Integrand size = 33, antiderivative size = 77 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {\left (c d^2-a e^2\right )^2 (d+e x)^4}{4 e^3}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^5}{5 e^3}+\frac {c^2 d^2 (d+e x)^6}{6 e^3} \] Output:
1/4*(-a*e^2+c*d^2)^2*(e*x+d)^4/e^3-2/5*c*d*(-a*e^2+c*d^2)*(e*x+d)^5/e^3+1/ 6*c^2*d^2*(e*x+d)^6/e^3
Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.56 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{60} x \left (15 a^2 e^2 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+6 a c d e x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+c^2 d^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right ) \] Input:
Integrate[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
(x*(15*a^2*e^2*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 6*a*c*d*e*x*( 10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + c^2*d^2*x^2*(20*d^3 + 45 *d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3)))/60
Time = 0.48 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, 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.
\(\displaystyle \int (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (-\frac {2 c d (d+e x)^4 \left (c d^2-a e^2\right )}{e^2}+\frac {(d+e x)^3 \left (a e^2-c d^2\right )^2}{e^2}+\frac {c^2 d^2 (d+e x)^5}{e^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 c d (d+e x)^5 \left (c d^2-a e^2\right )}{5 e^3}+\frac {(d+e x)^4 \left (c d^2-a e^2\right )^2}{4 e^3}+\frac {c^2 d^2 (d+e x)^6}{6 e^3}\) |
Input:
Int[(d + e*x)*(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)^4)/(4*e^3) - (2*c*d*(c*d^2 - a*e^2)*(d + e*x) ^5)/(5*e^3) + (c^2*d^2*(d + e*x)^6)/(6*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]))
Time = 1.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.78
method | result | size |
norman | \(\frac {c^{2} d^{2} e^{3} x^{6}}{6}+\left (\frac {2}{5} a c d \,e^{4}+\frac {3}{5} c^{2} d^{3} e^{2}\right ) x^{5}+\left (\frac {1}{4} a^{2} e^{5}+\frac {3}{2} a c \,d^{2} e^{3}+\frac {3}{4} c^{2} d^{4} e \right ) x^{4}+\left (a^{2} d \,e^{4}+2 a c \,d^{3} e^{2}+\frac {1}{3} c^{2} d^{5}\right ) x^{3}+\left (\frac {3}{2} e^{3} a^{2} d^{2}+a c \,d^{4} e \right ) x^{2}+e^{2} a^{2} d^{3} x\) | \(137\) |
risch | \(\frac {1}{6} c^{2} d^{2} e^{3} x^{6}+\frac {2}{5} x^{5} a c d \,e^{4}+\frac {3}{5} x^{5} c^{2} d^{3} e^{2}+\frac {1}{4} x^{4} a^{2} e^{5}+\frac {3}{2} x^{4} a c \,d^{2} e^{3}+\frac {3}{4} x^{4} c^{2} d^{4} e +x^{3} a^{2} d \,e^{4}+2 x^{3} a c \,d^{3} e^{2}+\frac {1}{3} x^{3} c^{2} d^{5}+\frac {3}{2} x^{2} e^{3} a^{2} d^{2}+x^{2} a c \,d^{4} e +e^{2} a^{2} d^{3} x\) | \(147\) |
parallelrisch | \(\frac {1}{6} c^{2} d^{2} e^{3} x^{6}+\frac {2}{5} x^{5} a c d \,e^{4}+\frac {3}{5} x^{5} c^{2} d^{3} e^{2}+\frac {1}{4} x^{4} a^{2} e^{5}+\frac {3}{2} x^{4} a c \,d^{2} e^{3}+\frac {3}{4} x^{4} c^{2} d^{4} e +x^{3} a^{2} d \,e^{4}+2 x^{3} a c \,d^{3} e^{2}+\frac {1}{3} x^{3} c^{2} d^{5}+\frac {3}{2} x^{2} e^{3} a^{2} d^{2}+x^{2} a c \,d^{4} e +e^{2} a^{2} d^{3} x\) | \(147\) |
gosper | \(\frac {x \left (10 c^{2} d^{2} e^{3} x^{5}+24 x^{4} a c d \,e^{4}+36 x^{4} c^{2} d^{3} e^{2}+15 x^{3} a^{2} e^{5}+90 x^{3} a c \,d^{2} e^{3}+45 x^{3} c^{2} d^{4} e +60 x^{2} a^{2} d \,e^{4}+120 x^{2} a c \,d^{3} e^{2}+20 x^{2} c^{2} d^{5}+90 x \,e^{3} a^{2} d^{2}+60 x a c \,d^{4} e +60 e^{2} a^{2} d^{3}\right )}{60}\) | \(148\) |
orering | \(\frac {x \left (10 c^{2} d^{2} e^{3} x^{5}+24 x^{4} a c d \,e^{4}+36 x^{4} c^{2} d^{3} e^{2}+15 x^{3} a^{2} e^{5}+90 x^{3} a c \,d^{2} e^{3}+45 x^{3} c^{2} d^{4} e +60 x^{2} a^{2} d \,e^{4}+120 x^{2} a c \,d^{3} e^{2}+20 x^{2} c^{2} d^{5}+90 x \,e^{3} a^{2} d^{2}+60 x a c \,d^{4} e +60 e^{2} a^{2} d^{3}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{2}}{60 \left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}\) | \(192\) |
default | \(\frac {c^{2} d^{2} e^{3} x^{6}}{6}+\frac {\left (c^{2} d^{3} e^{2}+2 e^{2} d c \left (a \,e^{2}+c \,d^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 d^{2} e c \left (a \,e^{2}+c \,d^{2}\right )+e \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right ) x^{4}}{4}+\frac {\left (d \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )+2 e^{2} a d \left (a \,e^{2}+c \,d^{2}\right )\right ) x^{3}}{3}+\frac {\left (2 d^{2} a e \left (a \,e^{2}+c \,d^{2}\right )+e^{3} a^{2} d^{2}\right ) x^{2}}{2}+e^{2} a^{2} d^{3} x\) | \(195\) |
Input:
int((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2,x,method=_RETURNVERBOSE)
Output:
1/6*c^2*d^2*e^3*x^6+(2/5*a*c*d*e^4+3/5*c^2*d^3*e^2)*x^5+(1/4*a^2*e^5+3/2*a *c*d^2*e^3+3/4*c^2*d^4*e)*x^4+(a^2*d*e^4+2*a*c*d^3*e^2+1/3*c^2*d^5)*x^3+(3 /2*e^3*a^2*d^2+a*c*d^4*e)*x^2+e^2*a^2*d^3*x
Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.82 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} d^{2} e^{3} x^{6} + a^{2} d^{3} e^{2} x + \frac {1}{5} \, {\left (3 \, c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x^{4} + \frac {1}{3} \, {\left (c^{2} d^{5} + 6 \, a c d^{3} e^{2} + 3 \, a^{2} d e^{4}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a c d^{4} e + 3 \, a^{2} d^{2} e^{3}\right )} x^{2} \] Input:
integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fricas ")
Output:
1/6*c^2*d^2*e^3*x^6 + a^2*d^3*e^2*x + 1/5*(3*c^2*d^3*e^2 + 2*a*c*d*e^4)*x^ 5 + 1/4*(3*c^2*d^4*e + 6*a*c*d^2*e^3 + a^2*e^5)*x^4 + 1/3*(c^2*d^5 + 6*a*c *d^3*e^2 + 3*a^2*d*e^4)*x^3 + 1/2*(2*a*c*d^4*e + 3*a^2*d^2*e^3)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (68) = 136\).
Time = 0.03 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.95 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=a^{2} d^{3} e^{2} x + \frac {c^{2} d^{2} e^{3} x^{6}}{6} + x^{5} \cdot \left (\frac {2 a c d e^{4}}{5} + \frac {3 c^{2} d^{3} e^{2}}{5}\right ) + x^{4} \left (\frac {a^{2} e^{5}}{4} + \frac {3 a c d^{2} e^{3}}{2} + \frac {3 c^{2} d^{4} e}{4}\right ) + x^{3} \left (a^{2} d e^{4} + 2 a c d^{3} e^{2} + \frac {c^{2} d^{5}}{3}\right ) + x^{2} \cdot \left (\frac {3 a^{2} d^{2} e^{3}}{2} + a c d^{4} e\right ) \] Input:
integrate((e*x+d)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
Output:
a**2*d**3*e**2*x + c**2*d**2*e**3*x**6/6 + x**5*(2*a*c*d*e**4/5 + 3*c**2*d **3*e**2/5) + x**4*(a**2*e**5/4 + 3*a*c*d**2*e**3/2 + 3*c**2*d**4*e/4) + x **3*(a**2*d*e**4 + 2*a*c*d**3*e**2 + c**2*d**5/3) + x**2*(3*a**2*d**2*e**3 /2 + a*c*d**4*e)
Time = 0.03 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.82 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} d^{2} e^{3} x^{6} + a^{2} d^{3} e^{2} x + \frac {1}{5} \, {\left (3 \, c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x^{4} + \frac {1}{3} \, {\left (c^{2} d^{5} + 6 \, a c d^{3} e^{2} + 3 \, a^{2} d e^{4}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a c d^{4} e + 3 \, a^{2} d^{2} e^{3}\right )} x^{2} \] Input:
integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxima ")
Output:
1/6*c^2*d^2*e^3*x^6 + a^2*d^3*e^2*x + 1/5*(3*c^2*d^3*e^2 + 2*a*c*d*e^4)*x^ 5 + 1/4*(3*c^2*d^4*e + 6*a*c*d^2*e^3 + a^2*e^5)*x^4 + 1/3*(c^2*d^5 + 6*a*c *d^3*e^2 + 3*a^2*d*e^4)*x^3 + 1/2*(2*a*c*d^4*e + 3*a^2*d^2*e^3)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (71) = 142\).
Time = 0.10 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.90 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} d^{2} e^{3} x^{6} + \frac {3}{5} \, c^{2} d^{3} e^{2} x^{5} + \frac {2}{5} \, a c d e^{4} x^{5} + \frac {3}{4} \, c^{2} d^{4} e x^{4} + \frac {3}{2} \, a c d^{2} e^{3} x^{4} + \frac {1}{4} \, a^{2} e^{5} x^{4} + \frac {1}{3} \, c^{2} d^{5} x^{3} + 2 \, a c d^{3} e^{2} x^{3} + a^{2} d e^{4} x^{3} + a c d^{4} e x^{2} + \frac {3}{2} \, a^{2} d^{2} e^{3} x^{2} + a^{2} d^{3} e^{2} x \] Input:
integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")
Output:
1/6*c^2*d^2*e^3*x^6 + 3/5*c^2*d^3*e^2*x^5 + 2/5*a*c*d*e^4*x^5 + 3/4*c^2*d^ 4*e*x^4 + 3/2*a*c*d^2*e^3*x^4 + 1/4*a^2*e^5*x^4 + 1/3*c^2*d^5*x^3 + 2*a*c* d^3*e^2*x^3 + a^2*d*e^4*x^3 + a*c*d^4*e*x^2 + 3/2*a^2*d^2*e^3*x^2 + a^2*d^ 3*e^2*x
Time = 5.57 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.75 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=x^3\,\left (a^2\,d\,e^4+2\,a\,c\,d^3\,e^2+\frac {c^2\,d^5}{3}\right )+x^4\,\left (\frac {a^2\,e^5}{4}+\frac {3\,a\,c\,d^2\,e^3}{2}+\frac {3\,c^2\,d^4\,e}{4}\right )+a^2\,d^3\,e^2\,x+\frac {c^2\,d^2\,e^3\,x^6}{6}+\frac {a\,d^2\,e\,x^2\,\left (2\,c\,d^2+3\,a\,e^2\right )}{2}+\frac {c\,d\,e^2\,x^5\,\left (3\,c\,d^2+2\,a\,e^2\right )}{5} \] Input:
int((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)
Output:
x^3*((c^2*d^5)/3 + a^2*d*e^4 + 2*a*c*d^3*e^2) + x^4*((a^2*e^5)/4 + (3*c^2* d^4*e)/4 + (3*a*c*d^2*e^3)/2) + a^2*d^3*e^2*x + (c^2*d^2*e^3*x^6)/6 + (a*d ^2*e*x^2*(3*a*e^2 + 2*c*d^2))/2 + (c*d*e^2*x^5*(2*a*e^2 + 3*c*d^2))/5
Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.91 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {x \left (10 c^{2} d^{2} e^{3} x^{5}+24 a c d \,e^{4} x^{4}+36 c^{2} d^{3} e^{2} x^{4}+15 a^{2} e^{5} x^{3}+90 a c \,d^{2} e^{3} x^{3}+45 c^{2} d^{4} e \,x^{3}+60 a^{2} d \,e^{4} x^{2}+120 a c \,d^{3} e^{2} x^{2}+20 c^{2} d^{5} x^{2}+90 a^{2} d^{2} e^{3} x +60 a c \,d^{4} e x +60 a^{2} d^{3} e^{2}\right )}{60} \] Input:
int((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
Output:
(x*(60*a**2*d**3*e**2 + 90*a**2*d**2*e**3*x + 60*a**2*d*e**4*x**2 + 15*a** 2*e**5*x**3 + 60*a*c*d**4*e*x + 120*a*c*d**3*e**2*x**2 + 90*a*c*d**2*e**3* x**3 + 24*a*c*d*e**4*x**4 + 20*c**2*d**5*x**2 + 45*c**2*d**4*e*x**3 + 36*c **2*d**3*e**2*x**4 + 10*c**2*d**2*e**3*x**5))/60