Integrand size = 32, antiderivative size = 156 \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}{3 e^5}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{2 e^5}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^5}{5 e^5}-\frac {c (2 c d-b e) (d+e x)^6}{3 e^5}+\frac {c^2 (d+e x)^7}{7 e^5} \] Output:
1/3*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^3/e^5-1/2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^ 2)*(e*x+d)^4/e^5+1/5*(6*c^2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))*(e*x+d)^5/e^5- 1/3*c*(-b*e+2*c*d)*(e*x+d)^6/e^5+1/7*c^2*(e*x+d)^7/e^5
Time = 0.05 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.98 \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=a^2 d^2 x+a d (b d+a e) x^2+\frac {1}{3} \left (b^2 d^2+2 a c d^2+4 a b d e+a^2 e^2\right ) x^3+\frac {1}{2} \left (b c d^2+b^2 d e+2 a c d e+a b e^2\right ) x^4+\frac {1}{5} \left (c^2 d^2+4 b c d e+b^2 e^2+2 a c e^2\right ) x^5+\frac {1}{3} c e (c d+b e) x^6+\frac {1}{7} c^2 e^2 x^7 \] Input:
Integrate[(a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c*e*x^3)^2,x]
Output:
a^2*d^2*x + a*d*(b*d + a*e)*x^2 + ((b^2*d^2 + 2*a*c*d^2 + 4*a*b*d*e + a^2* e^2)*x^3)/3 + ((b*c*d^2 + b^2*d*e + 2*a*c*d*e + a*b*e^2)*x^4)/2 + ((c^2*d^ 2 + 4*b*c*d*e + b^2*e^2 + 2*a*c*e^2)*x^5)/5 + (c*e*(c*d + b*e)*x^6)/3 + (c ^2*e^2*x^7)/7
Time = 0.43 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2464, 1140, 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.
| \(\displaystyle \int \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^2 \, dx\) |
| \(\Big \downarrow \) 2464 |
| \(\displaystyle \int (d+e x)^2 \left (a+b x+c x^2\right )^2dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x)^4 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac {2 (d+e x)^3 (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^4}+\frac {(d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}{e^4}-\frac {2 c (d+e x)^5 (2 c d-b e)}{e^4}+\frac {c^2 (d+e x)^6}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^5}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}{3 e^5}-\frac {c (d+e x)^6 (2 c d-b e)}{3 e^5}+\frac {c^2 (d+e x)^7}{7 e^5}\) |
Input:
Int[(a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c*e*x^3)^2,x]
Output:
((c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^3)/(3*e^5) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4)/(2*e^5) + ((6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^5)/(5*e^5) - (c*(2*c*d - b*e)*(d + e*x)^6)/(3*e^5) + (c ^2*(d + e*x)^7)/(7*e^5)
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
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 = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.82
| method | result | size |
| default | \(\frac {c^{2} e^{2} x^{7}}{7}+\frac {\left (e b +c d \right ) c e \,x^{6}}{3}+\frac {\left (2 c e \left (a e +b d \right )+\left (e b +c d \right )^{2}\right ) x^{5}}{5}+\frac {\left (2 a c d e +2 \left (a e +b d \right ) \left (e b +c d \right )\right ) x^{4}}{4}+\frac {\left (2 a d \left (e b +c d \right )+\left (a e +b d \right )^{2}\right ) x^{3}}{3}+a d \left (a e +b d \right ) x^{2}+x \,a^{2} d^{2}\) | \(128\) |
| norman | \(\frac {c^{2} e^{2} x^{7}}{7}+\left (\frac {1}{3} b c \,e^{2}+\frac {1}{3} c^{2} d e \right ) x^{6}+\left (\frac {2}{5} a c \,e^{2}+\frac {1}{5} b^{2} e^{2}+\frac {4}{5} b c d e +\frac {1}{5} c^{2} d^{2}\right ) x^{5}+\left (\frac {1}{2} a b \,e^{2}+a c d e +\frac {1}{2} b^{2} d e +\frac {1}{2} b c \,d^{2}\right ) x^{4}+\left (\frac {1}{3} a^{2} e^{2}+\frac {4}{3} a b d e +\frac {2}{3} a c \,d^{2}+\frac {1}{3} b^{2} d^{2}\right ) x^{3}+\left (a^{2} d e +a b \,d^{2}\right ) x^{2}+x \,a^{2} d^{2}\) | \(156\) |
| risch | \(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} b c \,e^{2} x^{6}+\frac {1}{3} c^{2} d e \,x^{6}+\frac {2}{5} x^{5} a c \,e^{2}+\frac {1}{5} x^{5} b^{2} e^{2}+\frac {4}{5} x^{5} b c d e +\frac {1}{5} x^{5} c^{2} d^{2}+\frac {1}{2} x^{4} a b \,e^{2}+x^{4} a c d e +\frac {1}{2} x^{4} b^{2} d e +\frac {1}{2} b c \,d^{2} x^{4}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {4}{3} x^{3} a b d e +\frac {2}{3} x^{3} a c \,d^{2}+\frac {1}{3} x^{3} b^{2} d^{2}+a^{2} d e \,x^{2}+a b \,d^{2} x^{2}+x \,a^{2} d^{2}\) | \(179\) |
| parallelrisch | \(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} b c \,e^{2} x^{6}+\frac {1}{3} c^{2} d e \,x^{6}+\frac {2}{5} x^{5} a c \,e^{2}+\frac {1}{5} x^{5} b^{2} e^{2}+\frac {4}{5} x^{5} b c d e +\frac {1}{5} x^{5} c^{2} d^{2}+\frac {1}{2} x^{4} a b \,e^{2}+x^{4} a c d e +\frac {1}{2} x^{4} b^{2} d e +\frac {1}{2} b c \,d^{2} x^{4}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {4}{3} x^{3} a b d e +\frac {2}{3} x^{3} a c \,d^{2}+\frac {1}{3} x^{3} b^{2} d^{2}+a^{2} d e \,x^{2}+a b \,d^{2} x^{2}+x \,a^{2} d^{2}\) | \(179\) |
| gosper | \(\frac {x \left (30 c^{2} e^{2} x^{6}+70 x^{5} b c \,e^{2}+70 x^{5} c^{2} d e +84 x^{4} a c \,e^{2}+42 x^{4} b^{2} e^{2}+168 x^{4} b c d e +42 x^{4} c^{2} d^{2}+105 x^{3} a b \,e^{2}+210 x^{3} a c d e +105 x^{3} b^{2} d e +105 x^{3} b c \,d^{2}+70 a^{2} e^{2} x^{2}+280 a b d e \,x^{2}+140 x^{2} a c \,d^{2}+70 x^{2} b^{2} d^{2}+210 a^{2} d e x +210 a b \,d^{2} x +210 a^{2} d^{2}\right )}{210}\) | \(181\) |
| orering | \(\frac {x \left (30 c^{2} e^{2} x^{6}+70 x^{5} b c \,e^{2}+70 x^{5} c^{2} d e +84 x^{4} a c \,e^{2}+42 x^{4} b^{2} e^{2}+168 x^{4} b c d e +42 x^{4} c^{2} d^{2}+105 x^{3} a b \,e^{2}+210 x^{3} a c d e +105 x^{3} b^{2} d e +105 x^{3} b c \,d^{2}+70 a^{2} e^{2} x^{2}+280 a b d e \,x^{2}+140 x^{2} a c \,d^{2}+70 x^{2} b^{2} d^{2}+210 a^{2} d e x +210 a b \,d^{2} x +210 a^{2} d^{2}\right ) \left (a d +\left (a e +b d \right ) x +\left (e b +c d \right ) x^{2}+c e \,x^{3}\right )^{2}}{210 \left (c \,x^{2}+b x +a \right )^{2} \left (e x +d \right )^{2}}\) | \(232\) |
Input:
int((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^2,x,method=_RETURNVERBOSE)
Output:
1/7*c^2*e^2*x^7+1/3*(b*e+c*d)*c*e*x^6+1/5*(2*c*e*(a*e+b*d)+(b*e+c*d)^2)*x^ 5+1/4*(2*a*c*d*e+2*(a*e+b*d)*(b*e+c*d))*x^4+1/3*(2*a*d*(b*e+c*d)+(a*e+b*d) ^2)*x^3+a*d*(a*e+b*d)*x^2+x*a^2*d^2
Time = 0.09 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.94 \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, {\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{2} + 4 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac {1}{2} \, {\left (b c d^{2} + a b e^{2} + {\left (b^{2} + 2 \, a c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (4 \, a b d e + a^{2} e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{3} + {\left (a b d^{2} + a^{2} d e\right )} x^{2} \] Input:
integrate((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^2,x, algorithm="fricas")
Output:
1/7*c^2*e^2*x^7 + 1/3*(c^2*d*e + b*c*e^2)*x^6 + 1/5*(c^2*d^2 + 4*b*c*d*e + (b^2 + 2*a*c)*e^2)*x^5 + a^2*d^2*x + 1/2*(b*c*d^2 + a*b*e^2 + (b^2 + 2*a* c)*d*e)*x^4 + 1/3*(4*a*b*d*e + a^2*e^2 + (b^2 + 2*a*c)*d^2)*x^3 + (a*b*d^2 + a^2*d*e)*x^2
Time = 0.04 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.11 \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=a^{2} d^{2} x + \frac {c^{2} e^{2} x^{7}}{7} + x^{6} \left (\frac {b c e^{2}}{3} + \frac {c^{2} d e}{3}\right ) + x^{5} \cdot \left (\frac {2 a c e^{2}}{5} + \frac {b^{2} e^{2}}{5} + \frac {4 b c d e}{5} + \frac {c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac {a b e^{2}}{2} + a c d e + \frac {b^{2} d e}{2} + \frac {b c d^{2}}{2}\right ) + x^{3} \left (\frac {a^{2} e^{2}}{3} + \frac {4 a b d e}{3} + \frac {2 a c d^{2}}{3} + \frac {b^{2} d^{2}}{3}\right ) + x^{2} \left (a^{2} d e + a b d^{2}\right ) \] Input:
integrate((a*d+(a*e+b*d)*x+(b*e+c*d)*x**2+c*e*x**3)**2,x)
Output:
a**2*d**2*x + c**2*e**2*x**7/7 + x**6*(b*c*e**2/3 + c**2*d*e/3) + x**5*(2* a*c*e**2/5 + b**2*e**2/5 + 4*b*c*d*e/5 + c**2*d**2/5) + x**4*(a*b*e**2/2 + a*c*d*e + b**2*d*e/2 + b*c*d**2/2) + x**3*(a**2*e**2/3 + 4*a*b*d*e/3 + 2* a*c*d**2/3 + b**2*d**2/3) + x**2*(a**2*d*e + a*b*d**2)
Time = 0.03 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.81 \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, {\left (c d + b e\right )} c e x^{6} + \frac {1}{5} \, {\left (c d + b e\right )}^{2} x^{5} + a^{2} d^{2} x + \frac {1}{3} \, {\left (b d + a e\right )}^{2} x^{3} + \frac {1}{6} \, {\left (3 \, c e x^{4} + 4 \, {\left (c d + b e\right )} x^{3} + 6 \, {\left (b d + a e\right )} x^{2}\right )} a d + \frac {1}{10} \, {\left (4 \, c e x^{5} + 5 \, {\left (c d + b e\right )} x^{4}\right )} {\left (b d + a e\right )} \] Input:
integrate((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^2,x, algorithm="maxima")
Output:
1/7*c^2*e^2*x^7 + 1/3*(c*d + b*e)*c*e*x^6 + 1/5*(c*d + b*e)^2*x^5 + a^2*d^ 2*x + 1/3*(b*d + a*e)^2*x^3 + 1/6*(3*c*e*x^4 + 4*(c*d + b*e)*x^3 + 6*(b*d + a*e)*x^2)*a*d + 1/10*(4*c*e*x^5 + 5*(c*d + b*e)*x^4)*(b*d + a*e)
Time = 0.13 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.14 \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, c^{2} d e x^{6} + \frac {1}{3} \, b c e^{2} x^{6} + \frac {1}{5} \, c^{2} d^{2} x^{5} + \frac {4}{5} \, b c d e x^{5} + \frac {1}{5} \, b^{2} e^{2} x^{5} + \frac {2}{5} \, a c e^{2} x^{5} + \frac {1}{2} \, b c d^{2} x^{4} + \frac {1}{2} \, b^{2} d e x^{4} + a c d e x^{4} + \frac {1}{2} \, a b e^{2} x^{4} + \frac {1}{3} \, b^{2} d^{2} x^{3} + \frac {2}{3} \, a c d^{2} x^{3} + \frac {4}{3} \, a b d e x^{3} + \frac {1}{3} \, a^{2} e^{2} x^{3} + a b d^{2} x^{2} + a^{2} d e x^{2} + a^{2} d^{2} x \] Input:
integrate((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^2,x, algorithm="giac")
Output:
1/7*c^2*e^2*x^7 + 1/3*c^2*d*e*x^6 + 1/3*b*c*e^2*x^6 + 1/5*c^2*d^2*x^5 + 4/ 5*b*c*d*e*x^5 + 1/5*b^2*e^2*x^5 + 2/5*a*c*e^2*x^5 + 1/2*b*c*d^2*x^4 + 1/2* b^2*d*e*x^4 + a*c*d*e*x^4 + 1/2*a*b*e^2*x^4 + 1/3*b^2*d^2*x^3 + 2/3*a*c*d^ 2*x^3 + 4/3*a*b*d*e*x^3 + 1/3*a^2*e^2*x^3 + a*b*d^2*x^2 + a^2*d*e*x^2 + a^ 2*d^2*x
Time = 0.03 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.94 \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=x^3\,\left (\frac {a^2\,e^2}{3}+\frac {4\,a\,b\,d\,e}{3}+\frac {2\,c\,a\,d^2}{3}+\frac {b^2\,d^2}{3}\right )+x^5\,\left (\frac {b^2\,e^2}{5}+\frac {4\,b\,c\,d\,e}{5}+\frac {c^2\,d^2}{5}+\frac {2\,a\,c\,e^2}{5}\right )+x^4\,\left (\frac {b^2\,d\,e}{2}+\frac {c\,b\,d^2}{2}+\frac {a\,b\,e^2}{2}+a\,c\,d\,e\right )+a^2\,d^2\,x+\frac {c^2\,e^2\,x^7}{7}+a\,d\,x^2\,\left (a\,e+b\,d\right )+\frac {c\,e\,x^6\,\left (b\,e+c\,d\right )}{3} \] Input:
int((a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3)^2,x)
Output:
x^3*((a^2*e^2)/3 + (b^2*d^2)/3 + (2*a*c*d^2)/3 + (4*a*b*d*e)/3) + x^5*((b^ 2*e^2)/5 + (c^2*d^2)/5 + (2*a*c*e^2)/5 + (4*b*c*d*e)/5) + x^4*((a*b*e^2)/2 + (b*c*d^2)/2 + (b^2*d*e)/2 + a*c*d*e) + a^2*d^2*x + (c^2*e^2*x^7)/7 + a* d*x^2*(a*e + b*d) + (c*e*x^6*(b*e + c*d))/3
Time = 0.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.15 \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=\frac {x \left (30 c^{2} e^{2} x^{6}+70 b c \,e^{2} x^{5}+70 c^{2} d e \,x^{5}+84 a c \,e^{2} x^{4}+42 b^{2} e^{2} x^{4}+168 b c d e \,x^{4}+42 c^{2} d^{2} x^{4}+105 a b \,e^{2} x^{3}+210 a c d e \,x^{3}+105 b^{2} d e \,x^{3}+105 b c \,d^{2} x^{3}+70 a^{2} e^{2} x^{2}+280 a b d e \,x^{2}+140 a c \,d^{2} x^{2}+70 b^{2} d^{2} x^{2}+210 a^{2} d e x +210 a b \,d^{2} x +210 a^{2} d^{2}\right )}{210} \] Input:
int((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^2,x)
Output:
(x*(210*a**2*d**2 + 210*a**2*d*e*x + 70*a**2*e**2*x**2 + 210*a*b*d**2*x + 280*a*b*d*e*x**2 + 105*a*b*e**2*x**3 + 140*a*c*d**2*x**2 + 210*a*c*d*e*x** 3 + 84*a*c*e**2*x**4 + 70*b**2*d**2*x**2 + 105*b**2*d*e*x**3 + 42*b**2*e** 2*x**4 + 105*b*c*d**2*x**3 + 168*b*c*d*e*x**4 + 70*b*c*e**2*x**5 + 42*c**2 *d**2*x**4 + 70*c**2*d*e*x**5 + 30*c**2*e**2*x**6))/210