Integrand size = 17, antiderivative size = 94 \[ \int \left (a+b x^2\right )^4 \left (A+B x^2\right ) \, dx=a^4 A x+\frac {1}{3} a^3 (4 A b+a B) x^3+\frac {2}{5} a^2 b (3 A b+2 a B) x^5+\frac {2}{7} a b^2 (2 A b+3 a B) x^7+\frac {1}{9} b^3 (A b+4 a B) x^9+\frac {1}{11} b^4 B x^{11} \] Output:
a^4*A*x+1/3*a^3*(4*A*b+B*a)*x^3+2/5*a^2*b*(3*A*b+2*B*a)*x^5+2/7*a*b^2*(2*A *b+3*B*a)*x^7+1/9*b^3*(A*b+4*B*a)*x^9+1/11*b^4*B*x^11
Time = 0.01 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right )^4 \left (A+B x^2\right ) \, dx=a^4 A x+\frac {1}{3} a^3 (4 A b+a B) x^3+\frac {2}{5} a^2 b (3 A b+2 a B) x^5+\frac {2}{7} a b^2 (2 A b+3 a B) x^7+\frac {1}{9} b^3 (A b+4 a B) x^9+\frac {1}{11} b^4 B x^{11} \] Input:
Integrate[(a + b*x^2)^4*(A + B*x^2),x]
Output:
a^4*A*x + (a^3*(4*A*b + a*B)*x^3)/3 + (2*a^2*b*(3*A*b + 2*a*B)*x^5)/5 + (2 *a*b^2*(2*A*b + 3*a*B)*x^7)/7 + (b^3*(A*b + 4*a*B)*x^9)/9 + (b^4*B*x^11)/1 1
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {290, 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.
\(\displaystyle \int \left (a+b x^2\right )^4 \left (A+B x^2\right ) \, dx\) |
\(\Big \downarrow \) 290 |
\(\displaystyle \int \left (a^4 A+a^3 x^2 (a B+4 A b)+2 a^2 b x^4 (2 a B+3 A b)+b^3 x^8 (4 a B+A b)+2 a b^2 x^6 (3 a B+2 A b)+b^4 B x^{10}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^4 A x+\frac {1}{3} a^3 x^3 (a B+4 A b)+\frac {2}{5} a^2 b x^5 (2 a B+3 A b)+\frac {1}{9} b^3 x^9 (4 a B+A b)+\frac {2}{7} a b^2 x^7 (3 a B+2 A b)+\frac {1}{11} b^4 B x^{11}\) |
Input:
Int[(a + b*x^2)^4*(A + B*x^2),x]
Output:
a^4*A*x + (a^3*(4*A*b + a*B)*x^3)/3 + (2*a^2*b*(3*A*b + 2*a*B)*x^5)/5 + (2 *a*b^2*(2*A*b + 3*a*B)*x^7)/7 + (b^3*(A*b + 4*a*B)*x^9)/9 + (b^4*B*x^11)/1 1
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d }, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Time = 0.56 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01
method | result | size |
norman | \(\frac {b^{4} B \,x^{11}}{11}+\left (\frac {1}{9} A \,b^{4}+\frac {4}{9} B a \,b^{3}\right ) x^{9}+\left (\frac {4}{7} a \,b^{3} A +\frac {6}{7} a^{2} b^{2} B \right ) x^{7}+\left (\frac {6}{5} A \,a^{2} b^{2}+\frac {4}{5} B \,a^{3} b \right ) x^{5}+\left (\frac {4}{3} A \,a^{3} b +\frac {1}{3} B \,a^{4}\right ) x^{3}+a^{4} A x\) | \(95\) |
default | \(\frac {b^{4} B \,x^{11}}{11}+\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) x^{9}}{9}+\frac {\left (4 a \,b^{3} A +6 a^{2} b^{2} B \right ) x^{7}}{7}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) x^{5}}{5}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) x^{3}}{3}+a^{4} A x\) | \(97\) |
gosper | \(\frac {1}{11} b^{4} B \,x^{11}+\frac {1}{9} x^{9} A \,b^{4}+\frac {4}{9} x^{9} B a \,b^{3}+\frac {4}{7} x^{7} a \,b^{3} A +\frac {6}{7} x^{7} a^{2} b^{2} B +\frac {6}{5} x^{5} A \,a^{2} b^{2}+\frac {4}{5} x^{5} B \,a^{3} b +\frac {4}{3} x^{3} A \,a^{3} b +\frac {1}{3} x^{3} B \,a^{4}+a^{4} A x\) | \(99\) |
risch | \(\frac {1}{11} b^{4} B \,x^{11}+\frac {1}{9} x^{9} A \,b^{4}+\frac {4}{9} x^{9} B a \,b^{3}+\frac {4}{7} x^{7} a \,b^{3} A +\frac {6}{7} x^{7} a^{2} b^{2} B +\frac {6}{5} x^{5} A \,a^{2} b^{2}+\frac {4}{5} x^{5} B \,a^{3} b +\frac {4}{3} x^{3} A \,a^{3} b +\frac {1}{3} x^{3} B \,a^{4}+a^{4} A x\) | \(99\) |
parallelrisch | \(\frac {1}{11} b^{4} B \,x^{11}+\frac {1}{9} x^{9} A \,b^{4}+\frac {4}{9} x^{9} B a \,b^{3}+\frac {4}{7} x^{7} a \,b^{3} A +\frac {6}{7} x^{7} a^{2} b^{2} B +\frac {6}{5} x^{5} A \,a^{2} b^{2}+\frac {4}{5} x^{5} B \,a^{3} b +\frac {4}{3} x^{3} A \,a^{3} b +\frac {1}{3} x^{3} B \,a^{4}+a^{4} A x\) | \(99\) |
orering | \(\frac {x \left (315 B \,b^{4} x^{10}+385 A \,x^{8} b^{4}+1540 B \,x^{8} a \,b^{3}+1980 A \,x^{6} a \,b^{3}+2970 B \,x^{6} a^{2} b^{2}+4158 A \,x^{4} a^{2} b^{2}+2772 B \,x^{4} a^{3} b +4620 A \,x^{2} a^{3} b +1155 B \,x^{2} a^{4}+3465 A \,a^{4}\right )}{3465}\) | \(102\) |
Input:
int((b*x^2+a)^4*(B*x^2+A),x,method=_RETURNVERBOSE)
Output:
1/11*b^4*B*x^11+(1/9*A*b^4+4/9*B*a*b^3)*x^9+(4/7*a*b^3*A+6/7*a^2*b^2*B)*x^ 7+(6/5*A*a^2*b^2+4/5*B*a^3*b)*x^5+(4/3*A*a^3*b+1/3*B*a^4)*x^3+a^4*A*x
Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^2\right )^4 \left (A+B x^2\right ) \, dx=\frac {1}{11} \, B b^{4} x^{11} + \frac {1}{9} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{9} + \frac {2}{7} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{7} + A a^{4} x + \frac {2}{5} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{5} + \frac {1}{3} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{3} \] Input:
integrate((b*x^2+a)^4*(B*x^2+A),x, algorithm="fricas")
Output:
1/11*B*b^4*x^11 + 1/9*(4*B*a*b^3 + A*b^4)*x^9 + 2/7*(3*B*a^2*b^2 + 2*A*a*b ^3)*x^7 + A*a^4*x + 2/5*(2*B*a^3*b + 3*A*a^2*b^2)*x^5 + 1/3*(B*a^4 + 4*A*a ^3*b)*x^3
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.14 \[ \int \left (a+b x^2\right )^4 \left (A+B x^2\right ) \, dx=A a^{4} x + \frac {B b^{4} x^{11}}{11} + x^{9} \left (\frac {A b^{4}}{9} + \frac {4 B a b^{3}}{9}\right ) + x^{7} \cdot \left (\frac {4 A a b^{3}}{7} + \frac {6 B a^{2} b^{2}}{7}\right ) + x^{5} \cdot \left (\frac {6 A a^{2} b^{2}}{5} + \frac {4 B a^{3} b}{5}\right ) + x^{3} \cdot \left (\frac {4 A a^{3} b}{3} + \frac {B a^{4}}{3}\right ) \] Input:
integrate((b*x**2+a)**4*(B*x**2+A),x)
Output:
A*a**4*x + B*b**4*x**11/11 + x**9*(A*b**4/9 + 4*B*a*b**3/9) + x**7*(4*A*a* b**3/7 + 6*B*a**2*b**2/7) + x**5*(6*A*a**2*b**2/5 + 4*B*a**3*b/5) + x**3*( 4*A*a**3*b/3 + B*a**4/3)
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^2\right )^4 \left (A+B x^2\right ) \, dx=\frac {1}{11} \, B b^{4} x^{11} + \frac {1}{9} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{9} + \frac {2}{7} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{7} + A a^{4} x + \frac {2}{5} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{5} + \frac {1}{3} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{3} \] Input:
integrate((b*x^2+a)^4*(B*x^2+A),x, algorithm="maxima")
Output:
1/11*B*b^4*x^11 + 1/9*(4*B*a*b^3 + A*b^4)*x^9 + 2/7*(3*B*a^2*b^2 + 2*A*a*b ^3)*x^7 + A*a^4*x + 2/5*(2*B*a^3*b + 3*A*a^2*b^2)*x^5 + 1/3*(B*a^4 + 4*A*a ^3*b)*x^3
Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.04 \[ \int \left (a+b x^2\right )^4 \left (A+B x^2\right ) \, dx=\frac {1}{11} \, B b^{4} x^{11} + \frac {4}{9} \, B a b^{3} x^{9} + \frac {1}{9} \, A b^{4} x^{9} + \frac {6}{7} \, B a^{2} b^{2} x^{7} + \frac {4}{7} \, A a b^{3} x^{7} + \frac {4}{5} \, B a^{3} b x^{5} + \frac {6}{5} \, A a^{2} b^{2} x^{5} + \frac {1}{3} \, B a^{4} x^{3} + \frac {4}{3} \, A a^{3} b x^{3} + A a^{4} x \] Input:
integrate((b*x^2+a)^4*(B*x^2+A),x, algorithm="giac")
Output:
1/11*B*b^4*x^11 + 4/9*B*a*b^3*x^9 + 1/9*A*b^4*x^9 + 6/7*B*a^2*b^2*x^7 + 4/ 7*A*a*b^3*x^7 + 4/5*B*a^3*b*x^5 + 6/5*A*a^2*b^2*x^5 + 1/3*B*a^4*x^3 + 4/3* A*a^3*b*x^3 + A*a^4*x
Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.94 \[ \int \left (a+b x^2\right )^4 \left (A+B x^2\right ) \, dx=x^3\,\left (\frac {B\,a^4}{3}+\frac {4\,A\,b\,a^3}{3}\right )+x^9\,\left (\frac {A\,b^4}{9}+\frac {4\,B\,a\,b^3}{9}\right )+\frac {B\,b^4\,x^{11}}{11}+A\,a^4\,x+\frac {2\,a^2\,b\,x^5\,\left (3\,A\,b+2\,B\,a\right )}{5}+\frac {2\,a\,b^2\,x^7\,\left (2\,A\,b+3\,B\,a\right )}{7} \] Input:
int((A + B*x^2)*(a + b*x^2)^4,x)
Output:
x^3*((B*a^4)/3 + (4*A*a^3*b)/3) + x^9*((A*b^4)/9 + (4*B*a*b^3)/9) + (B*b^4 *x^11)/11 + A*a^4*x + (2*a^2*b*x^5*(3*A*b + 2*B*a))/5 + (2*a*b^2*x^7*(2*A* b + 3*B*a))/7
Time = 0.15 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.61 \[ \int \left (a+b x^2\right )^4 \left (A+B x^2\right ) \, dx=\frac {x \left (63 b^{5} x^{10}+385 a \,b^{4} x^{8}+990 a^{2} b^{3} x^{6}+1386 a^{3} b^{2} x^{4}+1155 a^{4} b \,x^{2}+693 a^{5}\right )}{693} \] Input:
int((b*x^2+a)^4*(B*x^2+A),x)
Output:
(x*(693*a**5 + 1155*a**4*b*x**2 + 1386*a**3*b**2*x**4 + 990*a**2*b**3*x**6 + 385*a*b**4*x**8 + 63*b**5*x**10))/693