Integrand size = 26, antiderivative size = 92 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(b d-a e)^3 (a+b x)^5}{5 b^4}+\frac {e (b d-a e)^2 (a+b x)^6}{2 b^4}+\frac {3 e^2 (b d-a e) (a+b x)^7}{7 b^4}+\frac {e^3 (a+b x)^8}{8 b^4} \] Output:
1/5*(-a*e+b*d)^3*(b*x+a)^5/b^4+1/2*e*(-a*e+b*d)^2*(b*x+a)^6/b^4+3/7*e^2*(- a*e+b*d)*(b*x+a)^7/b^4+1/8*e^3*(b*x+a)^8/b^4
Leaf count is larger than twice the leaf count of optimal. \(217\) vs. \(2(92)=184\).
Time = 0.03 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.36 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^4 d^3 x+\frac {1}{2} a^3 d^2 (4 b d+3 a e) x^2+a^2 d \left (2 b^2 d^2+4 a b d e+a^2 e^2\right ) x^3+\frac {1}{4} a \left (4 b^3 d^3+18 a b^2 d^2 e+12 a^2 b d e^2+a^3 e^3\right ) x^4+\frac {1}{5} b \left (b^3 d^3+12 a b^2 d^2 e+18 a^2 b d e^2+4 a^3 e^3\right ) x^5+\frac {1}{2} b^2 e \left (b^2 d^2+4 a b d e+2 a^2 e^2\right ) x^6+\frac {1}{7} b^3 e^2 (3 b d+4 a e) x^7+\frac {1}{8} b^4 e^3 x^8 \] Input:
Integrate[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
a^4*d^3*x + (a^3*d^2*(4*b*d + 3*a*e)*x^2)/2 + a^2*d*(2*b^2*d^2 + 4*a*b*d*e + a^2*e^2)*x^3 + (a*(4*b^3*d^3 + 18*a*b^2*d^2*e + 12*a^2*b*d*e^2 + a^3*e^ 3)*x^4)/4 + (b*(b^3*d^3 + 12*a*b^2*d^2*e + 18*a^2*b*d*e^2 + 4*a^3*e^3)*x^5 )/5 + (b^2*e*(b^2*d^2 + 4*a*b*d*e + 2*a^2*e^2)*x^6)/2 + (b^3*e^2*(3*b*d + 4*a*e)*x^7)/7 + (b^4*e^3*x^8)/8
Time = 0.54 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1098, 27, 49, 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.
\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^3 \, dx\) |
\(\Big \downarrow \) 1098 |
\(\displaystyle \frac {\int b^4 (a+b x)^4 (d+e x)^3dx}{b^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int (a+b x)^4 (d+e x)^3dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {3 e^2 (a+b x)^6 (b d-a e)}{b^3}+\frac {3 e (a+b x)^5 (b d-a e)^2}{b^3}+\frac {(a+b x)^4 (b d-a e)^3}{b^3}+\frac {e^3 (a+b x)^7}{b^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 e^2 (a+b x)^7 (b d-a e)}{7 b^4}+\frac {e (a+b x)^6 (b d-a e)^2}{2 b^4}+\frac {(a+b x)^5 (b d-a e)^3}{5 b^4}+\frac {e^3 (a+b x)^8}{8 b^4}\) |
Input:
Int[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
((b*d - a*e)^3*(a + b*x)^5)/(5*b^4) + (e*(b*d - a*e)^2*(a + b*x)^6)/(2*b^4 ) + (3*e^2*(b*d - a*e)*(a + b*x)^7)/(7*b^4) + (e^3*(a + b*x)^8)/(8*b^4)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(221\) vs. \(2(84)=168\).
Time = 1.04 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.41
method | result | size |
norman | \(\frac {e^{3} b^{4} x^{8}}{8}+\left (\frac {4}{7} a \,b^{3} e^{3}+\frac {3}{7} b^{4} d \,e^{2}\right ) x^{7}+\left (a^{2} b^{2} e^{3}+2 d \,e^{2} a \,b^{3}+\frac {1}{2} d^{2} e \,b^{4}\right ) x^{6}+\left (\frac {4}{5} a^{3} b \,e^{3}+\frac {18}{5} a^{2} b^{2} d \,e^{2}+\frac {12}{5} d^{2} e a \,b^{3}+\frac {1}{5} b^{4} d^{3}\right ) x^{5}+\left (\frac {1}{4} e^{3} a^{4}+3 d \,e^{2} a^{3} b +\frac {9}{2} d^{2} e \,a^{2} b^{2}+a \,b^{3} d^{3}\right ) x^{4}+\left (d \,e^{2} a^{4}+4 d^{2} e \,a^{3} b +2 a^{2} b^{2} d^{3}\right ) x^{3}+\left (\frac {3}{2} d^{2} e \,a^{4}+2 a^{3} b \,d^{3}\right ) x^{2}+a^{4} d^{3} x\) | \(222\) |
default | \(\frac {e^{3} b^{4} x^{8}}{8}+\frac {\left (4 a \,b^{3} e^{3}+3 b^{4} d \,e^{2}\right ) x^{7}}{7}+\frac {\left (6 a^{2} b^{2} e^{3}+12 d \,e^{2} a \,b^{3}+3 d^{2} e \,b^{4}\right ) x^{6}}{6}+\frac {\left (4 a^{3} b \,e^{3}+18 a^{2} b^{2} d \,e^{2}+12 d^{2} e a \,b^{3}+b^{4} d^{3}\right ) x^{5}}{5}+\frac {\left (e^{3} a^{4}+12 d \,e^{2} a^{3} b +18 d^{2} e \,a^{2} b^{2}+4 a \,b^{3} d^{3}\right ) x^{4}}{4}+\frac {\left (3 d \,e^{2} a^{4}+12 d^{2} e \,a^{3} b +6 a^{2} b^{2} d^{3}\right ) x^{3}}{3}+\frac {\left (3 d^{2} e \,a^{4}+4 a^{3} b \,d^{3}\right ) x^{2}}{2}+a^{4} d^{3} x\) | \(229\) |
risch | \(\frac {1}{8} e^{3} b^{4} x^{8}+\frac {4}{7} x^{7} a \,b^{3} e^{3}+\frac {3}{7} x^{7} b^{4} d \,e^{2}+x^{6} a^{2} b^{2} e^{3}+2 x^{6} d \,e^{2} a \,b^{3}+\frac {1}{2} x^{6} d^{2} e \,b^{4}+\frac {4}{5} x^{5} a^{3} b \,e^{3}+\frac {18}{5} x^{5} a^{2} b^{2} d \,e^{2}+\frac {12}{5} x^{5} d^{2} e a \,b^{3}+\frac {1}{5} x^{5} b^{4} d^{3}+\frac {1}{4} a^{4} e^{3} x^{4}+3 x^{4} d \,e^{2} a^{3} b +\frac {9}{2} x^{4} d^{2} e \,a^{2} b^{2}+x^{4} a \,b^{3} d^{3}+a^{4} d \,e^{2} x^{3}+4 a^{3} b \,d^{2} e \,x^{3}+2 a^{2} b^{2} d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{4} x^{2}+2 a^{3} b \,d^{3} x^{2}+a^{4} d^{3} x\) | \(246\) |
parallelrisch | \(\frac {1}{8} e^{3} b^{4} x^{8}+\frac {4}{7} x^{7} a \,b^{3} e^{3}+\frac {3}{7} x^{7} b^{4} d \,e^{2}+x^{6} a^{2} b^{2} e^{3}+2 x^{6} d \,e^{2} a \,b^{3}+\frac {1}{2} x^{6} d^{2} e \,b^{4}+\frac {4}{5} x^{5} a^{3} b \,e^{3}+\frac {18}{5} x^{5} a^{2} b^{2} d \,e^{2}+\frac {12}{5} x^{5} d^{2} e a \,b^{3}+\frac {1}{5} x^{5} b^{4} d^{3}+\frac {1}{4} a^{4} e^{3} x^{4}+3 x^{4} d \,e^{2} a^{3} b +\frac {9}{2} x^{4} d^{2} e \,a^{2} b^{2}+x^{4} a \,b^{3} d^{3}+a^{4} d \,e^{2} x^{3}+4 a^{3} b \,d^{2} e \,x^{3}+2 a^{2} b^{2} d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{4} x^{2}+2 a^{3} b \,d^{3} x^{2}+a^{4} d^{3} x\) | \(246\) |
gosper | \(\frac {x \left (35 e^{3} b^{4} x^{7}+160 x^{6} a \,b^{3} e^{3}+120 x^{6} b^{4} d \,e^{2}+280 x^{5} a^{2} b^{2} e^{3}+560 x^{5} d \,e^{2} a \,b^{3}+140 x^{5} d^{2} e \,b^{4}+224 x^{4} a^{3} b \,e^{3}+1008 x^{4} a^{2} b^{2} d \,e^{2}+672 x^{4} d^{2} e a \,b^{3}+56 x^{4} b^{4} d^{3}+70 x^{3} e^{3} a^{4}+840 x^{3} d \,e^{2} a^{3} b +1260 x^{3} d^{2} e \,a^{2} b^{2}+280 x^{3} a \,b^{3} d^{3}+280 a^{4} d \,e^{2} x^{2}+1120 a^{3} b \,d^{2} e \,x^{2}+560 a^{2} b^{2} d^{3} x^{2}+420 x \,d^{2} e \,a^{4}+560 a^{3} b \,d^{3} x +280 a^{4} d^{3}\right )}{280}\) | \(248\) |
orering | \(\frac {x \left (35 e^{3} b^{4} x^{7}+160 x^{6} a \,b^{3} e^{3}+120 x^{6} b^{4} d \,e^{2}+280 x^{5} a^{2} b^{2} e^{3}+560 x^{5} d \,e^{2} a \,b^{3}+140 x^{5} d^{2} e \,b^{4}+224 x^{4} a^{3} b \,e^{3}+1008 x^{4} a^{2} b^{2} d \,e^{2}+672 x^{4} d^{2} e a \,b^{3}+56 x^{4} b^{4} d^{3}+70 x^{3} e^{3} a^{4}+840 x^{3} d \,e^{2} a^{3} b +1260 x^{3} d^{2} e \,a^{2} b^{2}+280 x^{3} a \,b^{3} d^{3}+280 a^{4} d \,e^{2} x^{2}+1120 a^{3} b \,d^{2} e \,x^{2}+560 a^{2} b^{2} d^{3} x^{2}+420 x \,d^{2} e \,a^{4}+560 a^{3} b \,d^{3} x +280 a^{4} d^{3}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}{280 \left (b x +a \right )^{4}}\) | \(273\) |
Input:
int((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)
Output:
1/8*e^3*b^4*x^8+(4/7*a*b^3*e^3+3/7*b^4*d*e^2)*x^7+(a^2*b^2*e^3+2*d*e^2*a*b ^3+1/2*d^2*e*b^4)*x^6+(4/5*a^3*b*e^3+18/5*a^2*b^2*d*e^2+12/5*d^2*e*a*b^3+1 /5*b^4*d^3)*x^5+(1/4*e^3*a^4+3*d*e^2*a^3*b+9/2*d^2*e*a^2*b^2+a*b^3*d^3)*x^ 4+(a^4*d*e^2+4*a^3*b*d^2*e+2*a^2*b^2*d^3)*x^3+(3/2*d^2*e*a^4+2*a^3*b*d^3)* x^2+a^4*d^3*x
Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (84) = 168\).
Time = 0.08 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.45 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, b^{4} e^{3} x^{8} + a^{4} d^{3} x + \frac {1}{7} \, {\left (3 \, b^{4} d e^{2} + 4 \, a b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b^{4} d^{2} e + 4 \, a b^{3} d e^{2} + 2 \, a^{2} b^{2} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{3} + 12 \, a b^{3} d^{2} e + 18 \, a^{2} b^{2} d e^{2} + 4 \, a^{3} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} d^{3} + 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} + a^{4} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{3} + 4 \, a^{3} b d^{2} e + a^{4} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \] Input:
integrate((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
Output:
1/8*b^4*e^3*x^8 + a^4*d^3*x + 1/7*(3*b^4*d*e^2 + 4*a*b^3*e^3)*x^7 + 1/2*(b ^4*d^2*e + 4*a*b^3*d*e^2 + 2*a^2*b^2*e^3)*x^6 + 1/5*(b^4*d^3 + 12*a*b^3*d^ 2*e + 18*a^2*b^2*d*e^2 + 4*a^3*b*e^3)*x^5 + 1/4*(4*a*b^3*d^3 + 18*a^2*b^2* d^2*e + 12*a^3*b*d*e^2 + a^4*e^3)*x^4 + (2*a^2*b^2*d^3 + 4*a^3*b*d^2*e + a ^4*d*e^2)*x^3 + 1/2*(4*a^3*b*d^3 + 3*a^4*d^2*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (80) = 160\).
Time = 0.03 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.64 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^{4} d^{3} x + \frac {b^{4} e^{3} x^{8}}{8} + x^{7} \cdot \left (\frac {4 a b^{3} e^{3}}{7} + \frac {3 b^{4} d e^{2}}{7}\right ) + x^{6} \left (a^{2} b^{2} e^{3} + 2 a b^{3} d e^{2} + \frac {b^{4} d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {4 a^{3} b e^{3}}{5} + \frac {18 a^{2} b^{2} d e^{2}}{5} + \frac {12 a b^{3} d^{2} e}{5} + \frac {b^{4} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{4} e^{3}}{4} + 3 a^{3} b d e^{2} + \frac {9 a^{2} b^{2} d^{2} e}{2} + a b^{3} d^{3}\right ) + x^{3} \left (a^{4} d e^{2} + 4 a^{3} b d^{2} e + 2 a^{2} b^{2} d^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{4} d^{2} e}{2} + 2 a^{3} b d^{3}\right ) \] Input:
integrate((e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**2,x)
Output:
a**4*d**3*x + b**4*e**3*x**8/8 + x**7*(4*a*b**3*e**3/7 + 3*b**4*d*e**2/7) + x**6*(a**2*b**2*e**3 + 2*a*b**3*d*e**2 + b**4*d**2*e/2) + x**5*(4*a**3*b *e**3/5 + 18*a**2*b**2*d*e**2/5 + 12*a*b**3*d**2*e/5 + b**4*d**3/5) + x**4 *(a**4*e**3/4 + 3*a**3*b*d*e**2 + 9*a**2*b**2*d**2*e/2 + a*b**3*d**3) + x* *3*(a**4*d*e**2 + 4*a**3*b*d**2*e + 2*a**2*b**2*d**3) + x**2*(3*a**4*d**2* e/2 + 2*a**3*b*d**3)
Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (84) = 168\).
Time = 0.04 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.45 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, b^{4} e^{3} x^{8} + a^{4} d^{3} x + \frac {1}{7} \, {\left (3 \, b^{4} d e^{2} + 4 \, a b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b^{4} d^{2} e + 4 \, a b^{3} d e^{2} + 2 \, a^{2} b^{2} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{3} + 12 \, a b^{3} d^{2} e + 18 \, a^{2} b^{2} d e^{2} + 4 \, a^{3} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} d^{3} + 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} + a^{4} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{3} + 4 \, a^{3} b d^{2} e + a^{4} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \] Input:
integrate((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
Output:
1/8*b^4*e^3*x^8 + a^4*d^3*x + 1/7*(3*b^4*d*e^2 + 4*a*b^3*e^3)*x^7 + 1/2*(b ^4*d^2*e + 4*a*b^3*d*e^2 + 2*a^2*b^2*e^3)*x^6 + 1/5*(b^4*d^3 + 12*a*b^3*d^ 2*e + 18*a^2*b^2*d*e^2 + 4*a^3*b*e^3)*x^5 + 1/4*(4*a*b^3*d^3 + 18*a^2*b^2* d^2*e + 12*a^3*b*d*e^2 + a^4*e^3)*x^4 + (2*a^2*b^2*d^3 + 4*a^3*b*d^2*e + a ^4*d*e^2)*x^3 + 1/2*(4*a^3*b*d^3 + 3*a^4*d^2*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (84) = 168\).
Time = 0.23 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.66 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, b^{4} e^{3} x^{8} + \frac {3}{7} \, b^{4} d e^{2} x^{7} + \frac {4}{7} \, a b^{3} e^{3} x^{7} + \frac {1}{2} \, b^{4} d^{2} e x^{6} + 2 \, a b^{3} d e^{2} x^{6} + a^{2} b^{2} e^{3} x^{6} + \frac {1}{5} \, b^{4} d^{3} x^{5} + \frac {12}{5} \, a b^{3} d^{2} e x^{5} + \frac {18}{5} \, a^{2} b^{2} d e^{2} x^{5} + \frac {4}{5} \, a^{3} b e^{3} x^{5} + a b^{3} d^{3} x^{4} + \frac {9}{2} \, a^{2} b^{2} d^{2} e x^{4} + 3 \, a^{3} b d e^{2} x^{4} + \frac {1}{4} \, a^{4} e^{3} x^{4} + 2 \, a^{2} b^{2} d^{3} x^{3} + 4 \, a^{3} b d^{2} e x^{3} + a^{4} d e^{2} x^{3} + 2 \, a^{3} b d^{3} x^{2} + \frac {3}{2} \, a^{4} d^{2} e x^{2} + a^{4} d^{3} x \] Input:
integrate((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
Output:
1/8*b^4*e^3*x^8 + 3/7*b^4*d*e^2*x^7 + 4/7*a*b^3*e^3*x^7 + 1/2*b^4*d^2*e*x^ 6 + 2*a*b^3*d*e^2*x^6 + a^2*b^2*e^3*x^6 + 1/5*b^4*d^3*x^5 + 12/5*a*b^3*d^2 *e*x^5 + 18/5*a^2*b^2*d*e^2*x^5 + 4/5*a^3*b*e^3*x^5 + a*b^3*d^3*x^4 + 9/2* a^2*b^2*d^2*e*x^4 + 3*a^3*b*d*e^2*x^4 + 1/4*a^4*e^3*x^4 + 2*a^2*b^2*d^3*x^ 3 + 4*a^3*b*d^2*e*x^3 + a^4*d*e^2*x^3 + 2*a^3*b*d^3*x^2 + 3/2*a^4*d^2*e*x^ 2 + a^4*d^3*x
Time = 0.08 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.26 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^4\,\left (\frac {a^4\,e^3}{4}+3\,a^3\,b\,d\,e^2+\frac {9\,a^2\,b^2\,d^2\,e}{2}+a\,b^3\,d^3\right )+x^5\,\left (\frac {4\,a^3\,b\,e^3}{5}+\frac {18\,a^2\,b^2\,d\,e^2}{5}+\frac {12\,a\,b^3\,d^2\,e}{5}+\frac {b^4\,d^3}{5}\right )+a^4\,d^3\,x+\frac {b^4\,e^3\,x^8}{8}+\frac {a^3\,d^2\,x^2\,\left (3\,a\,e+4\,b\,d\right )}{2}+\frac {b^3\,e^2\,x^7\,\left (4\,a\,e+3\,b\,d\right )}{7}+a^2\,d\,x^3\,\left (a^2\,e^2+4\,a\,b\,d\,e+2\,b^2\,d^2\right )+\frac {b^2\,e\,x^6\,\left (2\,a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )}{2} \] Input:
int((d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
Output:
x^4*((a^4*e^3)/4 + a*b^3*d^3 + (9*a^2*b^2*d^2*e)/2 + 3*a^3*b*d*e^2) + x^5* ((b^4*d^3)/5 + (4*a^3*b*e^3)/5 + (18*a^2*b^2*d*e^2)/5 + (12*a*b^3*d^2*e)/5 ) + a^4*d^3*x + (b^4*e^3*x^8)/8 + (a^3*d^2*x^2*(3*a*e + 4*b*d))/2 + (b^3*e ^2*x^7*(4*a*e + 3*b*d))/7 + a^2*d*x^3*(a^2*e^2 + 2*b^2*d^2 + 4*a*b*d*e) + (b^2*e*x^6*(2*a^2*e^2 + b^2*d^2 + 4*a*b*d*e))/2
Time = 0.19 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.68 \[ \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {x \left (35 b^{4} e^{3} x^{7}+160 a \,b^{3} e^{3} x^{6}+120 b^{4} d \,e^{2} x^{6}+280 a^{2} b^{2} e^{3} x^{5}+560 a \,b^{3} d \,e^{2} x^{5}+140 b^{4} d^{2} e \,x^{5}+224 a^{3} b \,e^{3} x^{4}+1008 a^{2} b^{2} d \,e^{2} x^{4}+672 a \,b^{3} d^{2} e \,x^{4}+56 b^{4} d^{3} x^{4}+70 a^{4} e^{3} x^{3}+840 a^{3} b d \,e^{2} x^{3}+1260 a^{2} b^{2} d^{2} e \,x^{3}+280 a \,b^{3} d^{3} x^{3}+280 a^{4} d \,e^{2} x^{2}+1120 a^{3} b \,d^{2} e \,x^{2}+560 a^{2} b^{2} d^{3} x^{2}+420 a^{4} d^{2} e x +560 a^{3} b \,d^{3} x +280 a^{4} d^{3}\right )}{280} \] Input:
int((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x)
Output:
(x*(280*a**4*d**3 + 420*a**4*d**2*e*x + 280*a**4*d*e**2*x**2 + 70*a**4*e** 3*x**3 + 560*a**3*b*d**3*x + 1120*a**3*b*d**2*e*x**2 + 840*a**3*b*d*e**2*x **3 + 224*a**3*b*e**3*x**4 + 560*a**2*b**2*d**3*x**2 + 1260*a**2*b**2*d**2 *e*x**3 + 1008*a**2*b**2*d*e**2*x**4 + 280*a**2*b**2*e**3*x**5 + 280*a*b** 3*d**3*x**3 + 672*a*b**3*d**2*e*x**4 + 560*a*b**3*d*e**2*x**5 + 160*a*b**3 *e**3*x**6 + 56*b**4*d**3*x**4 + 140*b**4*d**2*e*x**5 + 120*b**4*d*e**2*x* *6 + 35*b**4*e**3*x**7))/280