Integrand size = 61, antiderivative size = 154 \[ \int \left (a+b x^2+c x^4\right ) \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{3} a (2 b d+a f) x^3+\frac {1}{2} a b e x^4+\frac {1}{5} \left (b^2 d+2 a c d+2 a b f\right ) x^5+\frac {1}{6} \left (b^2+2 a c\right ) e x^6+\frac {1}{7} \left (2 b c d+b^2 f+2 a c f\right ) x^7+\frac {1}{4} b c e x^8+\frac {1}{9} c (c d+2 b f) x^9+\frac {1}{10} c^2 e x^{10}+\frac {1}{11} c^2 f x^{11} \] Output:
a^2*d*x+1/2*a^2*e*x^2+1/3*a*(a*f+2*b*d)*x^3+1/2*a*b*e*x^4+1/5*(2*a*b*f+2*a *c*d+b^2*d)*x^5+1/6*(2*a*c+b^2)*e*x^6+1/7*(2*a*c*f+b^2*f+2*b*c*d)*x^7+1/4* b*c*e*x^8+1/9*c*(2*b*f+c*d)*x^9+1/10*c^2*e*x^10+1/11*c^2*f*x^11
Time = 0.04 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2+c x^4\right ) \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{3} a (2 b d+a f) x^3+\frac {1}{2} a b e x^4+\frac {1}{5} \left (b^2 d+2 a c d+2 a b f\right ) x^5+\frac {1}{6} \left (b^2+2 a c\right ) e x^6+\frac {1}{7} \left (2 b c d+b^2 f+2 a c f\right ) x^7+\frac {1}{4} b c e x^8+\frac {1}{9} c (c d+2 b f) x^9+\frac {1}{10} c^2 e x^{10}+\frac {1}{11} c^2 f x^{11} \] Input:
Integrate[(a + b*x^2 + c*x^4)*(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + ( c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6),x]
Output:
a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*d + a*f)*x^3)/3 + (a*b*e*x^4)/2 + ((b^2* d + 2*a*c*d + 2*a*b*f)*x^5)/5 + ((b^2 + 2*a*c)*e*x^6)/6 + ((2*b*c*d + b^2* f + 2*a*c*f)*x^7)/7 + (b*c*e*x^8)/4 + (c*(c*d + 2*b*f)*x^9)/9 + (c^2*e*x^1 0)/10 + (c^2*f*x^11)/11
Time = 0.38 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, 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.
| \(\displaystyle \int \left (a+b x^2+c x^4\right ) \left (x^2 (a f+b d)+a d+a e x+x^4 (b f+c d)+b e x^3+c e x^5+c f x^6\right ) \, dx\) |
| \(\Big \downarrow \) 2200 |
| \(\displaystyle \int \left (a^2 d+a^2 e x+x^6 \left (2 a c f+b^2 f+2 b c d\right )+x^4 \left (2 a b f+2 a c d+b^2 d\right )+e x^5 \left (2 a c+b^2\right )+a x^2 (a f+2 b d)+2 a b e x^3+c x^8 (2 b f+c d)+2 b c e x^7+c^2 e x^9+c^2 f x^{10}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{7} x^7 \left (2 a c f+b^2 f+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b f+2 a c d+b^2 d\right )+\frac {1}{6} e x^6 \left (2 a c+b^2\right )+\frac {1}{3} a x^3 (a f+2 b d)+\frac {1}{2} a b e x^4+\frac {1}{9} c x^9 (2 b f+c d)+\frac {1}{4} b c e x^8+\frac {1}{10} c^2 e x^{10}+\frac {1}{11} c^2 f x^{11}\) |
Input:
Int[(a + b*x^2 + c*x^4)*(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6),x]
Output:
a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*d + a*f)*x^3)/3 + (a*b*e*x^4)/2 + ((b^2* d + 2*a*c*d + 2*a*b*f)*x^5)/5 + ((b^2 + 2*a*c)*e*x^6)/6 + ((2*b*c*d + b^2* f + 2*a*c*f)*x^7)/7 + (b*c*e*x^8)/4 + (c*(c*d + 2*b*f)*x^9)/9 + (c^2*e*x^1 0)/10 + (c^2*f*x^11)/11
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 = 0.13 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.92
| method | result | size |
| norman | \(\frac {c^{2} f \,x^{11}}{11}+\frac {c^{2} e \,x^{10}}{10}+\left (\frac {2}{9} b c f +\frac {1}{9} d \,c^{2}\right ) x^{9}+\frac {b c e \,x^{8}}{4}+\left (\frac {2}{7} a c f +\frac {1}{7} b^{2} f +\frac {2}{7} b c d \right ) x^{7}+\left (\frac {1}{3} a c e +\frac {1}{6} b^{2} e \right ) x^{6}+\left (\frac {2}{5} a b f +\frac {2}{5} d a c +\frac {1}{5} b^{2} d \right ) x^{5}+\frac {a b e \,x^{4}}{2}+\left (\frac {1}{3} f \,a^{2}+\frac {2}{3} a b d \right ) x^{3}+\frac {a^{2} e \,x^{2}}{2}+a^{2} d x\) | \(141\) |
| risch | \(\frac {1}{11} c^{2} f \,x^{11}+\frac {1}{10} c^{2} e \,x^{10}+\frac {2}{9} x^{9} b c f +\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{4} b c e \,x^{8}+\frac {2}{7} x^{7} a c f +\frac {1}{7} x^{7} b^{2} f +\frac {2}{7} x^{7} b c d +\frac {1}{3} a c e \,x^{6}+\frac {1}{6} x^{6} b^{2} e +\frac {2}{5} x^{5} a b f +\frac {2}{5} a c d \,x^{5}+\frac {1}{5} x^{5} b^{2} d +\frac {1}{2} a b e \,x^{4}+\frac {1}{3} x^{3} f \,a^{2}+\frac {2}{3} x^{3} a b d +\frac {1}{2} a^{2} e \,x^{2}+a^{2} d x\) | \(152\) |
| parallelrisch | \(\frac {1}{11} c^{2} f \,x^{11}+\frac {1}{10} c^{2} e \,x^{10}+\frac {2}{9} x^{9} b c f +\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{4} b c e \,x^{8}+\frac {2}{7} x^{7} a c f +\frac {1}{7} x^{7} b^{2} f +\frac {2}{7} x^{7} b c d +\frac {1}{3} a c e \,x^{6}+\frac {1}{6} x^{6} b^{2} e +\frac {2}{5} x^{5} a b f +\frac {2}{5} a c d \,x^{5}+\frac {1}{5} x^{5} b^{2} d +\frac {1}{2} a b e \,x^{4}+\frac {1}{3} x^{3} f \,a^{2}+\frac {2}{3} x^{3} a b d +\frac {1}{2} a^{2} e \,x^{2}+a^{2} d x\) | \(152\) |
| gosper | \(\frac {x \left (1260 f \,c^{2} x^{10}+1386 c^{2} e \,x^{9}+3080 x^{8} b c f +1540 x^{8} d \,c^{2}+3465 b c e \,x^{7}+3960 a c f \,x^{6}+1980 x^{6} b^{2} f +3960 b c d \,x^{6}+4620 a c e \,x^{5}+2310 x^{5} b^{2} e +5544 x^{4} a b f +5544 a c d \,x^{4}+2772 b^{2} d \,x^{4}+6930 a b e \,x^{3}+4620 x^{2} f \,a^{2}+9240 a b d \,x^{2}+6930 a^{2} e x +13860 a^{2} d \right )}{13860}\) | \(153\) |
| default | \(\frac {c^{2} f \,x^{11}}{11}+\frac {c^{2} e \,x^{10}}{10}+\frac {\left (b c f +c \left (b f +c d \right )\right ) x^{9}}{9}+\frac {b c e \,x^{8}}{4}+\frac {\left (a c f +b \left (b f +c d \right )+c \left (a f +b d \right )\right ) x^{7}}{7}+\frac {\left (2 a c e +b^{2} e \right ) x^{6}}{6}+\frac {\left (a \left (b f +c d \right )+b \left (a f +b d \right )+d a c \right ) x^{5}}{5}+\frac {a b e \,x^{4}}{2}+\frac {\left (a \left (a f +b d \right )+a b d \right ) x^{3}}{3}+\frac {a^{2} e \,x^{2}}{2}+a^{2} d x\) | \(161\) |
| orering | \(\frac {x \left (1260 f \,c^{2} x^{10}+1386 c^{2} e \,x^{9}+3080 x^{8} b c f +1540 x^{8} d \,c^{2}+3465 b c e \,x^{7}+3960 a c f \,x^{6}+1980 x^{6} b^{2} f +3960 b c d \,x^{6}+4620 a c e \,x^{5}+2310 x^{5} b^{2} e +5544 x^{4} a b f +5544 a c d \,x^{4}+2772 b^{2} d \,x^{4}+6930 a b e \,x^{3}+4620 x^{2} f \,a^{2}+9240 a b d \,x^{2}+6930 a^{2} e x +13860 a^{2} d \right ) \left (a d +a e x +\left (a f +b d \right ) x^{2}+b e \,x^{3}+\left (b f +c d \right ) x^{4}+c e \,x^{5}+c f \,x^{6}\right )}{13860 \left (f \,x^{2}+e x +d \right ) \left (c \,x^{4}+b \,x^{2}+a \right )}\) | \(227\) |
Input:
int((c*x^4+b*x^2+a)*(a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5 +c*f*x^6),x,method=_RETURNVERBOSE)
Output:
1/11*c^2*f*x^11+1/10*c^2*e*x^10+(2/9*b*c*f+1/9*d*c^2)*x^9+1/4*b*c*e*x^8+(2 /7*a*c*f+1/7*b^2*f+2/7*b*c*d)*x^7+(1/3*a*c*e+1/6*b^2*e)*x^6+(2/5*a*b*f+2/5 *d*a*c+1/5*b^2*d)*x^5+1/2*a*b*e*x^4+(1/3*f*a^2+2/3*a*b*d)*x^3+1/2*a^2*e*x^ 2+a^2*d*x
Time = 0.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90 \[ \int \left (a+b x^2+c x^4\right ) \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=\frac {1}{11} \, c^{2} f x^{11} + \frac {1}{10} \, c^{2} e x^{10} + \frac {1}{4} \, b c e x^{8} + \frac {1}{9} \, {\left (c^{2} d + 2 \, b c f\right )} x^{9} + \frac {1}{6} \, {\left (b^{2} + 2 \, a c\right )} e x^{6} + \frac {1}{7} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} f\right )} x^{7} + \frac {1}{2} \, a b e x^{4} + \frac {1}{5} \, {\left (2 \, a b f + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b d + a^{2} f\right )} x^{3} \] Input:
integrate((c*x^4+b*x^2+a)*(a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c *e*x^5+c*f*x^6),x, algorithm="fricas")
Output:
1/11*c^2*f*x^11 + 1/10*c^2*e*x^10 + 1/4*b*c*e*x^8 + 1/9*(c^2*d + 2*b*c*f)* x^9 + 1/6*(b^2 + 2*a*c)*e*x^6 + 1/7*(2*b*c*d + (b^2 + 2*a*c)*f)*x^7 + 1/2* a*b*e*x^4 + 1/5*(2*a*b*f + (b^2 + 2*a*c)*d)*x^5 + 1/2*a^2*e*x^2 + a^2*d*x + 1/3*(2*a*b*d + a^2*f)*x^3
Time = 0.02 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.07 \[ \int \left (a+b x^2+c x^4\right ) \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=a^{2} d x + \frac {a^{2} e x^{2}}{2} + \frac {a b e x^{4}}{2} + \frac {b c e x^{8}}{4} + \frac {c^{2} e x^{10}}{10} + \frac {c^{2} f x^{11}}{11} + x^{9} \cdot \left (\frac {2 b c f}{9} + \frac {c^{2} d}{9}\right ) + x^{7} \cdot \left (\frac {2 a c f}{7} + \frac {b^{2} f}{7} + \frac {2 b c d}{7}\right ) + x^{6} \left (\frac {a c e}{3} + \frac {b^{2} e}{6}\right ) + x^{5} \cdot \left (\frac {2 a b f}{5} + \frac {2 a c d}{5} + \frac {b^{2} d}{5}\right ) + x^{3} \left (\frac {a^{2} f}{3} + \frac {2 a b d}{3}\right ) \] Input:
integrate((c*x**4+b*x**2+a)*(a*d+a*e*x+(a*f+b*d)*x**2+b*e*x**3+(b*f+c*d)*x **4+c*e*x**5+c*f*x**6),x)
Output:
a**2*d*x + a**2*e*x**2/2 + a*b*e*x**4/2 + b*c*e*x**8/4 + c**2*e*x**10/10 + c**2*f*x**11/11 + x**9*(2*b*c*f/9 + c**2*d/9) + x**7*(2*a*c*f/7 + b**2*f/ 7 + 2*b*c*d/7) + x**6*(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**3*(a**2*f/3 + 2*a*b*d/3)
Time = 0.04 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90 \[ \int \left (a+b x^2+c x^4\right ) \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=\frac {1}{11} \, c^{2} f x^{11} + \frac {1}{10} \, c^{2} e x^{10} + \frac {1}{4} \, b c e x^{8} + \frac {1}{9} \, {\left (c^{2} d + 2 \, b c f\right )} x^{9} + \frac {1}{6} \, {\left (b^{2} + 2 \, a c\right )} e x^{6} + \frac {1}{7} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} f\right )} x^{7} + \frac {1}{2} \, a b e x^{4} + \frac {1}{5} \, {\left (2 \, a b f + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b d + a^{2} f\right )} x^{3} \] Input:
integrate((c*x^4+b*x^2+a)*(a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c *e*x^5+c*f*x^6),x, algorithm="maxima")
Output:
1/11*c^2*f*x^11 + 1/10*c^2*e*x^10 + 1/4*b*c*e*x^8 + 1/9*(c^2*d + 2*b*c*f)* x^9 + 1/6*(b^2 + 2*a*c)*e*x^6 + 1/7*(2*b*c*d + (b^2 + 2*a*c)*f)*x^7 + 1/2* a*b*e*x^4 + 1/5*(2*a*b*f + (b^2 + 2*a*c)*d)*x^5 + 1/2*a^2*e*x^2 + a^2*d*x + 1/3*(2*a*b*d + a^2*f)*x^3
Time = 0.15 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.98 \[ \int \left (a+b x^2+c x^4\right ) \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=\frac {1}{11} \, c^{2} f x^{11} + \frac {1}{10} \, c^{2} e x^{10} + \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{9} \, b c f x^{9} + \frac {1}{4} \, b c e x^{8} + \frac {2}{7} \, b c d x^{7} + \frac {1}{7} \, b^{2} f x^{7} + \frac {2}{7} \, a c f x^{7} + \frac {1}{6} \, b^{2} e x^{6} + \frac {1}{3} \, a c e x^{6} + \frac {1}{5} \, b^{2} d x^{5} + \frac {2}{5} \, a c d x^{5} + \frac {2}{5} \, a b f x^{5} + \frac {1}{2} \, a b e x^{4} + \frac {2}{3} \, a b d x^{3} + \frac {1}{3} \, a^{2} f x^{3} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x \] Input:
integrate((c*x^4+b*x^2+a)*(a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c *e*x^5+c*f*x^6),x, algorithm="giac")
Output:
1/11*c^2*f*x^11 + 1/10*c^2*e*x^10 + 1/9*c^2*d*x^9 + 2/9*b*c*f*x^9 + 1/4*b* c*e*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 + 1/5*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 + 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 = 17.98 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90 \[ \int \left (a+b x^2+c x^4\right ) \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=x^5\,\left (\frac {d\,b^2}{5}+\frac {2\,a\,f\,b}{5}+\frac {2\,a\,c\,d}{5}\right )+x^7\,\left (\frac {f\,b^2}{7}+\frac {2\,c\,d\,b}{7}+\frac {2\,a\,c\,f}{7}\right )+x^3\,\left (\frac {f\,a^2}{3}+\frac {2\,b\,d\,a}{3}\right )+x^9\,\left (\frac {d\,c^2}{9}+\frac {2\,b\,f\,c}{9}\right )+\frac {a^2\,e\,x^2}{2}+\frac {c^2\,e\,x^{10}}{10}+\frac {c^2\,f\,x^{11}}{11}+\frac {e\,x^6\,\left (b^2+2\,a\,c\right )}{6}+a^2\,d\,x+\frac {a\,b\,e\,x^4}{2}+\frac {b\,c\,e\,x^8}{4} \] Input:
int((a + b*x^2 + c*x^4)*(a*d + x^2*(b*d + a*f) + x^4*(c*d + b*f) + a*e*x + b*e*x^3 + c*e*x^5 + c*f*x^6),x)
Output:
x^5*((b^2*d)/5 + (2*a*c*d)/5 + (2*a*b*f)/5) + x^7*((b^2*f)/7 + (2*b*c*d)/7 + (2*a*c*f)/7) + x^3*((a^2*f)/3 + (2*a*b*d)/3) + x^9*((c^2*d)/9 + (2*b*c* f)/9) + (a^2*e*x^2)/2 + (c^2*e*x^10)/10 + (c^2*f*x^11)/11 + (e*x^6*(2*a*c + b^2))/6 + a^2*d*x + (a*b*e*x^4)/2 + (b*c*e*x^8)/4
Time = 0.17 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.99 \[ \int \left (a+b x^2+c x^4\right ) \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx=\frac {x \left (1260 c^{2} f \,x^{10}+1386 c^{2} e \,x^{9}+3080 b c f \,x^{8}+1540 c^{2} d \,x^{8}+3465 b c e \,x^{7}+3960 a c f \,x^{6}+1980 b^{2} f \,x^{6}+3960 b c d \,x^{6}+4620 a c e \,x^{5}+2310 b^{2} e \,x^{5}+5544 a b f \,x^{4}+5544 a c d \,x^{4}+2772 b^{2} d \,x^{4}+6930 a b e \,x^{3}+4620 a^{2} f \,x^{2}+9240 a b d \,x^{2}+6930 a^{2} e x +13860 a^{2} d \right )}{13860} \] Input:
int((c*x^4+b*x^2+a)*(a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5 +c*f*x^6),x)
Output:
(x*(13860*a**2*d + 6930*a**2*e*x + 4620*a**2*f*x**2 + 9240*a*b*d*x**2 + 69 30*a*b*e*x**3 + 5544*a*b*f*x**4 + 5544*a*c*d*x**4 + 4620*a*c*e*x**5 + 3960 *a*c*f*x**6 + 2772*b**2*d*x**4 + 2310*b**2*e*x**5 + 1980*b**2*f*x**6 + 396 0*b*c*d*x**6 + 3465*b*c*e*x**7 + 3080*b*c*f*x**8 + 1540*c**2*d*x**8 + 1386 *c**2*e*x**9 + 1260*c**2*f*x**10))/13860