Integrand size = 30, antiderivative size = 196 \[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{3} a (2 b d+a f) x^3+\frac {1}{4} a (2 b e+a g) x^4+\frac {1}{5} \left (b^2 d+2 a c d+2 a b f\right ) x^5+\frac {1}{6} \left (b^2 e+2 a c e+2 a b g\right ) x^6+\frac {1}{7} \left (2 b c d+b^2 f+2 a c f\right ) x^7+\frac {1}{8} \left (2 b c e+b^2 g+2 a c g\right ) x^8+\frac {1}{9} c (c d+2 b f) x^9+\frac {1}{10} c (c e+2 b g) x^{10}+\frac {1}{11} c^2 f x^{11}+\frac {1}{12} c^2 g x^{12} \]
a^2*d*x+1/2*a^2*e*x^2+1/3*a*(a*f+2*b*d)*x^3+1/4*a*(a*g+2*b*e)*x^4+1/5*(2*a *b*f+2*a*c*d+b^2*d)*x^5+1/6*(2*a*b*g+2*a*c*e+b^2*e)*x^6+1/7*(2*a*c*f+b^2*f +2*b*c*d)*x^7+1/8*(2*a*c*g+b^2*g+2*b*c*e)*x^8+1/9*c*(2*b*f+c*d)*x^9+1/10*c *(2*b*g+c*e)*x^10+1/11*c^2*f*x^11+1/12*c^2*g*x^12
Time = 0.04 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00 \[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{3} a (2 b d+a f) x^3+\frac {1}{4} a (2 b e+a g) x^4+\frac {1}{5} \left (b^2 d+2 a c d+2 a b f\right ) x^5+\frac {1}{6} \left (b^2 e+2 a c e+2 a b g\right ) x^6+\frac {1}{7} \left (2 b c d+b^2 f+2 a c f\right ) x^7+\frac {1}{8} \left (2 b c e+b^2 g+2 a c g\right ) x^8+\frac {1}{9} c (c d+2 b f) x^9+\frac {1}{10} c (c e+2 b g) x^{10}+\frac {1}{11} c^2 f x^{11}+\frac {1}{12} c^2 g x^{12} \]
a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*d + a*f)*x^3)/3 + (a*(2*b*e + a*g)*x^4)/ 4 + ((b^2*d + 2*a*c*d + 2*a*b*f)*x^5)/5 + ((b^2*e + 2*a*c*e + 2*a*b*g)*x^6 )/6 + ((2*b*c*d + b^2*f + 2*a*c*f)*x^7)/7 + ((2*b*c*e + b^2*g + 2*a*c*g)*x ^8)/8 + (c*(c*d + 2*b*f)*x^9)/9 + (c*(c*e + 2*b*g)*x^10)/10 + (c^2*f*x^11) /11 + (c^2*g*x^12)/12
Time = 0.42 (sec) , antiderivative size = 196, 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 = {2200, 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.
| \(\displaystyle \int \left (a+b x^2+c x^4\right )^2 \left (d+e x+f x^2+g x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2200 |
| \(\displaystyle \int \left (a^2 d+a^2 e x+x^6 \left (2 a c f+b^2 f+2 b c d\right )+x^4 \left (2 a b f+2 a c d+b^2 d\right )+x^7 \left (2 a c g+b^2 g+2 b c e\right )+x^5 \left (2 a b g+2 a c e+b^2 e\right )+a x^2 (a f+2 b d)+a x^3 (a g+2 b e)+c x^8 (2 b f+c d)+c x^9 (2 b g+c e)+c^2 f x^{10}+c^2 g x^{11}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{7} x^7 \left (2 a c f+b^2 f+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b f+2 a c d+b^2 d\right )+\frac {1}{8} x^8 \left (2 a c g+b^2 g+2 b c e\right )+\frac {1}{6} x^6 \left (2 a b g+2 a c e+b^2 e\right )+\frac {1}{3} a x^3 (a f+2 b d)+\frac {1}{4} a x^4 (a g+2 b e)+\frac {1}{9} c x^9 (2 b f+c d)+\frac {1}{10} c x^{10} (2 b g+c e)+\frac {1}{11} c^2 f x^{11}+\frac {1}{12} c^2 g x^{12}\) |
a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*d + a*f)*x^3)/3 + (a*(2*b*e + a*g)*x^4)/ 4 + ((b^2*d + 2*a*c*d + 2*a*b*f)*x^5)/5 + ((b^2*e + 2*a*c*e + 2*a*b*g)*x^6 )/6 + ((2*b*c*d + b^2*f + 2*a*c*f)*x^7)/7 + ((2*b*c*e + b^2*g + 2*a*c*g)*x ^8)/8 + (c*(c*d + 2*b*f)*x^9)/9 + (c*(c*e + 2*b*g)*x^10)/10 + (c^2*f*x^11) /11 + (c^2*g*x^12)/12
3.1.8.3.1 Defintions of rubi rules used
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[Expa ndIntegrand[Px*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c}, x] && Poly Q[Px, x] && IGtQ[p, 0]
Time = 0.15 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\frac {c^{2} g \,x^{12}}{12}+\frac {c^{2} f \,x^{11}}{11}+\frac {\left (2 g b c +e \,c^{2}\right ) x^{10}}{10}+\frac {\left (2 f b c +c^{2} d \right ) x^{9}}{9}+\frac {\left (2 e b c +g \left (2 a c +b^{2}\right )\right ) x^{8}}{8}+\frac {\left (2 b c d +f \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (e \left (2 a c +b^{2}\right )+2 a b g \right ) x^{6}}{6}+\frac {\left (d \left (2 a c +b^{2}\right )+2 a b f \right ) x^{5}}{5}+\frac {\left (g \,a^{2}+2 a b e \right ) x^{4}}{4}+\frac {\left (f \,a^{2}+2 d a b \right ) x^{3}}{3}+\frac {a^{2} e \,x^{2}}{2}+a^{2} d x\) | \(183\) |
| norman | \(\frac {c^{2} g \,x^{12}}{12}+\frac {c^{2} f \,x^{11}}{11}+\left (\frac {1}{5} g b c +\frac {1}{10} e \,c^{2}\right ) x^{10}+\left (\frac {2}{9} f b c +\frac {1}{9} c^{2} d \right ) x^{9}+\left (\frac {1}{4} a c g +\frac {1}{8} b^{2} g +\frac {1}{4} e b c \right ) x^{8}+\left (\frac {2}{7} a c f +\frac {1}{7} b^{2} f +\frac {2}{7} b c d \right ) x^{7}+\left (\frac {1}{3} a b g +\frac {1}{3} a c e +\frac {1}{6} b^{2} e \right ) x^{6}+\left (\frac {2}{5} a b f +\frac {2}{5} a c d +\frac {1}{5} b^{2} d \right ) x^{5}+\left (\frac {1}{4} g \,a^{2}+\frac {1}{2} a b e \right ) x^{4}+\left (\frac {1}{3} f \,a^{2}+\frac {2}{3} d a b \right ) x^{3}+\frac {a^{2} e \,x^{2}}{2}+a^{2} d x\) | \(183\) |
| gosper | \(\frac {1}{12} c^{2} g \,x^{12}+\frac {1}{11} c^{2} f \,x^{11}+\frac {1}{5} x^{10} g b c +\frac {1}{10} c^{2} e \,x^{10}+\frac {2}{9} x^{9} f b c +\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{4} x^{8} a c g +\frac {1}{8} x^{8} b^{2} g +\frac {1}{4} b c e \,x^{8}+\frac {2}{7} x^{7} a c f +\frac {1}{7} x^{7} b^{2} f +\frac {2}{7} x^{7} b c d +\frac {1}{3} x^{6} a b g +\frac {1}{3} x^{6} a c e +\frac {1}{6} x^{6} b^{2} e +\frac {2}{5} x^{5} a b f +\frac {2}{5} a c d \,x^{5}+\frac {1}{5} x^{5} b^{2} d +\frac {1}{4} x^{4} g \,a^{2}+\frac {1}{2} a b e \,x^{4}+\frac {1}{3} x^{3} f \,a^{2}+\frac {2}{3} x^{3} d a b +\frac {1}{2} a^{2} e \,x^{2}+a^{2} d x\) | \(203\) |
| risch | \(\frac {1}{12} c^{2} g \,x^{12}+\frac {1}{11} c^{2} f \,x^{11}+\frac {1}{5} x^{10} g b c +\frac {1}{10} c^{2} e \,x^{10}+\frac {2}{9} x^{9} f b c +\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{4} x^{8} a c g +\frac {1}{8} x^{8} b^{2} g +\frac {1}{4} b c e \,x^{8}+\frac {2}{7} x^{7} a c f +\frac {1}{7} x^{7} b^{2} f +\frac {2}{7} x^{7} b c d +\frac {1}{3} x^{6} a b g +\frac {1}{3} x^{6} a c e +\frac {1}{6} x^{6} b^{2} e +\frac {2}{5} x^{5} a b f +\frac {2}{5} a c d \,x^{5}+\frac {1}{5} x^{5} b^{2} d +\frac {1}{4} x^{4} g \,a^{2}+\frac {1}{2} a b e \,x^{4}+\frac {1}{3} x^{3} f \,a^{2}+\frac {2}{3} x^{3} d a b +\frac {1}{2} a^{2} e \,x^{2}+a^{2} d x\) | \(203\) |
| parallelrisch | \(\frac {1}{12} c^{2} g \,x^{12}+\frac {1}{11} c^{2} f \,x^{11}+\frac {1}{5} x^{10} g b c +\frac {1}{10} c^{2} e \,x^{10}+\frac {2}{9} x^{9} f b c +\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{4} x^{8} a c g +\frac {1}{8} x^{8} b^{2} g +\frac {1}{4} b c e \,x^{8}+\frac {2}{7} x^{7} a c f +\frac {1}{7} x^{7} b^{2} f +\frac {2}{7} x^{7} b c d +\frac {1}{3} x^{6} a b g +\frac {1}{3} x^{6} a c e +\frac {1}{6} x^{6} b^{2} e +\frac {2}{5} x^{5} a b f +\frac {2}{5} a c d \,x^{5}+\frac {1}{5} x^{5} b^{2} d +\frac {1}{4} x^{4} g \,a^{2}+\frac {1}{2} a b e \,x^{4}+\frac {1}{3} x^{3} f \,a^{2}+\frac {2}{3} x^{3} d a b +\frac {1}{2} a^{2} e \,x^{2}+a^{2} d x\) | \(203\) |
1/12*c^2*g*x^12+1/11*c^2*f*x^11+1/10*(2*b*c*g+c^2*e)*x^10+1/9*(2*b*c*f+c^2 *d)*x^9+1/8*(2*e*b*c+g*(2*a*c+b^2))*x^8+1/7*(2*b*c*d+f*(2*a*c+b^2))*x^7+1/ 6*(e*(2*a*c+b^2)+2*a*b*g)*x^6+1/5*(d*(2*a*c+b^2)+2*a*b*f)*x^5+1/4*(a^2*g+2 *a*b*e)*x^4+1/3*(a^2*f+2*a*b*d)*x^3+1/2*a^2*e*x^2+a^2*d*x
Time = 0.25 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.93 \[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{12} \, c^{2} g x^{12} + \frac {1}{11} \, c^{2} f x^{11} + \frac {1}{10} \, {\left (c^{2} e + 2 \, b c g\right )} x^{10} + \frac {1}{9} \, {\left (c^{2} d + 2 \, b c f\right )} x^{9} + \frac {1}{8} \, {\left (2 \, b c e + {\left (b^{2} + 2 \, a c\right )} g\right )} x^{8} + \frac {1}{7} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} f\right )} x^{7} + \frac {1}{6} \, {\left (2 \, a b g + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{6} + \frac {1}{5} \, {\left (2 \, a b f + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + \frac {1}{2} \, a^{2} e x^{2} + \frac {1}{4} \, {\left (2 \, a b e + a^{2} g\right )} x^{4} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b d + a^{2} f\right )} x^{3} \]
1/12*c^2*g*x^12 + 1/11*c^2*f*x^11 + 1/10*(c^2*e + 2*b*c*g)*x^10 + 1/9*(c^2 *d + 2*b*c*f)*x^9 + 1/8*(2*b*c*e + (b^2 + 2*a*c)*g)*x^8 + 1/7*(2*b*c*d + ( b^2 + 2*a*c)*f)*x^7 + 1/6*(2*a*b*g + (b^2 + 2*a*c)*e)*x^6 + 1/5*(2*a*b*f + (b^2 + 2*a*c)*d)*x^5 + 1/2*a^2*e*x^2 + 1/4*(2*a*b*e + a^2*g)*x^4 + a^2*d* x + 1/3*(2*a*b*d + a^2*f)*x^3
Time = 0.03 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.07 \[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^{2} d x + \frac {a^{2} e x^{2}}{2} + \frac {c^{2} f x^{11}}{11} + \frac {c^{2} g x^{12}}{12} + x^{10} \left (\frac {b c g}{5} + \frac {c^{2} e}{10}\right ) + x^{9} \cdot \left (\frac {2 b c f}{9} + \frac {c^{2} d}{9}\right ) + x^{8} \left (\frac {a c g}{4} + \frac {b^{2} g}{8} + \frac {b c e}{4}\right ) + x^{7} \cdot \left (\frac {2 a c f}{7} + \frac {b^{2} f}{7} + \frac {2 b c d}{7}\right ) + x^{6} \left (\frac {a b g}{3} + \frac {a c e}{3} + \frac {b^{2} e}{6}\right ) + x^{5} \cdot \left (\frac {2 a b f}{5} + \frac {2 a c d}{5} + \frac {b^{2} d}{5}\right ) + x^{4} \left (\frac {a^{2} g}{4} + \frac {a b e}{2}\right ) + x^{3} \left (\frac {a^{2} f}{3} + \frac {2 a b d}{3}\right ) \]
a**2*d*x + a**2*e*x**2/2 + c**2*f*x**11/11 + c**2*g*x**12/12 + x**10*(b*c* g/5 + c**2*e/10) + x**9*(2*b*c*f/9 + c**2*d/9) + x**8*(a*c*g/4 + b**2*g/8 + b*c*e/4) + x**7*(2*a*c*f/7 + b**2*f/7 + 2*b*c*d/7) + x**6*(a*b*g/3 + a*c *e/3 + b**2*e/6) + x**5*(2*a*b*f/5 + 2*a*c*d/5 + b**2*d/5) + x**4*(a**2*g/ 4 + a*b*e/2) + x**3*(a**2*f/3 + 2*a*b*d/3)
Time = 0.19 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.93 \[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{12} \, c^{2} g x^{12} + \frac {1}{11} \, c^{2} f x^{11} + \frac {1}{10} \, {\left (c^{2} e + 2 \, b c g\right )} x^{10} + \frac {1}{9} \, {\left (c^{2} d + 2 \, b c f\right )} x^{9} + \frac {1}{8} \, {\left (2 \, b c e + {\left (b^{2} + 2 \, a c\right )} g\right )} x^{8} + \frac {1}{7} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} f\right )} x^{7} + \frac {1}{6} \, {\left (2 \, a b g + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{6} + \frac {1}{5} \, {\left (2 \, a b f + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + \frac {1}{2} \, a^{2} e x^{2} + \frac {1}{4} \, {\left (2 \, a b e + a^{2} g\right )} x^{4} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b d + a^{2} f\right )} x^{3} \]
1/12*c^2*g*x^12 + 1/11*c^2*f*x^11 + 1/10*(c^2*e + 2*b*c*g)*x^10 + 1/9*(c^2 *d + 2*b*c*f)*x^9 + 1/8*(2*b*c*e + (b^2 + 2*a*c)*g)*x^8 + 1/7*(2*b*c*d + ( b^2 + 2*a*c)*f)*x^7 + 1/6*(2*a*b*g + (b^2 + 2*a*c)*e)*x^6 + 1/5*(2*a*b*f + (b^2 + 2*a*c)*d)*x^5 + 1/2*a^2*e*x^2 + 1/4*(2*a*b*e + a^2*g)*x^4 + a^2*d* x + 1/3*(2*a*b*d + a^2*f)*x^3
Time = 0.38 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.03 \[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{12} \, c^{2} g x^{12} + \frac {1}{11} \, c^{2} f x^{11} + \frac {1}{10} \, c^{2} e x^{10} + \frac {1}{5} \, b c g x^{10} + \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{9} \, b c f x^{9} + \frac {1}{4} \, b c e x^{8} + \frac {1}{8} \, b^{2} g x^{8} + \frac {1}{4} \, a c g x^{8} + \frac {2}{7} \, b c d x^{7} + \frac {1}{7} \, b^{2} f x^{7} + \frac {2}{7} \, a c f x^{7} + \frac {1}{6} \, b^{2} e x^{6} + \frac {1}{3} \, a c e x^{6} + \frac {1}{3} \, a b g x^{6} + \frac {1}{5} \, b^{2} d x^{5} + \frac {2}{5} \, a c d x^{5} + \frac {2}{5} \, a b f x^{5} + \frac {1}{2} \, a b e x^{4} + \frac {1}{4} \, a^{2} g x^{4} + \frac {2}{3} \, a b d x^{3} + \frac {1}{3} \, a^{2} f x^{3} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x \]
1/12*c^2*g*x^12 + 1/11*c^2*f*x^11 + 1/10*c^2*e*x^10 + 1/5*b*c*g*x^10 + 1/9 *c^2*d*x^9 + 2/9*b*c*f*x^9 + 1/4*b*c*e*x^8 + 1/8*b^2*g*x^8 + 1/4*a*c*g*x^8 + 2/7*b*c*d*x^7 + 1/7*b^2*f*x^7 + 2/7*a*c*f*x^7 + 1/6*b^2*e*x^6 + 1/3*a*c *e*x^6 + 1/3*a*b*g*x^6 + 1/5*b^2*d*x^5 + 2/5*a*c*d*x^5 + 2/5*a*b*f*x^5 + 1 /2*a*b*e*x^4 + 1/4*a^2*g*x^4 + 2/3*a*b*d*x^3 + 1/3*a^2*f*x^3 + 1/2*a^2*e*x ^2 + a^2*d*x
Time = 7.88 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.93 \[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^2 \, dx=x^5\,\left (\frac {d\,b^2}{5}+\frac {2\,a\,f\,b}{5}+\frac {2\,a\,c\,d}{5}\right )+x^6\,\left (\frac {e\,b^2}{6}+\frac {a\,g\,b}{3}+\frac {a\,c\,e}{3}\right )+x^7\,\left (\frac {f\,b^2}{7}+\frac {2\,c\,d\,b}{7}+\frac {2\,a\,c\,f}{7}\right )+x^8\,\left (\frac {g\,b^2}{8}+\frac {c\,e\,b}{4}+\frac {a\,c\,g}{4}\right )+x^3\,\left (\frac {f\,a^2}{3}+\frac {2\,b\,d\,a}{3}\right )+x^4\,\left (\frac {g\,a^2}{4}+\frac {b\,e\,a}{2}\right )+x^9\,\left (\frac {d\,c^2}{9}+\frac {2\,b\,f\,c}{9}\right )+x^{10}\,\left (\frac {e\,c^2}{10}+\frac {b\,g\,c}{5}\right )+\frac {a^2\,e\,x^2}{2}+\frac {c^2\,f\,x^{11}}{11}+\frac {c^2\,g\,x^{12}}{12}+a^2\,d\,x \]
x^5*((b^2*d)/5 + (2*a*c*d)/5 + (2*a*b*f)/5) + x^6*((b^2*e)/6 + (a*c*e)/3 + (a*b*g)/3) + x^7*((b^2*f)/7 + (2*b*c*d)/7 + (2*a*c*f)/7) + x^8*((b^2*g)/8 + (b*c*e)/4 + (a*c*g)/4) + x^3*((a^2*f)/3 + (2*a*b*d)/3) + x^4*((a^2*g)/4 + (a*b*e)/2) + x^9*((c^2*d)/9 + (2*b*c*f)/9) + x^10*((c^2*e)/10 + (b*c*g) /5) + (a^2*e*x^2)/2 + (c^2*f*x^11)/11 + (c^2*g*x^12)/12 + a^2*d*x