Integrand size = 30, antiderivative size = 101 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {1}{3} a^2 A x^3+\frac {1}{5} a (2 A b+a B) x^5+\frac {1}{7} \left (A b^2+a (2 b B+a C)\right ) x^7+\frac {1}{9} \left (b^2 B+2 a b C+a^2 D\right ) x^9+\frac {1}{11} b (b C+2 a D) x^{11}+\frac {1}{13} b^2 D x^{13} \] Output:
1/3*a^2*A*x^3+1/5*a*(2*A*b+B*a)*x^5+1/7*(A*b^2+a*(2*B*b+C*a))*x^7+1/9*(B*b ^2+2*C*a*b+D*a^2)*x^9+1/11*b*(C*b+2*D*a)*x^11+1/13*b^2*D*x^13
Time = 0.02 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {1}{3} a^2 A x^3+\frac {1}{5} a (2 A b+a B) x^5+\frac {1}{7} \left (A b^2+2 a b B+a^2 C\right ) x^7+\frac {1}{9} \left (b^2 B+2 a b C+a^2 D\right ) x^9+\frac {1}{11} b (b C+2 a D) x^{11}+\frac {1}{13} b^2 D x^{13} \] Input:
Integrate[x^2*(a + b*x^2)^2*(A + B*x^2 + C*x^4 + D*x^6),x]
Output:
(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^5)/5 + ((A*b^2 + 2*a*b*B + a^2*C)*x^7)/ 7 + ((b^2*B + 2*a*b*C + a^2*D)*x^9)/9 + (b*(b*C + 2*a*D)*x^11)/11 + (b^2*D *x^13)/13
Time = 0.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2333, 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.
\(\displaystyle \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx\) |
\(\Big \downarrow \) 2333 |
\(\displaystyle \int \left (a^2 A x^2+x^8 \left (a^2 D+2 a b C+b^2 B\right )+x^6 \left (a (a C+2 b B)+A b^2\right )+a x^4 (a B+2 A b)+b x^{10} (2 a D+b C)+b^2 D x^{12}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} a^2 A x^3+\frac {1}{9} x^9 \left (a^2 D+2 a b C+b^2 B\right )+\frac {1}{7} x^7 \left (a (a C+2 b B)+A b^2\right )+\frac {1}{5} a x^5 (a B+2 A b)+\frac {1}{11} b x^{11} (2 a D+b C)+\frac {1}{13} b^2 D x^{13}\) |
Input:
Int[x^2*(a + b*x^2)^2*(A + B*x^2 + C*x^4 + D*x^6),x]
Output:
(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^5)/5 + ((A*b^2 + a*(2*b*B + a*C))*x^7)/ 7 + ((b^2*B + 2*a*b*C + a^2*D)*x^9)/9 + (b*(b*C + 2*a*D)*x^11)/11 + (b^2*D *x^13)/13
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.39 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {b^{2} D x^{13}}{13}+\frac {\left (b^{2} C +2 a b D\right ) x^{11}}{11}+\frac {\left (B \,b^{2}+2 C a b +D a^{2}\right ) x^{9}}{9}+\frac {\left (b^{2} A +2 a b B +a^{2} C \right ) x^{7}}{7}+\frac {\left (2 a b A +a^{2} B \right ) x^{5}}{5}+\frac {a^{2} A \,x^{3}}{3}\) | \(94\) |
norman | \(\frac {b^{2} D x^{13}}{13}+\left (\frac {1}{11} b^{2} C +\frac {2}{11} a b D\right ) x^{11}+\left (\frac {1}{9} B \,b^{2}+\frac {2}{9} C a b +\frac {1}{9} D a^{2}\right ) x^{9}+\left (\frac {1}{7} b^{2} A +\frac {2}{7} a b B +\frac {1}{7} a^{2} C \right ) x^{7}+\left (\frac {2}{5} a b A +\frac {1}{5} a^{2} B \right ) x^{5}+\frac {a^{2} A \,x^{3}}{3}\) | \(96\) |
gosper | \(\frac {1}{13} b^{2} D x^{13}+\frac {1}{11} x^{11} b^{2} C +\frac {2}{11} x^{11} a b D+\frac {1}{9} b^{2} B \,x^{9}+\frac {2}{9} x^{9} C a b +\frac {1}{9} x^{9} D a^{2}+\frac {1}{7} x^{7} b^{2} A +\frac {2}{7} x^{7} a b B +\frac {1}{7} x^{7} a^{2} C +\frac {2}{5} x^{5} a b A +\frac {1}{5} a^{2} B \,x^{5}+\frac {1}{3} a^{2} A \,x^{3}\) | \(106\) |
parallelrisch | \(\frac {1}{13} b^{2} D x^{13}+\frac {1}{11} x^{11} b^{2} C +\frac {2}{11} x^{11} a b D+\frac {1}{9} b^{2} B \,x^{9}+\frac {2}{9} x^{9} C a b +\frac {1}{9} x^{9} D a^{2}+\frac {1}{7} x^{7} b^{2} A +\frac {2}{7} x^{7} a b B +\frac {1}{7} x^{7} a^{2} C +\frac {2}{5} x^{5} a b A +\frac {1}{5} a^{2} B \,x^{5}+\frac {1}{3} a^{2} A \,x^{3}\) | \(106\) |
orering | \(\frac {x^{3} \left (3465 b^{2} x^{10} D+4095 C \,b^{2} x^{8}+8190 D a b \,x^{8}+5005 b^{2} B \,x^{6}+10010 b \,x^{6} C a +5005 D a^{2} x^{6}+6435 A \,b^{2} x^{4}+12870 B a b \,x^{4}+6435 C \,a^{2} x^{4}+18018 a A b \,x^{2}+9009 B \,a^{2} x^{2}+15015 a^{2} A \right )}{45045}\) | \(108\) |
Input:
int(x^2*(b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
1/13*b^2*D*x^13+1/11*(C*b^2+2*D*a*b)*x^11+1/9*(B*b^2+2*C*a*b+D*a^2)*x^9+1/ 7*(A*b^2+2*B*a*b+C*a^2)*x^7+1/5*(2*A*a*b+B*a^2)*x^5+1/3*a^2*A*x^3
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {1}{13} \, D b^{2} x^{13} + \frac {1}{11} \, {\left (2 \, D a b + C b^{2}\right )} x^{11} + \frac {1}{9} \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} x^{9} + \frac {1}{7} \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} x^{7} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (B a^{2} + 2 \, A a b\right )} x^{5} \] Input:
integrate(x^2*(b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x, algorithm="fricas")
Output:
1/13*D*b^2*x^13 + 1/11*(2*D*a*b + C*b^2)*x^11 + 1/9*(D*a^2 + 2*C*a*b + B*b ^2)*x^9 + 1/7*(C*a^2 + 2*B*a*b + A*b^2)*x^7 + 1/3*A*a^2*x^3 + 1/5*(B*a^2 + 2*A*a*b)*x^5
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.06 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {A a^{2} x^{3}}{3} + \frac {D b^{2} x^{13}}{13} + x^{11} \left (\frac {C b^{2}}{11} + \frac {2 D a b}{11}\right ) + x^{9} \left (\frac {B b^{2}}{9} + \frac {2 C a b}{9} + \frac {D a^{2}}{9}\right ) + x^{7} \left (\frac {A b^{2}}{7} + \frac {2 B a b}{7} + \frac {C a^{2}}{7}\right ) + x^{5} \cdot \left (\frac {2 A a b}{5} + \frac {B a^{2}}{5}\right ) \] Input:
integrate(x**2*(b*x**2+a)**2*(D*x**6+C*x**4+B*x**2+A),x)
Output:
A*a**2*x**3/3 + D*b**2*x**13/13 + x**11*(C*b**2/11 + 2*D*a*b/11) + x**9*(B *b**2/9 + 2*C*a*b/9 + D*a**2/9) + x**7*(A*b**2/7 + 2*B*a*b/7 + C*a**2/7) + x**5*(2*A*a*b/5 + B*a**2/5)
Time = 0.02 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {1}{13} \, D b^{2} x^{13} + \frac {1}{11} \, {\left (2 \, D a b + C b^{2}\right )} x^{11} + \frac {1}{9} \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} x^{9} + \frac {1}{7} \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} x^{7} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (B a^{2} + 2 \, A a b\right )} x^{5} \] Input:
integrate(x^2*(b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x, algorithm="maxima")
Output:
1/13*D*b^2*x^13 + 1/11*(2*D*a*b + C*b^2)*x^11 + 1/9*(D*a^2 + 2*C*a*b + B*b ^2)*x^9 + 1/7*(C*a^2 + 2*B*a*b + A*b^2)*x^7 + 1/3*A*a^2*x^3 + 1/5*(B*a^2 + 2*A*a*b)*x^5
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {1}{13} \, D b^{2} x^{13} + \frac {2}{11} \, D a b x^{11} + \frac {1}{11} \, C b^{2} x^{11} + \frac {1}{9} \, D a^{2} x^{9} + \frac {2}{9} \, C a b x^{9} + \frac {1}{9} \, B b^{2} x^{9} + \frac {1}{7} \, C a^{2} x^{7} + \frac {2}{7} \, B a b x^{7} + \frac {1}{7} \, A b^{2} x^{7} + \frac {1}{5} \, B a^{2} x^{5} + \frac {2}{5} \, A a b x^{5} + \frac {1}{3} \, A a^{2} x^{3} \] Input:
integrate(x^2*(b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x, algorithm="giac")
Output:
1/13*D*b^2*x^13 + 2/11*D*a*b*x^11 + 1/11*C*b^2*x^11 + 1/9*D*a^2*x^9 + 2/9* C*a*b*x^9 + 1/9*B*b^2*x^9 + 1/7*C*a^2*x^7 + 2/7*B*a*b*x^7 + 1/7*A*b^2*x^7 + 1/5*B*a^2*x^5 + 2/5*A*a*b*x^5 + 1/3*A*a^2*x^3
Time = 2.33 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.07 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {a^2\,x^9\,D}{9}+\frac {b^2\,x^{13}\,D}{13}+\frac {A\,x^3\,\left (35\,a^2+42\,a\,b\,x^2+15\,b^2\,x^4\right )}{105}+\frac {B\,x^5\,\left (63\,a^2+90\,a\,b\,x^2+35\,b^2\,x^4\right )}{315}+\frac {C\,x^7\,\left (99\,a^2+154\,a\,b\,x^2+63\,b^2\,x^4\right )}{693}+\frac {2\,a\,b\,x^{11}\,D}{11} \] Input:
int(x^2*(a + b*x^2)^2*(A + B*x^2 + C*x^4 + x^6*D),x)
Output:
(a^2*x^9*D)/9 + (b^2*x^13*D)/13 + (A*x^3*(35*a^2 + 15*b^2*x^4 + 42*a*b*x^2 ))/105 + (B*x^5*(63*a^2 + 35*b^2*x^4 + 90*a*b*x^2))/315 + (C*x^7*(99*a^2 + 63*b^2*x^4 + 154*a*b*x^2))/693 + (2*a*b*x^11*D)/11
Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {x^{3} \left (3465 b^{2} d \,x^{10}+8190 a b d \,x^{8}+4095 b^{2} c \,x^{8}+5005 a^{2} d \,x^{6}+10010 a b c \,x^{6}+5005 b^{3} x^{6}+6435 a^{2} c \,x^{4}+19305 a \,b^{2} x^{4}+27027 a^{2} b \,x^{2}+15015 a^{3}\right )}{45045} \] Input:
int(x^2*(b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x)
Output:
(x**3*(15015*a**3 + 27027*a**2*b*x**2 + 6435*a**2*c*x**4 + 5005*a**2*d*x** 6 + 19305*a*b**2*x**4 + 10010*a*b*c*x**6 + 8190*a*b*d*x**8 + 5005*b**3*x** 6 + 4095*b**2*c*x**8 + 3465*b**2*d*x**10))/45045