9 + (a*b^2*d^2)/3) + (a^3*c^2*x^3)/3 + (b^3*d^2*x^11)/11 + (3*a*b*x^7*(a*d^2 + b*c^2))/7 + (a^3 *c*d*x^4)/2 + (b^3*c*d*x^10)/5 + a^2*b*c*d*x^6 + (3*a*b^2*c*d*x^8)/4
Time = 0.18 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.98 \[ \int x^2 (c+d x)^2 \left (a+b x^2\right )^3 \, dx=\frac {x^{3} \left (1260 b^{3} d^{2} x^{8}+2772 b^{3} c d \,x^{7}+4620 a \,b^{2} d^{2} x^{6}+1540 b^{3} c^{2} x^{6}+10395 a \,b^{2} c d \,x^{5}+5940 a^{2} b \,d^{2} x^{4}+5940 a \,b^{2} c^{2} x^{4}+13860 a^{2} b c d \,x^{3}+2772 a^{3} d^{2} x^{2}+8316 a^{2} b \,c^{2} x^{2}+6930 a^{3} c d x +4620 a^{3} c^{2}\right )}{13860} \] Input:
int(x^2*(d*x+c)^2*(b*x^2+a)^3,x)
Output:
(x**3*(4620*a**3*c**2 + 6930*a**3*c*d*x + 2772*a**3*d**2*x**2 + 8316*a**2* b*c**2*x**2 + 13860*a**2*b*c*d*x**3 + 5940*a**2*b*d**2*x**4 + 5940*a*b**2* c**2*x**4 + 10395*a*b**2*c*d*x**5 + 4620*a*b**2*d**2*x**6 + 1540*b**3*c**2 *x**6 + 2772*b**3*c*d*x**7 + 1260*b**3*d**2*x**8))/13860