Integrand size = 46, antiderivative size = 193 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=\frac {(b c-a d)^2 (b e-a f)^2 (a+b x)^3}{3 b^5}+\frac {(b c-a d) (b e-a f) (b d e+b c f-2 a d f) (a+b x)^4}{2 b^5}+\frac {\left (6 a^2 d^2 f^2-6 a b d f (d e+c f)+b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) (a+b x)^5}{5 b^5}+\frac {d f (b d e+b c f-2 a d f) (a+b x)^6}{3 b^5}+\frac {d^2 f^2 (a+b x)^7}{7 b^5} \]
1/3*(-a*d+b*c)^2*(-a*f+b*e)^2*(b*x+a)^3/b^5+1/2*(-a*d+b*c)*(-a*f+b*e)*(-2* a*d*f+b*c*f+b*d*e)*(b*x+a)^4/b^5+1/5*(6*a^2*d^2*f^2-6*a*b*d*f*(c*f+d*e)+b^ 2*(c^2*f^2+4*c*d*e*f+d^2*e^2))*(b*x+a)^5/b^5+1/3*d*f*(-2*a*d*f+b*c*f+b*d*e )*(b*x+a)^6/b^5+1/7*d^2*f^2*(b*x+a)^7/b^5
Time = 0.06 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.25 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=a^2 c^2 e^2 x+a c e (b c e+a d e+a c f) x^2+\frac {1}{3} \left (b^2 c^2 e^2+4 a b c e (d e+c f)+a^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^3+\frac {1}{2} \left (b^2 c e (d e+c f)+a^2 d f (d e+c f)+a b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^4+\frac {1}{5} \left (a^2 d^2 f^2+4 a b d f (d e+c f)+b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{3} b d f (b d e+b c f+a d f) x^6+\frac {1}{7} b^2 d^2 f^2 x^7 \]
a^2*c^2*e^2*x + a*c*e*(b*c*e + a*d*e + a*c*f)*x^2 + ((b^2*c^2*e^2 + 4*a*b* c*e*(d*e + c*f) + a^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^3)/3 + ((b^2*c*e* (d*e + c*f) + a^2*d*f*(d*e + c*f) + a*b*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x ^4)/2 + ((a^2*d^2*f^2 + 4*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^5)/5 + (b*d*f*(b*d*e + b*c*f + a*d*f)*x^6)/3 + (b^2*d^2*f^2*x ^7)/7
Time = 0.50 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2464, 99, 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.
| \(\displaystyle \int \left (x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3\right )^2 \, dx\) |
| \(\Big \downarrow \) 2464 |
| \(\displaystyle \int (a+b x)^2 (c+d x)^2 (e+f x)^2dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {(a+b x)^4 \left (6 a^2 d^2 f^2-6 a b d f (c f+d e)+b^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )}{b^4}+\frac {2 d f (a+b x)^5 (-2 a d f+b c f+b d e)}{b^4}+\frac {2 (a+b x)^3 (b c-a d) (b e-a f) (-2 a d f+b c f+b d e)}{b^4}+\frac {(a+b x)^2 (b c-a d)^2 (b e-a f)^2}{b^4}+\frac {d^2 f^2 (a+b x)^6}{b^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b x)^5 \left (6 a^2 d^2 f^2-6 a b d f (c f+d e)+b^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )}{5 b^5}+\frac {d f (a+b x)^6 (-2 a d f+b c f+b d e)}{3 b^5}+\frac {(a+b x)^4 (b c-a d) (b e-a f) (-2 a d f+b c f+b d e)}{2 b^5}+\frac {(a+b x)^3 (b c-a d)^2 (b e-a f)^2}{3 b^5}+\frac {d^2 f^2 (a+b x)^7}{7 b^5}\) |
((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^3)/(3*b^5) + ((b*c - a*d)*(b*e - a* f)*(b*d*e + b*c*f - 2*a*d*f)*(a + b*x)^4)/(2*b^5) + ((6*a^2*d^2*f^2 - 6*a* b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*(a + b*x)^5)/(5*b ^5) + (d*f*(b*d*e + b*c*f - 2*a*d*f)*(a + b*x)^6)/(3*b^5) + (d^2*f^2*(a + b*x)^7)/(7*b^5)
3.1.16.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[u*Qx^p, x] / ; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && IGtQ[p, 1]
Time = 0.04 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(\frac {d^{2} f^{2} b^{2} x^{7}}{7}+\frac {\left (a d f +f b c +b d e \right ) b d f \,x^{6}}{3}+\frac {\left (2 \left (a c f +e d a +e b c \right ) b d f +\left (a d f +f b c +b d e \right )^{2}\right ) x^{5}}{5}+\frac {\left (2 a c e b d f +2 \left (a c f +e d a +e b c \right ) \left (a d f +f b c +b d e \right )\right ) x^{4}}{4}+\frac {\left (2 a c e \left (a d f +f b c +b d e \right )+\left (a c f +e d a +e b c \right )^{2}\right ) x^{3}}{3}+a c e \left (a c f +e d a +e b c \right ) x^{2}+a^{2} c^{2} e^{2} x\) | \(188\) |
| norman | \(\frac {d^{2} f^{2} b^{2} x^{7}}{7}+\left (\frac {1}{3} d^{2} f^{2} a b +\frac {1}{3} b^{2} c d \,f^{2}+\frac {1}{3} b^{2} d^{2} e f \right ) x^{6}+\left (\frac {1}{5} a^{2} d^{2} f^{2}+\frac {4}{5} a b c d \,f^{2}+\frac {4}{5} a b \,d^{2} e f +\frac {1}{5} b^{2} c^{2} f^{2}+\frac {4}{5} b^{2} c d e f +\frac {1}{5} b^{2} d^{2} e^{2}\right ) x^{5}+\left (\frac {1}{2} a^{2} c d \,f^{2}+\frac {1}{2} a^{2} d^{2} e f +\frac {1}{2} a b \,c^{2} f^{2}+2 a c e b d f +\frac {1}{2} a b \,d^{2} e^{2}+\frac {1}{2} b^{2} c^{2} e f +\frac {1}{2} b^{2} c d \,e^{2}\right ) x^{4}+\left (\frac {1}{3} a^{2} c^{2} f^{2}+\frac {4}{3} a^{2} c d e f +\frac {1}{3} a^{2} d^{2} e^{2}+\frac {4}{3} a b \,c^{2} e f +\frac {4}{3} a b c d \,e^{2}+\frac {1}{3} b^{2} c^{2} e^{2}\right ) x^{3}+\left (a^{2} c^{2} e f +a^{2} c d \,e^{2}+e^{2} c^{2} a b \right ) x^{2}+a^{2} c^{2} e^{2} x\) | \(297\) |
| gosper | \(\frac {x \left (30 d^{2} f^{2} b^{2} x^{6}+70 x^{5} d^{2} f^{2} a b +70 x^{5} b^{2} c d \,f^{2}+70 x^{5} b^{2} d^{2} e f +42 x^{4} a^{2} d^{2} f^{2}+168 x^{4} a b c d \,f^{2}+168 x^{4} a b \,d^{2} e f +42 x^{4} b^{2} c^{2} f^{2}+168 x^{4} b^{2} c d e f +42 x^{4} b^{2} d^{2} e^{2}+105 x^{3} a^{2} c d \,f^{2}+105 x^{3} a^{2} d^{2} e f +105 x^{3} a b \,c^{2} f^{2}+420 x^{3} a c e b d f +105 x^{3} a b \,d^{2} e^{2}+105 x^{3} b^{2} c^{2} e f +105 x^{3} b^{2} c d \,e^{2}+70 x^{2} a^{2} c^{2} f^{2}+280 x^{2} a^{2} c d e f +70 x^{2} a^{2} d^{2} e^{2}+280 x^{2} a b \,c^{2} e f +280 x^{2} a b c d \,e^{2}+70 x^{2} b^{2} c^{2} e^{2}+210 a^{2} c^{2} e f x +210 a^{2} c d \,e^{2} x +210 a b \,c^{2} e^{2} x +210 a^{2} c^{2} e^{2}\right )}{210}\) | \(347\) |
| risch | \(\frac {4}{5} x^{5} a b c d \,f^{2}+\frac {4}{5} x^{5} a b \,d^{2} e f +\frac {4}{5} x^{5} b^{2} c d e f +\frac {4}{3} x^{3} a^{2} c d e f +\frac {4}{3} x^{3} a b \,c^{2} e f +\frac {4}{3} x^{3} a b c d \,e^{2}+a^{2} c^{2} e^{2} x +\frac {1}{7} d^{2} f^{2} b^{2} x^{7}+2 x^{4} a c e b d f +\frac {1}{2} x^{4} a^{2} d^{2} e f +a^{2} c^{2} e f \,x^{2}+\frac {1}{2} x^{4} a b \,c^{2} f^{2}+\frac {1}{2} x^{4} a b \,d^{2} e^{2}+\frac {1}{2} x^{4} b^{2} c^{2} e f +\frac {1}{2} x^{4} b^{2} c d \,e^{2}+a^{2} c d \,e^{2} x^{2}+a b \,c^{2} e^{2} x^{2}+\frac {1}{3} a b \,d^{2} f^{2} x^{6}+\frac {1}{3} b^{2} c d \,f^{2} x^{6}+\frac {1}{3} b^{2} d^{2} e f \,x^{6}+\frac {1}{2} x^{4} a^{2} c d \,f^{2}+\frac {1}{5} x^{5} a^{2} d^{2} f^{2}+\frac {1}{5} x^{5} b^{2} c^{2} f^{2}+\frac {1}{5} x^{5} b^{2} d^{2} e^{2}+\frac {1}{3} x^{3} a^{2} c^{2} f^{2}+\frac {1}{3} x^{3} a^{2} d^{2} e^{2}+\frac {1}{3} x^{3} b^{2} c^{2} e^{2}\) | \(347\) |
| parallelrisch | \(\frac {4}{5} x^{5} a b c d \,f^{2}+\frac {4}{5} x^{5} a b \,d^{2} e f +\frac {4}{5} x^{5} b^{2} c d e f +\frac {4}{3} x^{3} a^{2} c d e f +\frac {4}{3} x^{3} a b \,c^{2} e f +\frac {4}{3} x^{3} a b c d \,e^{2}+a^{2} c^{2} e^{2} x +\frac {1}{7} d^{2} f^{2} b^{2} x^{7}+2 x^{4} a c e b d f +\frac {1}{2} x^{4} a^{2} d^{2} e f +a^{2} c^{2} e f \,x^{2}+\frac {1}{2} x^{4} a b \,c^{2} f^{2}+\frac {1}{2} x^{4} a b \,d^{2} e^{2}+\frac {1}{2} x^{4} b^{2} c^{2} e f +\frac {1}{2} x^{4} b^{2} c d \,e^{2}+a^{2} c d \,e^{2} x^{2}+a b \,c^{2} e^{2} x^{2}+\frac {1}{3} a b \,d^{2} f^{2} x^{6}+\frac {1}{3} b^{2} c d \,f^{2} x^{6}+\frac {1}{3} b^{2} d^{2} e f \,x^{6}+\frac {1}{2} x^{4} a^{2} c d \,f^{2}+\frac {1}{5} x^{5} a^{2} d^{2} f^{2}+\frac {1}{5} x^{5} b^{2} c^{2} f^{2}+\frac {1}{5} x^{5} b^{2} d^{2} e^{2}+\frac {1}{3} x^{3} a^{2} c^{2} f^{2}+\frac {1}{3} x^{3} a^{2} d^{2} e^{2}+\frac {1}{3} x^{3} b^{2} c^{2} e^{2}\) | \(347\) |
int((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^2,x,me thod=_RETURNVERBOSE)
1/7*d^2*f^2*b^2*x^7+1/3*(a*d*f+b*c*f+b*d*e)*b*d*f*x^6+1/5*(2*(a*c*f+a*d*e+ b*c*e)*b*d*f+(a*d*f+b*c*f+b*d*e)^2)*x^5+1/4*(2*a*c*e*b*d*f+2*(a*c*f+a*d*e+ b*c*e)*(a*d*f+b*c*f+b*d*e))*x^4+1/3*(2*a*c*e*(a*d*f+b*c*f+b*d*e)+(a*c*f+a* d*e+b*c*e)^2)*x^3+a*c*e*(a*c*f+a*d*e+b*c*e)*x^2+a^2*c^2*e^2*x
Time = 0.23 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.39 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=\frac {1}{7} \, b^{2} d^{2} f^{2} x^{7} + a^{2} c^{2} e^{2} x + \frac {1}{3} \, {\left (b^{2} d^{2} e f + {\left (b^{2} c d + a b d^{2}\right )} f^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{2} d^{2} e^{2} + 4 \, {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2}\right )} x^{5} + \frac {1}{2} \, {\left ({\left (b^{2} c d + a b d^{2}\right )} e^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e f + {\left (a b c^{2} + a^{2} c d\right )} f^{2}\right )} x^{4} + \frac {1}{3} \, {\left (a^{2} c^{2} f^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e^{2} + 4 \, {\left (a b c^{2} + a^{2} c d\right )} e f\right )} x^{3} + {\left (a^{2} c^{2} e f + {\left (a b c^{2} + a^{2} c d\right )} e^{2}\right )} x^{2} \]
integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^ 2,x, algorithm="fricas")
1/7*b^2*d^2*f^2*x^7 + a^2*c^2*e^2*x + 1/3*(b^2*d^2*e*f + (b^2*c*d + a*b*d^ 2)*f^2)*x^6 + 1/5*(b^2*d^2*e^2 + 4*(b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 + 4* a*b*c*d + a^2*d^2)*f^2)*x^5 + 1/2*((b^2*c*d + a*b*d^2)*e^2 + (b^2*c^2 + 4* a*b*c*d + a^2*d^2)*e*f + (a*b*c^2 + a^2*c*d)*f^2)*x^4 + 1/3*(a^2*c^2*f^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2 + 4*(a*b*c^2 + a^2*c*d)*e*f)*x^3 + (a ^2*c^2*e*f + (a*b*c^2 + a^2*c*d)*e^2)*x^2
Time = 0.04 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.79 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=a^{2} c^{2} e^{2} x + \frac {b^{2} d^{2} f^{2} x^{7}}{7} + x^{6} \left (\frac {a b d^{2} f^{2}}{3} + \frac {b^{2} c d f^{2}}{3} + \frac {b^{2} d^{2} e f}{3}\right ) + x^{5} \left (\frac {a^{2} d^{2} f^{2}}{5} + \frac {4 a b c d f^{2}}{5} + \frac {4 a b d^{2} e f}{5} + \frac {b^{2} c^{2} f^{2}}{5} + \frac {4 b^{2} c d e f}{5} + \frac {b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (\frac {a^{2} c d f^{2}}{2} + \frac {a^{2} d^{2} e f}{2} + \frac {a b c^{2} f^{2}}{2} + 2 a b c d e f + \frac {a b d^{2} e^{2}}{2} + \frac {b^{2} c^{2} e f}{2} + \frac {b^{2} c d e^{2}}{2}\right ) + x^{3} \left (\frac {a^{2} c^{2} f^{2}}{3} + \frac {4 a^{2} c d e f}{3} + \frac {a^{2} d^{2} e^{2}}{3} + \frac {4 a b c^{2} e f}{3} + \frac {4 a b c d e^{2}}{3} + \frac {b^{2} c^{2} e^{2}}{3}\right ) + x^{2} \left (a^{2} c^{2} e f + a^{2} c d e^{2} + a b c^{2} e^{2}\right ) \]
a**2*c**2*e**2*x + b**2*d**2*f**2*x**7/7 + x**6*(a*b*d**2*f**2/3 + b**2*c* d*f**2/3 + b**2*d**2*e*f/3) + x**5*(a**2*d**2*f**2/5 + 4*a*b*c*d*f**2/5 + 4*a*b*d**2*e*f/5 + b**2*c**2*f**2/5 + 4*b**2*c*d*e*f/5 + b**2*d**2*e**2/5) + x**4*(a**2*c*d*f**2/2 + a**2*d**2*e*f/2 + a*b*c**2*f**2/2 + 2*a*b*c*d*e *f + a*b*d**2*e**2/2 + b**2*c**2*e*f/2 + b**2*c*d*e**2/2) + x**3*(a**2*c** 2*f**2/3 + 4*a**2*c*d*e*f/3 + a**2*d**2*e**2/3 + 4*a*b*c**2*e*f/3 + 4*a*b* c*d*e**2/3 + b**2*c**2*e**2/3) + x**2*(a**2*c**2*e*f + a**2*c*d*e**2 + a*b *c**2*e**2)
Time = 0.21 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.93 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=\frac {1}{7} \, b^{2} d^{2} f^{2} x^{7} + \frac {1}{3} \, {\left (b d e + b c f + a d f\right )} b d f x^{6} + a^{2} c^{2} e^{2} x + \frac {1}{5} \, {\left (b d e + b c f + a d f\right )}^{2} x^{5} + \frac {1}{3} \, {\left (b c e + a d e + a c f\right )}^{2} x^{3} + \frac {1}{6} \, {\left (3 \, b d f x^{4} + 4 \, {\left (b d e + b c f + a d f\right )} x^{3} + 6 \, {\left (b c e + a d e + a c f\right )} x^{2}\right )} a c e + \frac {1}{10} \, {\left (4 \, b d f x^{5} + 5 \, {\left (b d e + {\left (b c + a d\right )} f\right )} x^{4}\right )} {\left (b c e + a d e + a c f\right )} \]
integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^ 2,x, algorithm="maxima")
1/7*b^2*d^2*f^2*x^7 + 1/3*(b*d*e + b*c*f + a*d*f)*b*d*f*x^6 + a^2*c^2*e^2* x + 1/5*(b*d*e + b*c*f + a*d*f)^2*x^5 + 1/3*(b*c*e + a*d*e + a*c*f)^2*x^3 + 1/6*(3*b*d*f*x^4 + 4*(b*d*e + b*c*f + a*d*f)*x^3 + 6*(b*c*e + a*d*e + a* c*f)*x^2)*a*c*e + 1/10*(4*b*d*f*x^5 + 5*(b*d*e + (b*c + a*d)*f)*x^4)*(b*c* e + a*d*e + a*c*f)
Time = 0.35 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.79 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=\frac {1}{7} \, b^{2} d^{2} f^{2} x^{7} + \frac {1}{3} \, b^{2} d^{2} e f x^{6} + \frac {1}{3} \, b^{2} c d f^{2} x^{6} + \frac {1}{3} \, a b d^{2} f^{2} x^{6} + \frac {1}{5} \, b^{2} d^{2} e^{2} x^{5} + \frac {4}{5} \, b^{2} c d e f x^{5} + \frac {4}{5} \, a b d^{2} e f x^{5} + \frac {1}{5} \, b^{2} c^{2} f^{2} x^{5} + \frac {4}{5} \, a b c d f^{2} x^{5} + \frac {1}{5} \, a^{2} d^{2} f^{2} x^{5} + \frac {1}{2} \, b^{2} c d e^{2} x^{4} + \frac {1}{2} \, a b d^{2} e^{2} x^{4} + \frac {1}{2} \, b^{2} c^{2} e f x^{4} + 2 \, a b c d e f x^{4} + \frac {1}{2} \, a^{2} d^{2} e f x^{4} + \frac {1}{2} \, a b c^{2} f^{2} x^{4} + \frac {1}{2} \, a^{2} c d f^{2} x^{4} + \frac {1}{3} \, b^{2} c^{2} e^{2} x^{3} + \frac {4}{3} \, a b c d e^{2} x^{3} + \frac {1}{3} \, a^{2} d^{2} e^{2} x^{3} + \frac {4}{3} \, a b c^{2} e f x^{3} + \frac {4}{3} \, a^{2} c d e f x^{3} + \frac {1}{3} \, a^{2} c^{2} f^{2} x^{3} + a b c^{2} e^{2} x^{2} + a^{2} c d e^{2} x^{2} + a^{2} c^{2} e f x^{2} + a^{2} c^{2} e^{2} x \]
integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^ 2,x, algorithm="giac")
1/7*b^2*d^2*f^2*x^7 + 1/3*b^2*d^2*e*f*x^6 + 1/3*b^2*c*d*f^2*x^6 + 1/3*a*b* d^2*f^2*x^6 + 1/5*b^2*d^2*e^2*x^5 + 4/5*b^2*c*d*e*f*x^5 + 4/5*a*b*d^2*e*f* x^5 + 1/5*b^2*c^2*f^2*x^5 + 4/5*a*b*c*d*f^2*x^5 + 1/5*a^2*d^2*f^2*x^5 + 1/ 2*b^2*c*d*e^2*x^4 + 1/2*a*b*d^2*e^2*x^4 + 1/2*b^2*c^2*e*f*x^4 + 2*a*b*c*d* e*f*x^4 + 1/2*a^2*d^2*e*f*x^4 + 1/2*a*b*c^2*f^2*x^4 + 1/2*a^2*c*d*f^2*x^4 + 1/3*b^2*c^2*e^2*x^3 + 4/3*a*b*c*d*e^2*x^3 + 1/3*a^2*d^2*e^2*x^3 + 4/3*a* b*c^2*e*f*x^3 + 4/3*a^2*c*d*e*f*x^3 + 1/3*a^2*c^2*f^2*x^3 + a*b*c^2*e^2*x^ 2 + a^2*c*d*e^2*x^2 + a^2*c^2*e*f*x^2 + a^2*c^2*e^2*x
Time = 0.05 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.40 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=x^4\,\left (\frac {a^2\,c\,d\,f^2}{2}+\frac {a^2\,d^2\,e\,f}{2}+\frac {a\,b\,c^2\,f^2}{2}+2\,a\,b\,c\,d\,e\,f+\frac {a\,b\,d^2\,e^2}{2}+\frac {b^2\,c^2\,e\,f}{2}+\frac {b^2\,c\,d\,e^2}{2}\right )+x^3\,\left (\frac {a^2\,c^2\,f^2}{3}+\frac {4\,a^2\,c\,d\,e\,f}{3}+\frac {a^2\,d^2\,e^2}{3}+\frac {4\,a\,b\,c^2\,e\,f}{3}+\frac {4\,a\,b\,c\,d\,e^2}{3}+\frac {b^2\,c^2\,e^2}{3}\right )+x^5\,\left (\frac {a^2\,d^2\,f^2}{5}+\frac {4\,a\,b\,c\,d\,f^2}{5}+\frac {4\,a\,b\,d^2\,e\,f}{5}+\frac {b^2\,c^2\,f^2}{5}+\frac {4\,b^2\,c\,d\,e\,f}{5}+\frac {b^2\,d^2\,e^2}{5}\right )+a^2\,c^2\,e^2\,x+\frac {b^2\,d^2\,f^2\,x^7}{7}+a\,c\,e\,x^2\,\left (a\,c\,f+a\,d\,e+b\,c\,e\right )+\frac {b\,d\,f\,x^6\,\left (a\,d\,f+b\,c\,f+b\,d\,e\right )}{3} \]
x^4*((a*b*c^2*f^2)/2 + (a*b*d^2*e^2)/2 + (a^2*c*d*f^2)/2 + (b^2*c*d*e^2)/2 + (a^2*d^2*e*f)/2 + (b^2*c^2*e*f)/2 + 2*a*b*c*d*e*f) + x^3*((a^2*c^2*f^2) /3 + (a^2*d^2*e^2)/3 + (b^2*c^2*e^2)/3 + (4*a*b*c*d*e^2)/3 + (4*a*b*c^2*e* f)/3 + (4*a^2*c*d*e*f)/3) + x^5*((a^2*d^2*f^2)/5 + (b^2*c^2*f^2)/5 + (b^2* d^2*e^2)/5 + (4*a*b*c*d*f^2)/5 + (4*a*b*d^2*e*f)/5 + (4*b^2*c*d*e*f)/5) + a^2*c^2*e^2*x + (b^2*d^2*f^2*x^7)/7 + a*c*e*x^2*(a*c*f + a*d*e + b*c*e) + (b*d*f*x^6*(a*d*f + b*c*f + b*d*e))/3