Integrand size = 25, antiderivative size = 154 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^2 A x+\frac {1}{2} a^2 B x^2+\frac {1}{3} a (2 A b+a C) x^3+\frac {1}{2} a b B x^4+\frac {1}{5} \left (A \left (b^2+2 a c\right )+2 a b C\right ) x^5+\frac {1}{6} B \left (b^2+2 a c\right ) x^6+\frac {1}{7} \left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^7+\frac {1}{4} b B c x^8+\frac {1}{9} c (A c+2 b C) x^9+\frac {1}{10} B c^2 x^{10}+\frac {1}{11} c^2 C x^{11} \]
a^2*A*x+1/2*a^2*B*x^2+1/3*a*(2*A*b+C*a)*x^3+1/2*a*b*B*x^4+1/5*(A*(2*a*c+b^ 2)+2*a*b*C)*x^5+1/6*B*(2*a*c+b^2)*x^6+1/7*(2*A*b*c+(2*a*c+b^2)*C)*x^7+1/4* b*B*c*x^8+1/9*c*(A*c+2*C*b)*x^9+1/10*B*c^2*x^10+1/11*c^2*C*x^11
Time = 0.02 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^2 A x+\frac {1}{2} a^2 B x^2+\frac {1}{3} a (2 A b+a C) x^3+\frac {1}{2} a b B x^4+\frac {1}{5} \left (A b^2+2 a A c+2 a b C\right ) x^5+\frac {1}{6} B \left (b^2+2 a c\right ) x^6+\frac {1}{7} \left (2 A b c+b^2 C+2 a c C\right ) x^7+\frac {1}{4} b B c x^8+\frac {1}{9} c (A c+2 b C) x^9+\frac {1}{10} B c^2 x^{10}+\frac {1}{11} c^2 C x^{11} \]
a^2*A*x + (a^2*B*x^2)/2 + (a*(2*A*b + a*C)*x^3)/3 + (a*b*B*x^4)/2 + ((A*b^ 2 + 2*a*A*c + 2*a*b*C)*x^5)/5 + (B*(b^2 + 2*a*c)*x^6)/6 + ((2*A*b*c + b^2* C + 2*a*c*C)*x^7)/7 + (b*B*c*x^8)/4 + (c*(A*c + 2*b*C)*x^9)/9 + (B*c^2*x^1 0)/10 + (c^2*C*x^11)/11
Time = 0.32 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2188, 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.
\(\displaystyle \int \left (a+b x^2+c x^4\right )^2 \left (A+B x+C x^2\right ) \, dx\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \int \left (a^2 A+a^2 B x+x^6 \left (C \left (2 a c+b^2\right )+2 A b c\right )+x^4 \left (A \left (2 a c+b^2\right )+2 a b C\right )+a x^2 (a C+2 A b)+B x^5 \left (2 a c+b^2\right )+2 a b B x^3+c x^8 (A c+2 b C)+2 b B c x^7+B c^2 x^9+c^2 C x^{10}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 A x+\frac {1}{2} a^2 B x^2+\frac {1}{7} x^7 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac {1}{5} x^5 \left (A \left (2 a c+b^2\right )+2 a b C\right )+\frac {1}{3} a x^3 (a C+2 A b)+\frac {1}{6} B x^6 \left (2 a c+b^2\right )+\frac {1}{2} a b B x^4+\frac {1}{9} c x^9 (A c+2 b C)+\frac {1}{4} b B c x^8+\frac {1}{10} B c^2 x^{10}+\frac {1}{11} c^2 C x^{11}\) |
a^2*A*x + (a^2*B*x^2)/2 + (a*(2*A*b + a*C)*x^3)/3 + (a*b*B*x^4)/2 + ((A*(b ^2 + 2*a*c) + 2*a*b*C)*x^5)/5 + (B*(b^2 + 2*a*c)*x^6)/6 + ((2*A*b*c + (b^2 + 2*a*c)*C)*x^7)/7 + (b*B*c*x^8)/4 + (c*(A*c + 2*b*C)*x^9)/9 + (B*c^2*x^1 0)/10 + (c^2*C*x^11)/11
3.1.13.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
Time = 0.14 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {c^{2} C \,x^{11}}{11}+\frac {B \,c^{2} x^{10}}{10}+\frac {\left (A \,c^{2}+2 C b c \right ) x^{9}}{9}+\frac {b B c \,x^{8}}{4}+\frac {\left (2 A b c +\left (2 a c +b^{2}\right ) C \right ) x^{7}}{7}+\frac {B \left (2 a c +b^{2}\right ) x^{6}}{6}+\frac {\left (A \left (2 a c +b^{2}\right )+2 a b C \right ) x^{5}}{5}+\frac {B a b \,x^{4}}{2}+\frac {\left (2 A a b +C \,a^{2}\right ) x^{3}}{3}+\frac {B \,a^{2} x^{2}}{2}+a^{2} A x\) | \(139\) |
norman | \(\frac {c^{2} C \,x^{11}}{11}+\frac {B \,c^{2} x^{10}}{10}+\left (\frac {1}{9} A \,c^{2}+\frac {2}{9} C b c \right ) x^{9}+\frac {b B c \,x^{8}}{4}+\left (\frac {2}{7} A b c +\frac {2}{7} a c C +\frac {1}{7} b^{2} C \right ) x^{7}+\left (\frac {1}{3} B a c +\frac {1}{6} B \,b^{2}\right ) x^{6}+\left (\frac {2}{5} A a c +\frac {1}{5} A \,b^{2}+\frac {2}{5} a b C \right ) x^{5}+\frac {B a b \,x^{4}}{2}+\left (\frac {2}{3} A a b +\frac {1}{3} C \,a^{2}\right ) x^{3}+\frac {B \,a^{2} x^{2}}{2}+a^{2} A x\) | \(141\) |
gosper | \(\frac {1}{11} c^{2} C \,x^{11}+\frac {1}{10} B \,c^{2} x^{10}+\frac {1}{9} x^{9} A \,c^{2}+\frac {2}{9} x^{9} C b c +\frac {1}{4} b B c \,x^{8}+\frac {2}{7} x^{7} A b c +\frac {2}{7} x^{7} a c C +\frac {1}{7} x^{7} b^{2} C +\frac {1}{3} x^{6} B a c +\frac {1}{6} x^{6} B \,b^{2}+\frac {2}{5} x^{5} A a c +\frac {1}{5} x^{5} A \,b^{2}+\frac {2}{5} x^{5} a b C +\frac {1}{2} B a b \,x^{4}+\frac {2}{3} x^{3} A a b +\frac {1}{3} x^{3} C \,a^{2}+\frac {1}{2} B \,a^{2} x^{2}+a^{2} A x\) | \(152\) |
risch | \(\frac {1}{11} c^{2} C \,x^{11}+\frac {1}{10} B \,c^{2} x^{10}+\frac {1}{9} x^{9} A \,c^{2}+\frac {2}{9} x^{9} C b c +\frac {1}{4} b B c \,x^{8}+\frac {2}{7} x^{7} A b c +\frac {2}{7} x^{7} a c C +\frac {1}{7} x^{7} b^{2} C +\frac {1}{3} x^{6} B a c +\frac {1}{6} x^{6} B \,b^{2}+\frac {2}{5} x^{5} A a c +\frac {1}{5} x^{5} A \,b^{2}+\frac {2}{5} x^{5} a b C +\frac {1}{2} B a b \,x^{4}+\frac {2}{3} x^{3} A a b +\frac {1}{3} x^{3} C \,a^{2}+\frac {1}{2} B \,a^{2} x^{2}+a^{2} A x\) | \(152\) |
parallelrisch | \(\frac {1}{11} c^{2} C \,x^{11}+\frac {1}{10} B \,c^{2} x^{10}+\frac {1}{9} x^{9} A \,c^{2}+\frac {2}{9} x^{9} C b c +\frac {1}{4} b B c \,x^{8}+\frac {2}{7} x^{7} A b c +\frac {2}{7} x^{7} a c C +\frac {1}{7} x^{7} b^{2} C +\frac {1}{3} x^{6} B a c +\frac {1}{6} x^{6} B \,b^{2}+\frac {2}{5} x^{5} A a c +\frac {1}{5} x^{5} A \,b^{2}+\frac {2}{5} x^{5} a b C +\frac {1}{2} B a b \,x^{4}+\frac {2}{3} x^{3} A a b +\frac {1}{3} x^{3} C \,a^{2}+\frac {1}{2} B \,a^{2} x^{2}+a^{2} A x\) | \(152\) |
1/11*c^2*C*x^11+1/10*B*c^2*x^10+1/9*(A*c^2+2*C*b*c)*x^9+1/4*b*B*c*x^8+1/7* (2*A*b*c+(2*a*c+b^2)*C)*x^7+1/6*B*(2*a*c+b^2)*x^6+1/5*(A*(2*a*c+b^2)+2*a*b *C)*x^5+1/2*B*a*b*x^4+1/3*(2*A*a*b+C*a^2)*x^3+1/2*B*a^2*x^2+a^2*A*x
Time = 0.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.91 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{11} \, C c^{2} x^{11} + \frac {1}{10} \, B c^{2} x^{10} + \frac {1}{4} \, B b c x^{8} + \frac {1}{9} \, {\left (2 \, C b c + A c^{2}\right )} x^{9} + \frac {1}{7} \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{7} + \frac {1}{2} \, B a b x^{4} + \frac {1}{6} \, {\left (B b^{2} + 2 \, B a c\right )} x^{6} + \frac {1}{5} \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{5} + \frac {1}{2} \, B a^{2} x^{2} + A a^{2} x + \frac {1}{3} \, {\left (C a^{2} + 2 \, A a b\right )} x^{3} \]
1/11*C*c^2*x^11 + 1/10*B*c^2*x^10 + 1/4*B*b*c*x^8 + 1/9*(2*C*b*c + A*c^2)* x^9 + 1/7*(C*b^2 + 2*(C*a + A*b)*c)*x^7 + 1/2*B*a*b*x^4 + 1/6*(B*b^2 + 2*B *a*c)*x^6 + 1/5*(2*C*a*b + A*b^2 + 2*A*a*c)*x^5 + 1/2*B*a^2*x^2 + A*a^2*x + 1/3*(C*a^2 + 2*A*a*b)*x^3
Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.07 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=A a^{2} x + \frac {B a^{2} x^{2}}{2} + \frac {B a b x^{4}}{2} + \frac {B b c x^{8}}{4} + \frac {B c^{2} x^{10}}{10} + \frac {C c^{2} x^{11}}{11} + x^{9} \left (\frac {A c^{2}}{9} + \frac {2 C b c}{9}\right ) + x^{7} \cdot \left (\frac {2 A b c}{7} + \frac {2 C a c}{7} + \frac {C b^{2}}{7}\right ) + x^{6} \left (\frac {B a c}{3} + \frac {B b^{2}}{6}\right ) + x^{5} \cdot \left (\frac {2 A a c}{5} + \frac {A b^{2}}{5} + \frac {2 C a b}{5}\right ) + x^{3} \cdot \left (\frac {2 A a b}{3} + \frac {C a^{2}}{3}\right ) \]
A*a**2*x + B*a**2*x**2/2 + B*a*b*x**4/2 + B*b*c*x**8/4 + B*c**2*x**10/10 + C*c**2*x**11/11 + x**9*(A*c**2/9 + 2*C*b*c/9) + x**7*(2*A*b*c/7 + 2*C*a*c /7 + C*b**2/7) + x**6*(B*a*c/3 + B*b**2/6) + x**5*(2*A*a*c/5 + A*b**2/5 + 2*C*a*b/5) + x**3*(2*A*a*b/3 + C*a**2/3)
Time = 0.19 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.91 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{11} \, C c^{2} x^{11} + \frac {1}{10} \, B c^{2} x^{10} + \frac {1}{4} \, B b c x^{8} + \frac {1}{9} \, {\left (2 \, C b c + A c^{2}\right )} x^{9} + \frac {1}{7} \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{7} + \frac {1}{2} \, B a b x^{4} + \frac {1}{6} \, {\left (B b^{2} + 2 \, B a c\right )} x^{6} + \frac {1}{5} \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{5} + \frac {1}{2} \, B a^{2} x^{2} + A a^{2} x + \frac {1}{3} \, {\left (C a^{2} + 2 \, A a b\right )} x^{3} \]
1/11*C*c^2*x^11 + 1/10*B*c^2*x^10 + 1/4*B*b*c*x^8 + 1/9*(2*C*b*c + A*c^2)* x^9 + 1/7*(C*b^2 + 2*(C*a + A*b)*c)*x^7 + 1/2*B*a*b*x^4 + 1/6*(B*b^2 + 2*B *a*c)*x^6 + 1/5*(2*C*a*b + A*b^2 + 2*A*a*c)*x^5 + 1/2*B*a^2*x^2 + A*a^2*x + 1/3*(C*a^2 + 2*A*a*b)*x^3
Time = 0.29 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.98 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{11} \, C c^{2} x^{11} + \frac {1}{10} \, B c^{2} x^{10} + \frac {2}{9} \, C b c x^{9} + \frac {1}{9} \, A c^{2} x^{9} + \frac {1}{4} \, B b c x^{8} + \frac {1}{7} \, C b^{2} x^{7} + \frac {2}{7} \, C a c x^{7} + \frac {2}{7} \, A b c x^{7} + \frac {1}{6} \, B b^{2} x^{6} + \frac {1}{3} \, B a c x^{6} + \frac {2}{5} \, C a b x^{5} + \frac {1}{5} \, A b^{2} x^{5} + \frac {2}{5} \, A a c x^{5} + \frac {1}{2} \, B a b x^{4} + \frac {1}{3} \, C a^{2} x^{3} + \frac {2}{3} \, A a b x^{3} + \frac {1}{2} \, B a^{2} x^{2} + A a^{2} x \]
1/11*C*c^2*x^11 + 1/10*B*c^2*x^10 + 2/9*C*b*c*x^9 + 1/9*A*c^2*x^9 + 1/4*B* b*c*x^8 + 1/7*C*b^2*x^7 + 2/7*C*a*c*x^7 + 2/7*A*b*c*x^7 + 1/6*B*b^2*x^6 + 1/3*B*a*c*x^6 + 2/5*C*a*b*x^5 + 1/5*A*b^2*x^5 + 2/5*A*a*c*x^5 + 1/2*B*a*b* x^4 + 1/3*C*a^2*x^3 + 2/3*A*a*b*x^3 + 1/2*B*a^2*x^2 + A*a^2*x
Time = 0.05 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90 \[ \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=x^3\,\left (\frac {C\,a^2}{3}+\frac {2\,A\,b\,a}{3}\right )+x^9\,\left (\frac {A\,c^2}{9}+\frac {2\,C\,b\,c}{9}\right )+x^5\,\left (\frac {A\,b^2}{5}+\frac {2\,C\,a\,b}{5}+\frac {2\,A\,a\,c}{5}\right )+x^7\,\left (\frac {C\,b^2}{7}+\frac {2\,A\,c\,b}{7}+\frac {2\,C\,a\,c}{7}\right )+\frac {B\,a^2\,x^2}{2}+\frac {B\,c^2\,x^{10}}{10}+\frac {C\,c^2\,x^{11}}{11}+\frac {B\,x^6\,\left (b^2+2\,a\,c\right )}{6}+A\,a^2\,x+\frac {B\,a\,b\,x^4}{2}+\frac {B\,b\,c\,x^8}{4} \]