Integrand size = 27, antiderivative size = 96 \[ \int \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=a^2 A x+\frac {1}{3} a (2 A b+a B) x^3+\frac {1}{5} \left (A b^2+a (2 b B+a C)\right ) x^5+\frac {1}{7} \left (b^2 B+2 a b C+a^2 D\right ) x^7+\frac {1}{9} b (b C+2 a D) x^9+\frac {1}{11} b^2 D x^{11} \] Output:
a^2*A*x+1/3*a*(2*A*b+B*a)*x^3+1/5*(A*b^2+a*(2*B*b+C*a))*x^5+1/7*(B*b^2+2*C *a*b+D*a^2)*x^7+1/9*b*(C*b+2*D*a)*x^9+1/11*b^2*D*x^11
Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=a^2 A x+\frac {1}{3} a (2 A b+a B) x^3+\frac {1}{5} \left (A b^2+2 a b B+a^2 C\right ) x^5+\frac {1}{7} \left (b^2 B+2 a b C+a^2 D\right ) x^7+\frac {1}{9} b (b C+2 a D) x^9+\frac {1}{11} b^2 D x^{11} \] Input:
Integrate[(a + b*x^2)^2*(A + B*x^2 + C*x^4 + D*x^6),x]
Output:
a^2*A*x + (a*(2*A*b + a*B)*x^3)/3 + ((A*b^2 + 2*a*b*B + a^2*C)*x^5)/5 + (( b^2*B + 2*a*b*C + a^2*D)*x^7)/7 + (b*(b*C + 2*a*D)*x^9)/9 + (b^2*D*x^11)/1 1
Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2341, 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.
\(\displaystyle \int \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx\) |
\(\Big \downarrow \) 2341 |
\(\displaystyle \int \left (a^2 A+x^6 \left (a^2 D+2 a b C+b^2 B\right )+x^4 \left (a (a C+2 b B)+A b^2\right )+a x^2 (a B+2 A b)+b x^8 (2 a D+b C)+b^2 D x^{10}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 A x+\frac {1}{7} x^7 \left (a^2 D+2 a b C+b^2 B\right )+\frac {1}{5} x^5 \left (a (a C+2 b B)+A b^2\right )+\frac {1}{3} a x^3 (a B+2 A b)+\frac {1}{9} b x^9 (2 a D+b C)+\frac {1}{11} b^2 D x^{11}\) |
Input:
Int[(a + b*x^2)^2*(A + B*x^2 + C*x^4 + D*x^6),x]
Output:
a^2*A*x + (a*(2*A*b + a*B)*x^3)/3 + ((A*b^2 + a*(2*b*B + a*C))*x^5)/5 + (( b^2*B + 2*a*b*C + a^2*D)*x^7)/7 + (b*(b*C + 2*a*D)*x^9)/9 + (b^2*D*x^11)/1 1
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* (a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.40 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {b^{2} D x^{11}}{11}+\frac {\left (b^{2} C +2 a b D\right ) x^{9}}{9}+\frac {\left (B \,b^{2}+2 C a b +D a^{2}\right ) x^{7}}{7}+\frac {\left (b^{2} A +2 a b B +a^{2} C \right ) x^{5}}{5}+\frac {\left (2 a b A +a^{2} B \right ) x^{3}}{3}+a^{2} A x\) | \(91\) |
norman | \(\frac {b^{2} D x^{11}}{11}+\left (\frac {1}{9} b^{2} C +\frac {2}{9} a b D\right ) x^{9}+\left (\frac {1}{7} B \,b^{2}+\frac {2}{7} C a b +\frac {1}{7} D a^{2}\right ) x^{7}+\left (\frac {1}{5} b^{2} A +\frac {2}{5} a b B +\frac {1}{5} a^{2} C \right ) x^{5}+\left (\frac {2}{3} a b A +\frac {1}{3} a^{2} B \right ) x^{3}+a^{2} A x\) | \(93\) |
gosper | \(\frac {1}{11} b^{2} D x^{11}+\frac {1}{9} b^{2} C \,x^{9}+\frac {2}{9} x^{9} a b D+\frac {1}{7} b^{2} B \,x^{7}+\frac {2}{7} x^{7} C a b +\frac {1}{7} x^{7} D a^{2}+\frac {1}{5} A \,b^{2} x^{5}+\frac {2}{5} x^{5} a b B +\frac {1}{5} x^{5} a^{2} C +\frac {2}{3} a A b \,x^{3}+\frac {1}{3} x^{3} a^{2} B +a^{2} A x\) | \(103\) |
parallelrisch | \(\frac {1}{11} b^{2} D x^{11}+\frac {1}{9} b^{2} C \,x^{9}+\frac {2}{9} x^{9} a b D+\frac {1}{7} b^{2} B \,x^{7}+\frac {2}{7} x^{7} C a b +\frac {1}{7} x^{7} D a^{2}+\frac {1}{5} A \,b^{2} x^{5}+\frac {2}{5} x^{5} a b B +\frac {1}{5} x^{5} a^{2} C +\frac {2}{3} a A b \,x^{3}+\frac {1}{3} x^{3} a^{2} B +a^{2} A x\) | \(103\) |
orering | \(\frac {x \left (315 b^{2} x^{10} D+385 C \,b^{2} x^{8}+770 D a b \,x^{8}+495 b^{2} B \,x^{6}+990 b \,x^{6} C a +495 D a^{2} x^{6}+693 A \,b^{2} x^{4}+1386 B a b \,x^{4}+693 C \,a^{2} x^{4}+2310 a A b \,x^{2}+1155 B \,a^{2} x^{2}+3465 a^{2} A \right )}{3465}\) | \(106\) |
Input:
int((b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
1/11*b^2*D*x^11+1/9*(C*b^2+2*D*a*b)*x^9+1/7*(B*b^2+2*C*a*b+D*a^2)*x^7+1/5* (A*b^2+2*B*a*b+C*a^2)*x^5+1/3*(2*A*a*b+B*a^2)*x^3+a^2*A*x
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {1}{11} \, D b^{2} x^{11} + \frac {1}{9} \, {\left (2 \, D a b + C b^{2}\right )} x^{9} + \frac {1}{7} \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} x^{7} + \frac {1}{5} \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} x^{5} + A a^{2} x + \frac {1}{3} \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} \] Input:
integrate((b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x, algorithm="fricas")
Output:
1/11*D*b^2*x^11 + 1/9*(2*D*a*b + C*b^2)*x^9 + 1/7*(D*a^2 + 2*C*a*b + B*b^2 )*x^7 + 1/5*(C*a^2 + 2*B*a*b + A*b^2)*x^5 + A*a^2*x + 1/3*(B*a^2 + 2*A*a*b )*x^3
Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08 \[ \int \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=A a^{2} x + \frac {D b^{2} x^{11}}{11} + x^{9} \left (\frac {C b^{2}}{9} + \frac {2 D a b}{9}\right ) + x^{7} \left (\frac {B b^{2}}{7} + \frac {2 C a b}{7} + \frac {D a^{2}}{7}\right ) + x^{5} \left (\frac {A b^{2}}{5} + \frac {2 B a b}{5} + \frac {C a^{2}}{5}\right ) + x^{3} \cdot \left (\frac {2 A a b}{3} + \frac {B a^{2}}{3}\right ) \] Input:
integrate((b*x**2+a)**2*(D*x**6+C*x**4+B*x**2+A),x)
Output:
A*a**2*x + D*b**2*x**11/11 + x**9*(C*b**2/9 + 2*D*a*b/9) + x**7*(B*b**2/7 + 2*C*a*b/7 + D*a**2/7) + x**5*(A*b**2/5 + 2*B*a*b/5 + C*a**2/5) + x**3*(2 *A*a*b/3 + B*a**2/3)
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {1}{11} \, D b^{2} x^{11} + \frac {1}{9} \, {\left (2 \, D a b + C b^{2}\right )} x^{9} + \frac {1}{7} \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} x^{7} + \frac {1}{5} \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} x^{5} + A a^{2} x + \frac {1}{3} \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} \] Input:
integrate((b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x, algorithm="maxima")
Output:
1/11*D*b^2*x^11 + 1/9*(2*D*a*b + C*b^2)*x^9 + 1/7*(D*a^2 + 2*C*a*b + B*b^2 )*x^7 + 1/5*(C*a^2 + 2*B*a*b + A*b^2)*x^5 + A*a^2*x + 1/3*(B*a^2 + 2*A*a*b )*x^3
Time = 0.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {1}{11} \, D b^{2} x^{11} + \frac {2}{9} \, D a b x^{9} + \frac {1}{9} \, C b^{2} x^{9} + \frac {1}{7} \, D a^{2} x^{7} + \frac {2}{7} \, C a b x^{7} + \frac {1}{7} \, B b^{2} x^{7} + \frac {1}{5} \, C a^{2} x^{5} + \frac {2}{5} \, B a b x^{5} + \frac {1}{5} \, A b^{2} x^{5} + \frac {1}{3} \, B a^{2} x^{3} + \frac {2}{3} \, A a b x^{3} + A a^{2} x \] Input:
integrate((b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x, algorithm="giac")
Output:
1/11*D*b^2*x^11 + 2/9*D*a*b*x^9 + 1/9*C*b^2*x^9 + 1/7*D*a^2*x^7 + 2/7*C*a* b*x^7 + 1/7*B*b^2*x^7 + 1/5*C*a^2*x^5 + 2/5*B*a*b*x^5 + 1/5*A*b^2*x^5 + 1/ 3*B*a^2*x^3 + 2/3*A*a*b*x^3 + A*a^2*x
Time = 1.36 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.10 \[ \int \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {A\,x\,\left (15\,a^2+10\,a\,b\,x^2+3\,b^2\,x^4\right )}{15}+\frac {a^2\,x^7\,D}{7}+\frac {b^2\,x^{11}\,D}{11}+\frac {B\,x^3\,\left (35\,a^2+42\,a\,b\,x^2+15\,b^2\,x^4\right )}{105}+\frac {C\,x^5\,\left (63\,a^2+90\,a\,b\,x^2+35\,b^2\,x^4\right )}{315}+\frac {2\,a\,b\,x^9\,D}{9} \] Input:
int((a + b*x^2)^2*(A + B*x^2 + C*x^4 + x^6*D),x)
Output:
(A*x*(15*a^2 + 3*b^2*x^4 + 10*a*b*x^2))/15 + (a^2*x^7*D)/7 + (b^2*x^11*D)/ 11 + (B*x^3*(35*a^2 + 15*b^2*x^4 + 42*a*b*x^2))/105 + (C*x^5*(63*a^2 + 35* b^2*x^4 + 90*a*b*x^2))/315 + (2*a*b*x^9*D)/9
Time = 0.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {x \left (315 b^{2} d \,x^{10}+770 a b d \,x^{8}+385 b^{2} c \,x^{8}+495 a^{2} d \,x^{6}+990 a b c \,x^{6}+495 b^{3} x^{6}+693 a^{2} c \,x^{4}+2079 a \,b^{2} x^{4}+3465 a^{2} b \,x^{2}+3465 a^{3}\right )}{3465} \] Input:
int((b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x)
Output:
(x*(3465*a**3 + 3465*a**2*b*x**2 + 693*a**2*c*x**4 + 495*a**2*d*x**6 + 207 9*a*b**2*x**4 + 990*a*b*c*x**6 + 770*a*b*d*x**8 + 495*b**3*x**6 + 385*b**2 *c*x**8 + 315*b**2*d*x**10))/3465