Integrand size = 35, antiderivative size = 77 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {\left (c d^2-a e^2\right )^2 (d+e x)^5}{5 e^3}-\frac {c d \left (c d^2-a e^2\right ) (d+e x)^6}{3 e^3}+\frac {c^2 d^2 (d+e x)^7}{7 e^3} \] Output:
1/5*(-a*e^2+c*d^2)^2*(e*x+d)^5/e^3-1/3*c*d*(-a*e^2+c*d^2)*(e*x+d)^6/e^3+1/ 7*c^2*d^2*(e*x+d)^7/e^3
Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(77)=154\).
Time = 0.02 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.08 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{15} a c d e x^2 \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+\frac {1}{105} c^2 d^2 x^3 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+a^2 \left (d^4 e^2 x+2 d^3 e^3 x^2+2 d^2 e^4 x^3+d e^5 x^4+\frac {e^6 x^5}{5}\right ) \] Input:
Integrate[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
(a*c*d*e*x^2*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4* x^4))/15 + (c^2*d^2*x^3*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3 *x^3 + 15*e^4*x^4))/105 + a^2*(d^4*e^2*x + 2*d^3*e^3*x^2 + 2*d^2*e^4*x^3 + d*e^5*x^4 + (e^6*x^5)/5)
Time = 0.51 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1121, 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.
\(\displaystyle \int (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {2 c d (d+e x)^5 \left (c d^2-a e^2\right )}{e^2}+\frac {(d+e x)^4 \left (a e^2-c d^2\right )^2}{e^2}+\frac {c^2 d^2 (d+e x)^6}{e^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {c d (d+e x)^6 \left (c d^2-a e^2\right )}{3 e^3}+\frac {(d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e^3}+\frac {c^2 d^2 (d+e x)^7}{7 e^3}\) |
Input:
Int[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
((c*d^2 - a*e^2)^2*(d + e*x)^5)/(5*e^3) - (c*d*(c*d^2 - a*e^2)*(d + e*x)^6 )/(3*e^3) + (c^2*d^2*(d + e*x)^7)/(7*e^3)
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(172\) vs. \(2(71)=142\).
Time = 1.25 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.25
method | result | size |
norman | \(\frac {c^{2} d^{2} e^{4} x^{7}}{7}+\left (\frac {1}{3} a c d \,e^{5}+\frac {2}{3} c^{2} d^{3} e^{3}\right ) x^{6}+\left (\frac {1}{5} a^{2} e^{6}+\frac {8}{5} a c \,d^{2} e^{4}+\frac {6}{5} c^{2} d^{4} e^{2}\right ) x^{5}+\left (e^{5} a^{2} d +3 a c \,d^{3} e^{3}+c^{2} d^{5} e \right ) x^{4}+\left (2 a^{2} d^{2} e^{4}+\frac {8}{3} a \,d^{4} e^{2} c +\frac {1}{3} d^{6} c^{2}\right ) x^{3}+\left (2 e^{3} a^{2} d^{3}+a c \,d^{5} e \right ) x^{2}+e^{2} a^{2} d^{4} x\) | \(173\) |
risch | \(\frac {1}{7} c^{2} d^{2} e^{4} x^{7}+\frac {1}{3} x^{6} a c d \,e^{5}+\frac {2}{3} x^{6} c^{2} d^{3} e^{3}+\frac {1}{5} x^{5} a^{2} e^{6}+\frac {8}{5} x^{5} a c \,d^{2} e^{4}+\frac {6}{5} x^{5} c^{2} d^{4} e^{2}+a^{2} d \,e^{5} x^{4}+3 a c \,d^{3} e^{3} x^{4}+c^{2} d^{5} e \,x^{4}+2 x^{3} a^{2} d^{2} e^{4}+\frac {8}{3} x^{3} a \,d^{4} e^{2} c +\frac {1}{3} x^{3} d^{6} c^{2}+2 a^{2} d^{3} e^{3} x^{2}+a c \,d^{5} e \,x^{2}+e^{2} a^{2} d^{4} x\) | \(187\) |
parallelrisch | \(\frac {1}{7} c^{2} d^{2} e^{4} x^{7}+\frac {1}{3} x^{6} a c d \,e^{5}+\frac {2}{3} x^{6} c^{2} d^{3} e^{3}+\frac {1}{5} x^{5} a^{2} e^{6}+\frac {8}{5} x^{5} a c \,d^{2} e^{4}+\frac {6}{5} x^{5} c^{2} d^{4} e^{2}+a^{2} d \,e^{5} x^{4}+3 a c \,d^{3} e^{3} x^{4}+c^{2} d^{5} e \,x^{4}+2 x^{3} a^{2} d^{2} e^{4}+\frac {8}{3} x^{3} a \,d^{4} e^{2} c +\frac {1}{3} x^{3} d^{6} c^{2}+2 a^{2} d^{3} e^{3} x^{2}+a c \,d^{5} e \,x^{2}+e^{2} a^{2} d^{4} x\) | \(187\) |
gosper | \(\frac {x \left (15 c^{2} d^{2} e^{4} x^{6}+35 x^{5} a c d \,e^{5}+70 x^{5} c^{2} d^{3} e^{3}+21 x^{4} a^{2} e^{6}+168 x^{4} a c \,d^{2} e^{4}+126 x^{4} c^{2} d^{4} e^{2}+105 a^{2} d \,e^{5} x^{3}+315 a c \,d^{3} e^{3} x^{3}+105 c^{2} d^{5} e \,x^{3}+210 x^{2} a^{2} d^{2} e^{4}+280 x^{2} a \,d^{4} e^{2} c +35 x^{2} d^{6} c^{2}+210 a^{2} d^{3} e^{3} x +105 a c \,d^{5} e x +105 e^{2} a^{2} d^{4}\right )}{105}\) | \(189\) |
orering | \(\frac {x \left (15 c^{2} d^{2} e^{4} x^{6}+35 x^{5} a c d \,e^{5}+70 x^{5} c^{2} d^{3} e^{3}+21 x^{4} a^{2} e^{6}+168 x^{4} a c \,d^{2} e^{4}+126 x^{4} c^{2} d^{4} e^{2}+105 a^{2} d \,e^{5} x^{3}+315 a c \,d^{3} e^{3} x^{3}+105 c^{2} d^{5} e \,x^{3}+210 x^{2} a^{2} d^{2} e^{4}+280 x^{2} a \,d^{4} e^{2} c +35 x^{2} d^{6} c^{2}+210 a^{2} d^{3} e^{3} x +105 a c \,d^{5} e x +105 e^{2} a^{2} d^{4}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{2}}{105 \left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}\) | \(233\) |
default | \(\frac {c^{2} d^{2} e^{4} x^{7}}{7}+\frac {\left (2 c^{2} d^{3} e^{3}+2 e^{3} d c \left (a \,e^{2}+c \,d^{2}\right )\right ) x^{6}}{6}+\frac {\left (c^{2} d^{4} e^{2}+4 d^{2} e^{2} c \left (a \,e^{2}+c \,d^{2}\right )+e^{2} \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 d^{3} e c \left (a \,e^{2}+c \,d^{2}\right )+2 d e \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )+2 e^{3} a d \left (a \,e^{2}+c \,d^{2}\right )\right ) x^{4}}{4}+\frac {\left (d^{2} \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )+4 d^{2} e^{2} a \left (a \,e^{2}+c \,d^{2}\right )+a^{2} d^{2} e^{4}\right ) x^{3}}{3}+\frac {\left (2 d^{3} a e \left (a \,e^{2}+c \,d^{2}\right )+2 e^{3} a^{2} d^{3}\right ) x^{2}}{2}+e^{2} a^{2} d^{4} x\) | \(295\) |
Input:
int((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2,x,method=_RETURNVERBOSE)
Output:
1/7*c^2*d^2*e^4*x^7+(1/3*a*c*d*e^5+2/3*c^2*d^3*e^3)*x^6+(1/5*a^2*e^6+8/5*a *c*d^2*e^4+6/5*c^2*d^4*e^2)*x^5+(a^2*d*e^5+3*a*c*d^3*e^3+c^2*d^5*e)*x^4+(2 *a^2*d^2*e^4+8/3*a*d^4*e^2*c+1/3*d^6*c^2)*x^3+(2*a^2*d^3*e^3+a*c*d^5*e)*x^ 2+e^2*a^2*d^4*x
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (71) = 142\).
Time = 0.08 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.23 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} d^{2} e^{4} x^{7} + a^{2} d^{4} e^{2} x + \frac {1}{3} \, {\left (2 \, c^{2} d^{3} e^{3} + a c d e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, c^{2} d^{4} e^{2} + 8 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{5} + {\left (c^{2} d^{5} e + 3 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x^{4} + \frac {1}{3} \, {\left (c^{2} d^{6} + 8 \, a c d^{4} e^{2} + 6 \, a^{2} d^{2} e^{4}\right )} x^{3} + {\left (a c d^{5} e + 2 \, a^{2} d^{3} e^{3}\right )} x^{2} \] Input:
integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fric as")
Output:
1/7*c^2*d^2*e^4*x^7 + a^2*d^4*e^2*x + 1/3*(2*c^2*d^3*e^3 + a*c*d*e^5)*x^6 + 1/5*(6*c^2*d^4*e^2 + 8*a*c*d^2*e^4 + a^2*e^6)*x^5 + (c^2*d^5*e + 3*a*c*d ^3*e^3 + a^2*d*e^5)*x^4 + 1/3*(c^2*d^6 + 8*a*c*d^4*e^2 + 6*a^2*d^2*e^4)*x^ 3 + (a*c*d^5*e + 2*a^2*d^3*e^3)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (66) = 132\).
Time = 0.04 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.40 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=a^{2} d^{4} e^{2} x + \frac {c^{2} d^{2} e^{4} x^{7}}{7} + x^{6} \left (\frac {a c d e^{5}}{3} + \frac {2 c^{2} d^{3} e^{3}}{3}\right ) + x^{5} \left (\frac {a^{2} e^{6}}{5} + \frac {8 a c d^{2} e^{4}}{5} + \frac {6 c^{2} d^{4} e^{2}}{5}\right ) + x^{4} \left (a^{2} d e^{5} + 3 a c d^{3} e^{3} + c^{2} d^{5} e\right ) + x^{3} \cdot \left (2 a^{2} d^{2} e^{4} + \frac {8 a c d^{4} e^{2}}{3} + \frac {c^{2} d^{6}}{3}\right ) + x^{2} \cdot \left (2 a^{2} d^{3} e^{3} + a c d^{5} e\right ) \] Input:
integrate((e*x+d)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
Output:
a**2*d**4*e**2*x + c**2*d**2*e**4*x**7/7 + x**6*(a*c*d*e**5/3 + 2*c**2*d** 3*e**3/3) + x**5*(a**2*e**6/5 + 8*a*c*d**2*e**4/5 + 6*c**2*d**4*e**2/5) + x**4*(a**2*d*e**5 + 3*a*c*d**3*e**3 + c**2*d**5*e) + x**3*(2*a**2*d**2*e** 4 + 8*a*c*d**4*e**2/3 + c**2*d**6/3) + x**2*(2*a**2*d**3*e**3 + a*c*d**5*e )
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (71) = 142\).
Time = 0.03 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.23 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} d^{2} e^{4} x^{7} + a^{2} d^{4} e^{2} x + \frac {1}{3} \, {\left (2 \, c^{2} d^{3} e^{3} + a c d e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, c^{2} d^{4} e^{2} + 8 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{5} + {\left (c^{2} d^{5} e + 3 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x^{4} + \frac {1}{3} \, {\left (c^{2} d^{6} + 8 \, a c d^{4} e^{2} + 6 \, a^{2} d^{2} e^{4}\right )} x^{3} + {\left (a c d^{5} e + 2 \, a^{2} d^{3} e^{3}\right )} x^{2} \] Input:
integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxi ma")
Output:
1/7*c^2*d^2*e^4*x^7 + a^2*d^4*e^2*x + 1/3*(2*c^2*d^3*e^3 + a*c*d*e^5)*x^6 + 1/5*(6*c^2*d^4*e^2 + 8*a*c*d^2*e^4 + a^2*e^6)*x^5 + (c^2*d^5*e + 3*a*c*d ^3*e^3 + a^2*d*e^5)*x^4 + 1/3*(c^2*d^6 + 8*a*c*d^4*e^2 + 6*a^2*d^2*e^4)*x^ 3 + (a*c*d^5*e + 2*a^2*d^3*e^3)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (71) = 142\).
Time = 0.16 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.42 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} d^{2} e^{4} x^{7} + \frac {2}{3} \, c^{2} d^{3} e^{3} x^{6} + \frac {1}{3} \, a c d e^{5} x^{6} + \frac {6}{5} \, c^{2} d^{4} e^{2} x^{5} + \frac {8}{5} \, a c d^{2} e^{4} x^{5} + \frac {1}{5} \, a^{2} e^{6} x^{5} + c^{2} d^{5} e x^{4} + 3 \, a c d^{3} e^{3} x^{4} + a^{2} d e^{5} x^{4} + \frac {1}{3} \, c^{2} d^{6} x^{3} + \frac {8}{3} \, a c d^{4} e^{2} x^{3} + 2 \, a^{2} d^{2} e^{4} x^{3} + a c d^{5} e x^{2} + 2 \, a^{2} d^{3} e^{3} x^{2} + a^{2} d^{4} e^{2} x \] Input:
integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac ")
Output:
1/7*c^2*d^2*e^4*x^7 + 2/3*c^2*d^3*e^3*x^6 + 1/3*a*c*d*e^5*x^6 + 6/5*c^2*d^ 4*e^2*x^5 + 8/5*a*c*d^2*e^4*x^5 + 1/5*a^2*e^6*x^5 + c^2*d^5*e*x^4 + 3*a*c* d^3*e^3*x^4 + a^2*d*e^5*x^4 + 1/3*c^2*d^6*x^3 + 8/3*a*c*d^4*e^2*x^3 + 2*a^ 2*d^2*e^4*x^3 + a*c*d^5*e*x^2 + 2*a^2*d^3*e^3*x^2 + a^2*d^4*e^2*x
Time = 0.04 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.18 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=x^3\,\left (2\,a^2\,d^2\,e^4+\frac {8\,a\,c\,d^4\,e^2}{3}+\frac {c^2\,d^6}{3}\right )+x^5\,\left (\frac {a^2\,e^6}{5}+\frac {8\,a\,c\,d^2\,e^4}{5}+\frac {6\,c^2\,d^4\,e^2}{5}\right )+x^4\,\left (a^2\,d\,e^5+3\,a\,c\,d^3\,e^3+c^2\,d^5\,e\right )+a^2\,d^4\,e^2\,x+\frac {c^2\,d^2\,e^4\,x^7}{7}+a\,d^3\,e\,x^2\,\left (c\,d^2+2\,a\,e^2\right )+\frac {c\,d\,e^3\,x^6\,\left (2\,c\,d^2+a\,e^2\right )}{3} \] Input:
int((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)
Output:
x^3*((c^2*d^6)/3 + 2*a^2*d^2*e^4 + (8*a*c*d^4*e^2)/3) + x^5*((a^2*e^6)/5 + (6*c^2*d^4*e^2)/5 + (8*a*c*d^2*e^4)/5) + x^4*(a^2*d*e^5 + c^2*d^5*e + 3*a *c*d^3*e^3) + a^2*d^4*e^2*x + (c^2*d^2*e^4*x^7)/7 + a*d^3*e*x^2*(2*a*e^2 + c*d^2) + (c*d*e^3*x^6*(a*e^2 + 2*c*d^2))/3
Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.44 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {x \left (15 c^{2} d^{2} e^{4} x^{6}+35 a c d \,e^{5} x^{5}+70 c^{2} d^{3} e^{3} x^{5}+21 a^{2} e^{6} x^{4}+168 a c \,d^{2} e^{4} x^{4}+126 c^{2} d^{4} e^{2} x^{4}+105 a^{2} d \,e^{5} x^{3}+315 a c \,d^{3} e^{3} x^{3}+105 c^{2} d^{5} e \,x^{3}+210 a^{2} d^{2} e^{4} x^{2}+280 a c \,d^{4} e^{2} x^{2}+35 c^{2} d^{6} x^{2}+210 a^{2} d^{3} e^{3} x +105 a c \,d^{5} e x +105 a^{2} d^{4} e^{2}\right )}{105} \] Input:
int((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
Output:
(x*(105*a**2*d**4*e**2 + 210*a**2*d**3*e**3*x + 210*a**2*d**2*e**4*x**2 + 105*a**2*d*e**5*x**3 + 21*a**2*e**6*x**4 + 105*a*c*d**5*e*x + 280*a*c*d**4 *e**2*x**2 + 315*a*c*d**3*e**3*x**3 + 168*a*c*d**2*e**4*x**4 + 35*a*c*d*e* *5*x**5 + 35*c**2*d**6*x**2 + 105*c**2*d**5*e*x**3 + 126*c**2*d**4*e**2*x* *4 + 70*c**2*d**3*e**3*x**5 + 15*c**2*d**2*e**4*x**6))/105