Integrand size = 19, antiderivative size = 122 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=a^2 c^3 x+\frac {1}{3} a c^2 (2 b c+3 a d) x^3+\frac {1}{5} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^5+\frac {1}{7} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7+\frac {1}{9} b d^2 (3 b c+2 a d) x^9+\frac {1}{11} b^2 d^3 x^{11} \]
a^2*c^3*x+1/3*a*c^2*(3*a*d+2*b*c)*x^3+1/5*c*(3*a^2*d^2+6*a*b*c*d+b^2*c^2)* x^5+1/7*d*(a^2*d^2+6*a*b*c*d+3*b^2*c^2)*x^7+1/9*b*d^2*(2*a*d+3*b*c)*x^9+1/ 11*b^2*d^3*x^11
Time = 0.02 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=a^2 c^3 x+\frac {1}{3} a c^2 (2 b c+3 a d) x^3+\frac {1}{5} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^5+\frac {1}{7} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7+\frac {1}{9} b d^2 (3 b c+2 a d) x^9+\frac {1}{11} b^2 d^3 x^{11} \]
a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^3)/3 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^ 2*d^2)*x^5)/5 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^7)/7 + (b*d^2*(3*b* c + 2*a*d)*x^9)/9 + (b^2*d^3*x^11)/11
Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
\(\displaystyle \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx\) |
\(\Big \downarrow \) 290 |
\(\displaystyle \int \left (d x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+c x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3+a c^2 x^2 (3 a d+2 b c)+b d^2 x^8 (2 a d+3 b c)+b^2 d^3 x^{10}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{7} d x^7 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{5} c x^5 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac {1}{3} a c^2 x^3 (3 a d+2 b c)+\frac {1}{9} b d^2 x^9 (2 a d+3 b c)+\frac {1}{11} b^2 d^3 x^{11}\) |
a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^3)/3 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^ 2*d^2)*x^5)/5 + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^7)/7 + (b*d^2*(3*b* c + 2*a*d)*x^9)/9 + (b^2*d^3*x^11)/11
3.1.8.3.1 Defintions of rubi rules used
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 = 2.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00
method | result | size |
norman | \(\frac {b^{2} d^{3} x^{11}}{11}+\left (\frac {2}{9} a b \,d^{3}+\frac {1}{3} b^{2} c \,d^{2}\right ) x^{9}+\left (\frac {1}{7} a^{2} d^{3}+\frac {6}{7} a b c \,d^{2}+\frac {3}{7} b^{2} c^{2} d \right ) x^{7}+\left (\frac {3}{5} c \,a^{2} d^{2}+\frac {6}{5} a b \,c^{2} d +\frac {1}{5} b^{2} c^{3}\right ) x^{5}+\left (a^{2} c^{2} d +\frac {2}{3} a b \,c^{3}\right ) x^{3}+a^{2} c^{3} x\) | \(122\) |
default | \(\frac {b^{2} d^{3} x^{11}}{11}+\frac {\left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{9}}{9}+\frac {\left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{7}}{7}+\frac {\left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{5}}{5}+\frac {\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{3}}{3}+a^{2} c^{3} x\) | \(125\) |
gosper | \(\frac {1}{11} b^{2} d^{3} x^{11}+\frac {2}{9} x^{9} a b \,d^{3}+\frac {1}{3} x^{9} b^{2} c \,d^{2}+\frac {1}{7} x^{7} a^{2} d^{3}+\frac {6}{7} x^{7} a b c \,d^{2}+\frac {3}{7} x^{7} b^{2} c^{2} d +\frac {3}{5} x^{5} c \,a^{2} d^{2}+\frac {6}{5} x^{5} a b \,c^{2} d +\frac {1}{5} x^{5} b^{2} c^{3}+x^{3} a^{2} c^{2} d +\frac {2}{3} x^{3} a b \,c^{3}+a^{2} c^{3} x\) | \(132\) |
risch | \(\frac {1}{11} b^{2} d^{3} x^{11}+\frac {2}{9} x^{9} a b \,d^{3}+\frac {1}{3} x^{9} b^{2} c \,d^{2}+\frac {1}{7} x^{7} a^{2} d^{3}+\frac {6}{7} x^{7} a b c \,d^{2}+\frac {3}{7} x^{7} b^{2} c^{2} d +\frac {3}{5} x^{5} c \,a^{2} d^{2}+\frac {6}{5} x^{5} a b \,c^{2} d +\frac {1}{5} x^{5} b^{2} c^{3}+x^{3} a^{2} c^{2} d +\frac {2}{3} x^{3} a b \,c^{3}+a^{2} c^{3} x\) | \(132\) |
parallelrisch | \(\frac {1}{11} b^{2} d^{3} x^{11}+\frac {2}{9} x^{9} a b \,d^{3}+\frac {1}{3} x^{9} b^{2} c \,d^{2}+\frac {1}{7} x^{7} a^{2} d^{3}+\frac {6}{7} x^{7} a b c \,d^{2}+\frac {3}{7} x^{7} b^{2} c^{2} d +\frac {3}{5} x^{5} c \,a^{2} d^{2}+\frac {6}{5} x^{5} a b \,c^{2} d +\frac {1}{5} x^{5} b^{2} c^{3}+x^{3} a^{2} c^{2} d +\frac {2}{3} x^{3} a b \,c^{3}+a^{2} c^{3} x\) | \(132\) |
1/11*b^2*d^3*x^11+(2/9*a*b*d^3+1/3*b^2*c*d^2)*x^9+(1/7*a^2*d^3+6/7*a*b*c*d ^2+3/7*b^2*c^2*d)*x^7+(3/5*c*a^2*d^2+6/5*a*b*c^2*d+1/5*b^2*c^3)*x^5+(a^2*c ^2*d+2/3*a*b*c^3)*x^3+a^2*c^3*x
Time = 0.24 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {1}{11} \, b^{2} d^{3} x^{11} + \frac {1}{9} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{7} + a^{2} c^{3} x + \frac {1}{5} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{3} \]
1/11*b^2*d^3*x^11 + 1/9*(3*b^2*c*d^2 + 2*a*b*d^3)*x^9 + 1/7*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^7 + a^2*c^3*x + 1/5*(b^2*c^3 + 6*a*b*c^2*d + 3*a ^2*c*d^2)*x^5 + 1/3*(2*a*b*c^3 + 3*a^2*c^2*d)*x^3
Time = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=a^{2} c^{3} x + \frac {b^{2} d^{3} x^{11}}{11} + x^{9} \cdot \left (\frac {2 a b d^{3}}{9} + \frac {b^{2} c d^{2}}{3}\right ) + x^{7} \left (\frac {a^{2} d^{3}}{7} + \frac {6 a b c d^{2}}{7} + \frac {3 b^{2} c^{2} d}{7}\right ) + x^{5} \cdot \left (\frac {3 a^{2} c d^{2}}{5} + \frac {6 a b c^{2} d}{5} + \frac {b^{2} c^{3}}{5}\right ) + x^{3} \left (a^{2} c^{2} d + \frac {2 a b c^{3}}{3}\right ) \]
a**2*c**3*x + b**2*d**3*x**11/11 + x**9*(2*a*b*d**3/9 + b**2*c*d**2/3) + x **7*(a**2*d**3/7 + 6*a*b*c*d**2/7 + 3*b**2*c**2*d/7) + x**5*(3*a**2*c*d**2 /5 + 6*a*b*c**2*d/5 + b**2*c**3/5) + x**3*(a**2*c**2*d + 2*a*b*c**3/3)
Time = 0.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {1}{11} \, b^{2} d^{3} x^{11} + \frac {1}{9} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{7} + a^{2} c^{3} x + \frac {1}{5} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{3} \]
1/11*b^2*d^3*x^11 + 1/9*(3*b^2*c*d^2 + 2*a*b*d^3)*x^9 + 1/7*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^7 + a^2*c^3*x + 1/5*(b^2*c^3 + 6*a*b*c^2*d + 3*a ^2*c*d^2)*x^5 + 1/3*(2*a*b*c^3 + 3*a^2*c^2*d)*x^3
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {1}{11} \, b^{2} d^{3} x^{11} + \frac {1}{3} \, b^{2} c d^{2} x^{9} + \frac {2}{9} \, a b d^{3} x^{9} + \frac {3}{7} \, b^{2} c^{2} d x^{7} + \frac {6}{7} \, a b c d^{2} x^{7} + \frac {1}{7} \, a^{2} d^{3} x^{7} + \frac {1}{5} \, b^{2} c^{3} x^{5} + \frac {6}{5} \, a b c^{2} d x^{5} + \frac {3}{5} \, a^{2} c d^{2} x^{5} + \frac {2}{3} \, a b c^{3} x^{3} + a^{2} c^{2} d x^{3} + a^{2} c^{3} x \]
1/11*b^2*d^3*x^11 + 1/3*b^2*c*d^2*x^9 + 2/9*a*b*d^3*x^9 + 3/7*b^2*c^2*d*x^ 7 + 6/7*a*b*c*d^2*x^7 + 1/7*a^2*d^3*x^7 + 1/5*b^2*c^3*x^5 + 6/5*a*b*c^2*d* x^5 + 3/5*a^2*c*d^2*x^5 + 2/3*a*b*c^3*x^3 + a^2*c^2*d*x^3 + a^2*c^3*x
Time = 4.34 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.95 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=x^5\,\left (\frac {3\,a^2\,c\,d^2}{5}+\frac {6\,a\,b\,c^2\,d}{5}+\frac {b^2\,c^3}{5}\right )+x^7\,\left (\frac {a^2\,d^3}{7}+\frac {6\,a\,b\,c\,d^2}{7}+\frac {3\,b^2\,c^2\,d}{7}\right )+a^2\,c^3\,x+\frac {b^2\,d^3\,x^{11}}{11}+\frac {a\,c^2\,x^3\,\left (3\,a\,d+2\,b\,c\right )}{3}+\frac {b\,d^2\,x^9\,\left (2\,a\,d+3\,b\,c\right )}{9} \]