\(\int (c e+d e x)^3 (a+b (c+d x)^3)^2 \, dx\) [56]

Optimal result

Integrand size = 24, antiderivative size = 60 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=\frac {a^2 e^3 (c+d x)^4}{4 d}+\frac {2 a b e^3 (c+d x)^7}{7 d}+\frac {b^2 e^3 (c+d x)^{10}}{10 d} \] Output:

1/4*a^2*e^3*(d*x+c)^4/d+2/7*a*b*e^3*(d*x+c)^7/d+1/10*b^2*e^3*(d*x+c)^10/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(207\) vs. \(2(60)=120\).

Time = 0.01 (sec) , antiderivative size = 207, normalized size of antiderivative = 3.45 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=e^3 \left (c^3 \left (a+b c^3\right )^2 x+\frac {3}{2} c^2 \left (a^2+4 a b c^3+3 b^2 c^6\right ) d x^2+c \left (a^2+10 a b c^3+12 b^2 c^6\right ) d^2 x^3+\frac {1}{4} \left (a^2+40 a b c^3+84 b^2 c^6\right ) d^3 x^4+\frac {6}{5} b c^2 \left (5 a+21 b c^3\right ) d^4 x^5+b c \left (2 a+21 b c^3\right ) d^5 x^6+\frac {2}{7} b \left (a+42 b c^3\right ) d^6 x^7+\frac {9}{2} b^2 c^2 d^7 x^8+b^2 c d^8 x^9+\frac {1}{10} b^2 d^9 x^{10}\right ) \] Input:

Integrate[(c*e + d*e*x)^3*(a + b*(c + d*x)^3)^2,x]
 

Output:

e^3*(c^3*(a + b*c^3)^2*x + (3*c^2*(a^2 + 4*a*b*c^3 + 3*b^2*c^6)*d*x^2)/2 + 
 c*(a^2 + 10*a*b*c^3 + 12*b^2*c^6)*d^2*x^3 + ((a^2 + 40*a*b*c^3 + 84*b^2*c 
^6)*d^3*x^4)/4 + (6*b*c^2*(5*a + 21*b*c^3)*d^4*x^5)/5 + b*c*(2*a + 21*b*c^ 
3)*d^5*x^6 + (2*b*(a + 42*b*c^3)*d^6*x^7)/7 + (9*b^2*c^2*d^7*x^8)/2 + b^2* 
c*d^8*x^9 + (b^2*d^9*x^10)/10)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {895, 802, 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.

\(\displaystyle \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx\)

\(\Big \downarrow \) 895

\(\displaystyle \frac {e^3 \int (c+d x)^3 \left (b (c+d x)^3+a\right )^2d(c+d x)}{d}\)

\(\Big \downarrow \) 802

\(\displaystyle \frac {e^3 \int \left (b^2 (c+d x)^9+2 a b (c+d x)^6+a^2 (c+d x)^3\right )d(c+d x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^3 \left (\frac {1}{4} a^2 (c+d x)^4+\frac {2}{7} a b (c+d x)^7+\frac {1}{10} b^2 (c+d x)^{10}\right )}{d}\)

Input:

Int[(c*e + d*e*x)^3*(a + b*(c + d*x)^3)^2,x]
 

Output:

(e^3*((a^2*(c + d*x)^4)/4 + (2*a*b*(c + d*x)^7)/7 + (b^2*(c + d*x)^10)/10) 
)/d
 

Defintions of rubi rules used

rule 802

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp 
andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && 
IGtQ[p, 0]
 

rule 895

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff 
icient[v, x, 1]*v^m)   Subst[Int[x^m*(a + b*x^n)^p, x], x, v], x] /; FreeQ[ 
{a, b, m, n, p}, x] && LinearPairQ[u, v, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs. \(2(54)=108\).

Time = 0.65 (sec) , antiderivative size = 250, normalized size of antiderivative = 4.17

method	result	size
gosper	\(\frac {e^{3} x \left (14 d^{9} b^{2} x^{9}+140 c \,d^{8} b^{2} x^{8}+630 b^{2} c^{2} d^{7} x^{7}+1680 x^{6} c^{3} b^{2} d^{6}+2940 b^{2} c^{4} d^{5} x^{5}+3528 x^{4} c^{5} b^{2} d^{4}+40 x^{6} a b \,d^{6}+2940 x^{3} b^{2} c^{6} d^{3}+280 a b c \,d^{5} x^{5}+1680 b^{2} c^{7} d^{2} x^{2}+840 x^{4} a b \,c^{2} d^{4}+630 x \,b^{2} c^{8} d +1400 x^{3} a b \,c^{3} d^{3}+140 b^{2} c^{9}+1400 a b \,c^{4} d^{2} x^{2}+840 x a b \,c^{5} d +35 a^{2} d^{3} x^{3}+280 a b \,c^{6}+140 a^{2} c \,d^{2} x^{2}+210 a^{2} c^{2} d x +140 a^{2} c^{3}\right )}{140}\)	\(250\)
norman	\(\left (12 c^{3} e^{3} b^{2} d^{6}+\frac {2}{7} a b \,d^{6} e^{3}\right ) x^{7}+\left (\frac {126}{5} c^{5} e^{3} b^{2} d^{4}+6 a b \,c^{2} d^{4} e^{3}\right ) x^{5}+\left (21 b^{2} c^{6} d^{3} e^{3}+10 a b \,c^{3} d^{3} e^{3}+\frac {1}{4} a^{2} d^{3} e^{3}\right ) x^{4}+\left (\frac {9}{2} b^{2} c^{8} d \,e^{3}+6 a b \,c^{5} d \,e^{3}+\frac {3}{2} a^{2} c^{2} d \,e^{3}\right ) x^{2}+\left (21 c^{4} e^{3} b^{2} d^{5}+2 a b c \,d^{5} e^{3}\right ) x^{6}+\left (b^{2} c^{9} e^{3}+2 a b \,c^{6} e^{3}+a^{2} c^{3} e^{3}\right ) x +\left (12 b^{2} c^{7} d^{2} e^{3}+10 a b \,c^{4} d^{2} e^{3}+a^{2} c \,d^{2} e^{3}\right ) x^{3}+c \,e^{3} d^{8} b^{2} x^{9}+\frac {e^{3} d^{9} b^{2} x^{10}}{10}+\frac {9 c^{2} e^{3} d^{7} b^{2} x^{8}}{2}\)	\(297\)
risch	\(\frac {1}{10} e^{3} d^{9} b^{2} x^{10}+c \,e^{3} d^{8} b^{2} x^{9}+\frac {9}{2} c^{2} e^{3} d^{7} b^{2} x^{8}+12 e^{3} x^{7} c^{3} b^{2} d^{6}+\frac {2}{7} e^{3} x^{7} a b \,d^{6}+21 e^{3} b^{2} c^{4} d^{5} x^{6}+2 e^{3} a b c \,d^{5} x^{6}+\frac {126}{5} e^{3} x^{5} c^{5} b^{2} d^{4}+6 e^{3} x^{5} a b \,c^{2} d^{4}+21 e^{3} x^{4} b^{2} c^{6} d^{3}+10 e^{3} x^{4} a b \,c^{3} d^{3}+\frac {1}{4} e^{3} a^{2} d^{3} x^{4}+12 e^{3} b^{2} c^{7} d^{2} x^{3}+10 e^{3} a b \,c^{4} d^{2} x^{3}+e^{3} a^{2} c \,d^{2} x^{3}+\frac {9}{2} e^{3} x^{2} b^{2} c^{8} d +6 e^{3} x^{2} a b \,c^{5} d +\frac {3}{2} e^{3} a^{2} c^{2} d \,x^{2}+e^{3} b^{2} c^{9} x +2 e^{3} a b \,c^{6} x +a^{2} c^{3} e^{3} x\)	\(312\)
parallelrisch	\(\frac {1}{10} e^{3} d^{9} b^{2} x^{10}+c \,e^{3} d^{8} b^{2} x^{9}+\frac {9}{2} c^{2} e^{3} d^{7} b^{2} x^{8}+12 e^{3} x^{7} c^{3} b^{2} d^{6}+\frac {2}{7} e^{3} x^{7} a b \,d^{6}+21 e^{3} b^{2} c^{4} d^{5} x^{6}+2 e^{3} a b c \,d^{5} x^{6}+\frac {126}{5} e^{3} x^{5} c^{5} b^{2} d^{4}+6 e^{3} x^{5} a b \,c^{2} d^{4}+21 e^{3} x^{4} b^{2} c^{6} d^{3}+10 e^{3} x^{4} a b \,c^{3} d^{3}+\frac {1}{4} e^{3} a^{2} d^{3} x^{4}+12 e^{3} b^{2} c^{7} d^{2} x^{3}+10 e^{3} a b \,c^{4} d^{2} x^{3}+e^{3} a^{2} c \,d^{2} x^{3}+\frac {9}{2} e^{3} x^{2} b^{2} c^{8} d +6 e^{3} x^{2} a b \,c^{5} d +\frac {3}{2} e^{3} a^{2} c^{2} d \,x^{2}+e^{3} b^{2} c^{9} x +2 e^{3} a b \,c^{6} x +a^{2} c^{3} e^{3} x\)	\(312\)
orering	\(\frac {x \left (14 d^{9} b^{2} x^{9}+140 c \,d^{8} b^{2} x^{8}+630 b^{2} c^{2} d^{7} x^{7}+1680 x^{6} c^{3} b^{2} d^{6}+2940 b^{2} c^{4} d^{5} x^{5}+3528 x^{4} c^{5} b^{2} d^{4}+40 x^{6} a b \,d^{6}+2940 x^{3} b^{2} c^{6} d^{3}+280 a b c \,d^{5} x^{5}+1680 b^{2} c^{7} d^{2} x^{2}+840 x^{4} a b \,c^{2} d^{4}+630 x \,b^{2} c^{8} d +1400 x^{3} a b \,c^{3} d^{3}+140 b^{2} c^{9}+1400 a b \,c^{4} d^{2} x^{2}+840 x a b \,c^{5} d +35 a^{2} d^{3} x^{3}+280 a b \,c^{6}+140 a^{2} c \,d^{2} x^{2}+210 a^{2} c^{2} d x +140 a^{2} c^{3}\right ) \left (d e x +c e \right )^{3} \left (a +b \left (x d +c \right )^{3}\right )^{2}}{140 \left (x d +c \right )^{3} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )^{2}}\)	\(312\)
default	\(\frac {e^{3} d^{9} b^{2} x^{10}}{10}+c \,e^{3} d^{8} b^{2} x^{9}+\frac {9 c^{2} e^{3} d^{7} b^{2} x^{8}}{2}+\frac {\left (64 c^{3} e^{3} b^{2} d^{6}+e^{3} d^{3} \left (2 \left (c^{3} b +a \right ) b \,d^{3}+18 b^{2} c^{3} d^{3}\right )\right ) x^{7}}{7}+\frac {\left (51 c^{4} e^{3} b^{2} d^{5}+3 c \,e^{3} d^{2} \left (2 \left (c^{3} b +a \right ) b \,d^{3}+18 b^{2} c^{3} d^{3}\right )+e^{3} d^{3} \left (6 \left (c^{3} b +a \right ) b c \,d^{2}+9 b^{2} c^{4} d^{2}\right )\right ) x^{6}}{6}+\frac {\left (15 c^{5} e^{3} b^{2} d^{4}+3 c^{2} e^{3} d \left (2 \left (c^{3} b +a \right ) b \,d^{3}+18 b^{2} c^{3} d^{3}\right )+3 c \,e^{3} d^{2} \left (6 \left (c^{3} b +a \right ) b c \,d^{2}+9 b^{2} c^{4} d^{2}\right )+6 e^{3} d^{4} \left (c^{3} b +a \right ) b \,c^{2}\right ) x^{5}}{5}+\frac {\left (c^{3} e^{3} \left (2 \left (c^{3} b +a \right ) b \,d^{3}+18 b^{2} c^{3} d^{3}\right )+3 c^{2} e^{3} d \left (6 \left (c^{3} b +a \right ) b c \,d^{2}+9 b^{2} c^{4} d^{2}\right )+18 c^{3} e^{3} d^{3} \left (c^{3} b +a \right ) b +e^{3} d^{3} \left (c^{3} b +a \right )^{2}\right ) x^{4}}{4}+\frac {\left (c^{3} e^{3} \left (6 \left (c^{3} b +a \right ) b c \,d^{2}+9 b^{2} c^{4} d^{2}\right )+18 c^{4} e^{3} d^{2} \left (c^{3} b +a \right ) b +3 c \,e^{3} d^{2} \left (c^{3} b +a \right )^{2}\right ) x^{3}}{3}+\frac {\left (6 c^{5} e^{3} \left (c^{3} b +a \right ) b d +3 c^{2} e^{3} d \left (c^{3} b +a \right )^{2}\right ) x^{2}}{2}+c^{3} e^{3} \left (c^{3} b +a \right )^{2} x\)	\(536\)

Input:

int((d*e*x+c*e)^3*(a+b*(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
 

Output:

1/140*e^3*x*(14*b^2*d^9*x^9+140*b^2*c*d^8*x^8+630*b^2*c^2*d^7*x^7+1680*b^2 
*c^3*d^6*x^6+2940*b^2*c^4*d^5*x^5+3528*b^2*c^5*d^4*x^4+40*a*b*d^6*x^6+2940 
*b^2*c^6*d^3*x^3+280*a*b*c*d^5*x^5+1680*b^2*c^7*d^2*x^2+840*a*b*c^2*d^4*x^ 
4+630*b^2*c^8*d*x+1400*a*b*c^3*d^3*x^3+140*b^2*c^9+1400*a*b*c^4*d^2*x^2+84 
0*a*b*c^5*d*x+35*a^2*d^3*x^3+280*a*b*c^6+140*a^2*c*d^2*x^2+210*a^2*c^2*d*x 
+140*a^2*c^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (54) = 108\).

Time = 0.07 (sec) , antiderivative size = 240, normalized size of antiderivative = 4.00 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=\frac {1}{10} \, b^{2} d^{9} e^{3} x^{10} + b^{2} c d^{8} e^{3} x^{9} + \frac {9}{2} \, b^{2} c^{2} d^{7} e^{3} x^{8} + \frac {2}{7} \, {\left (42 \, b^{2} c^{3} + a b\right )} d^{6} e^{3} x^{7} + {\left (21 \, b^{2} c^{4} + 2 \, a b c\right )} d^{5} e^{3} x^{6} + \frac {6}{5} \, {\left (21 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{4} e^{3} x^{5} + \frac {1}{4} \, {\left (84 \, b^{2} c^{6} + 40 \, a b c^{3} + a^{2}\right )} d^{3} e^{3} x^{4} + {\left (12 \, b^{2} c^{7} + 10 \, a b c^{4} + a^{2} c\right )} d^{2} e^{3} x^{3} + \frac {3}{2} \, {\left (3 \, b^{2} c^{8} + 4 \, a b c^{5} + a^{2} c^{2}\right )} d e^{3} x^{2} + {\left (b^{2} c^{9} + 2 \, a b c^{6} + a^{2} c^{3}\right )} e^{3} x \] Input:

integrate((d*e*x+c*e)^3*(a+b*(d*x+c)^3)^2,x, algorithm="fricas")
 

Output:

1/10*b^2*d^9*e^3*x^10 + b^2*c*d^8*e^3*x^9 + 9/2*b^2*c^2*d^7*e^3*x^8 + 2/7* 
(42*b^2*c^3 + a*b)*d^6*e^3*x^7 + (21*b^2*c^4 + 2*a*b*c)*d^5*e^3*x^6 + 6/5* 
(21*b^2*c^5 + 5*a*b*c^2)*d^4*e^3*x^5 + 1/4*(84*b^2*c^6 + 40*a*b*c^3 + a^2) 
*d^3*e^3*x^4 + (12*b^2*c^7 + 10*a*b*c^4 + a^2*c)*d^2*e^3*x^3 + 3/2*(3*b^2* 
c^8 + 4*a*b*c^5 + a^2*c^2)*d*e^3*x^2 + (b^2*c^9 + 2*a*b*c^6 + a^2*c^3)*e^3 
*x
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (51) = 102\).

Time = 0.06 (sec) , antiderivative size = 323, normalized size of antiderivative = 5.38 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=\frac {9 b^{2} c^{2} d^{7} e^{3} x^{8}}{2} + b^{2} c d^{8} e^{3} x^{9} + \frac {b^{2} d^{9} e^{3} x^{10}}{10} + x^{7} \cdot \left (\frac {2 a b d^{6} e^{3}}{7} + 12 b^{2} c^{3} d^{6} e^{3}\right ) + x^{6} \cdot \left (2 a b c d^{5} e^{3} + 21 b^{2} c^{4} d^{5} e^{3}\right ) + x^{5} \cdot \left (6 a b c^{2} d^{4} e^{3} + \frac {126 b^{2} c^{5} d^{4} e^{3}}{5}\right ) + x^{4} \left (\frac {a^{2} d^{3} e^{3}}{4} + 10 a b c^{3} d^{3} e^{3} + 21 b^{2} c^{6} d^{3} e^{3}\right ) + x^{3} \left (a^{2} c d^{2} e^{3} + 10 a b c^{4} d^{2} e^{3} + 12 b^{2} c^{7} d^{2} e^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c^{2} d e^{3}}{2} + 6 a b c^{5} d e^{3} + \frac {9 b^{2} c^{8} d e^{3}}{2}\right ) + x \left (a^{2} c^{3} e^{3} + 2 a b c^{6} e^{3} + b^{2} c^{9} e^{3}\right ) \] Input:

integrate((d*e*x+c*e)**3*(a+b*(d*x+c)**3)**2,x)
 

Output:

9*b**2*c**2*d**7*e**3*x**8/2 + b**2*c*d**8*e**3*x**9 + b**2*d**9*e**3*x**1 
0/10 + x**7*(2*a*b*d**6*e**3/7 + 12*b**2*c**3*d**6*e**3) + x**6*(2*a*b*c*d 
**5*e**3 + 21*b**2*c**4*d**5*e**3) + x**5*(6*a*b*c**2*d**4*e**3 + 126*b**2 
*c**5*d**4*e**3/5) + x**4*(a**2*d**3*e**3/4 + 10*a*b*c**3*d**3*e**3 + 21*b 
**2*c**6*d**3*e**3) + x**3*(a**2*c*d**2*e**3 + 10*a*b*c**4*d**2*e**3 + 12* 
b**2*c**7*d**2*e**3) + x**2*(3*a**2*c**2*d*e**3/2 + 6*a*b*c**5*d*e**3 + 9* 
b**2*c**8*d*e**3/2) + x*(a**2*c**3*e**3 + 2*a*b*c**6*e**3 + b**2*c**9*e**3 
)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (54) = 108\).

Time = 0.03 (sec) , antiderivative size = 240, normalized size of antiderivative = 4.00 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=\frac {1}{10} \, b^{2} d^{9} e^{3} x^{10} + b^{2} c d^{8} e^{3} x^{9} + \frac {9}{2} \, b^{2} c^{2} d^{7} e^{3} x^{8} + \frac {2}{7} \, {\left (42 \, b^{2} c^{3} + a b\right )} d^{6} e^{3} x^{7} + {\left (21 \, b^{2} c^{4} + 2 \, a b c\right )} d^{5} e^{3} x^{6} + \frac {6}{5} \, {\left (21 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{4} e^{3} x^{5} + \frac {1}{4} \, {\left (84 \, b^{2} c^{6} + 40 \, a b c^{3} + a^{2}\right )} d^{3} e^{3} x^{4} + {\left (12 \, b^{2} c^{7} + 10 \, a b c^{4} + a^{2} c\right )} d^{2} e^{3} x^{3} + \frac {3}{2} \, {\left (3 \, b^{2} c^{8} + 4 \, a b c^{5} + a^{2} c^{2}\right )} d e^{3} x^{2} + {\left (b^{2} c^{9} + 2 \, a b c^{6} + a^{2} c^{3}\right )} e^{3} x \] Input:

integrate((d*e*x+c*e)^3*(a+b*(d*x+c)^3)^2,x, algorithm="maxima")
 

Output:

1/10*b^2*d^9*e^3*x^10 + b^2*c*d^8*e^3*x^9 + 9/2*b^2*c^2*d^7*e^3*x^8 + 2/7* 
(42*b^2*c^3 + a*b)*d^6*e^3*x^7 + (21*b^2*c^4 + 2*a*b*c)*d^5*e^3*x^6 + 6/5* 
(21*b^2*c^5 + 5*a*b*c^2)*d^4*e^3*x^5 + 1/4*(84*b^2*c^6 + 40*a*b*c^3 + a^2) 
*d^3*e^3*x^4 + (12*b^2*c^7 + 10*a*b*c^4 + a^2*c)*d^2*e^3*x^3 + 3/2*(3*b^2* 
c^8 + 4*a*b*c^5 + a^2*c^2)*d*e^3*x^2 + (b^2*c^9 + 2*a*b*c^6 + a^2*c^3)*e^3 
*x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (54) = 108\).

Time = 0.11 (sec) , antiderivative size = 311, normalized size of antiderivative = 5.18 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=\frac {1}{10} \, b^{2} d^{9} e^{3} x^{10} + b^{2} c d^{8} e^{3} x^{9} + \frac {9}{2} \, b^{2} c^{2} d^{7} e^{3} x^{8} + 12 \, b^{2} c^{3} d^{6} e^{3} x^{7} + 21 \, b^{2} c^{4} d^{5} e^{3} x^{6} + \frac {126}{5} \, b^{2} c^{5} d^{4} e^{3} x^{5} + 21 \, b^{2} c^{6} d^{3} e^{3} x^{4} + \frac {2}{7} \, a b d^{6} e^{3} x^{7} + 12 \, b^{2} c^{7} d^{2} e^{3} x^{3} + 2 \, a b c d^{5} e^{3} x^{6} + \frac {9}{2} \, b^{2} c^{8} d e^{3} x^{2} + 6 \, a b c^{2} d^{4} e^{3} x^{5} + b^{2} c^{9} e^{3} x + 10 \, a b c^{3} d^{3} e^{3} x^{4} + 10 \, a b c^{4} d^{2} e^{3} x^{3} + 6 \, a b c^{5} d e^{3} x^{2} + 2 \, a b c^{6} e^{3} x + \frac {1}{4} \, a^{2} d^{3} e^{3} x^{4} + a^{2} c d^{2} e^{3} x^{3} + \frac {3}{2} \, a^{2} c^{2} d e^{3} x^{2} + a^{2} c^{3} e^{3} x \] Input:

integrate((d*e*x+c*e)^3*(a+b*(d*x+c)^3)^2,x, algorithm="giac")
 

Output:

1/10*b^2*d^9*e^3*x^10 + b^2*c*d^8*e^3*x^9 + 9/2*b^2*c^2*d^7*e^3*x^8 + 12*b 
^2*c^3*d^6*e^3*x^7 + 21*b^2*c^4*d^5*e^3*x^6 + 126/5*b^2*c^5*d^4*e^3*x^5 + 
21*b^2*c^6*d^3*e^3*x^4 + 2/7*a*b*d^6*e^3*x^7 + 12*b^2*c^7*d^2*e^3*x^3 + 2* 
a*b*c*d^5*e^3*x^6 + 9/2*b^2*c^8*d*e^3*x^2 + 6*a*b*c^2*d^4*e^3*x^5 + b^2*c^ 
9*e^3*x + 10*a*b*c^3*d^3*e^3*x^4 + 10*a*b*c^4*d^2*e^3*x^3 + 6*a*b*c^5*d*e^ 
3*x^2 + 2*a*b*c^6*e^3*x + 1/4*a^2*d^3*e^3*x^4 + a^2*c*d^2*e^3*x^3 + 3/2*a^ 
2*c^2*d*e^3*x^2 + a^2*c^3*e^3*x
 

Mupad [B] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 221, normalized size of antiderivative = 3.68 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=c^3\,e^3\,x\,{\left (b\,c^3+a\right )}^2+\frac {b^2\,d^9\,e^3\,x^{10}}{10}+\frac {d^3\,e^3\,x^4\,\left (a^2+40\,a\,b\,c^3+84\,b^2\,c^6\right )}{4}+\frac {3\,c^2\,d\,e^3\,x^2\,\left (a^2+4\,a\,b\,c^3+3\,b^2\,c^6\right )}{2}+c\,d^2\,e^3\,x^3\,\left (a^2+10\,a\,b\,c^3+12\,b^2\,c^6\right )+\frac {9\,b^2\,c^2\,d^7\,e^3\,x^8}{2}+\frac {2\,b\,d^6\,e^3\,x^7\,\left (42\,b\,c^3+a\right )}{7}+b^2\,c\,d^8\,e^3\,x^9+b\,c\,d^5\,e^3\,x^6\,\left (21\,b\,c^3+2\,a\right )+\frac {6\,b\,c^2\,d^4\,e^3\,x^5\,\left (21\,b\,c^3+5\,a\right )}{5} \] Input:

int((c*e + d*e*x)^3*(a + b*(c + d*x)^3)^2,x)
 

Output:

c^3*e^3*x*(a + b*c^3)^2 + (b^2*d^9*e^3*x^10)/10 + (d^3*e^3*x^4*(a^2 + 84*b 
^2*c^6 + 40*a*b*c^3))/4 + (3*c^2*d*e^3*x^2*(a^2 + 3*b^2*c^6 + 4*a*b*c^3))/ 
2 + c*d^2*e^3*x^3*(a^2 + 12*b^2*c^6 + 10*a*b*c^3) + (9*b^2*c^2*d^7*e^3*x^8 
)/2 + (2*b*d^6*e^3*x^7*(a + 42*b*c^3))/7 + b^2*c*d^8*e^3*x^9 + b*c*d^5*e^3 
*x^6*(2*a + 21*b*c^3) + (6*b*c^2*d^4*e^3*x^5*(5*a + 21*b*c^3))/5
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 249, normalized size of antiderivative = 4.15 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=\frac {e^{3} x \left (14 b^{2} d^{9} x^{9}+140 b^{2} c \,d^{8} x^{8}+630 b^{2} c^{2} d^{7} x^{7}+1680 b^{2} c^{3} d^{6} x^{6}+2940 b^{2} c^{4} d^{5} x^{5}+3528 b^{2} c^{5} d^{4} x^{4}+40 a b \,d^{6} x^{6}+2940 b^{2} c^{6} d^{3} x^{3}+280 a b c \,d^{5} x^{5}+1680 b^{2} c^{7} d^{2} x^{2}+840 a b \,c^{2} d^{4} x^{4}+630 b^{2} c^{8} d x +1400 a b \,c^{3} d^{3} x^{3}+140 b^{2} c^{9}+1400 a b \,c^{4} d^{2} x^{2}+840 a b \,c^{5} d x +35 a^{2} d^{3} x^{3}+280 a b \,c^{6}+140 a^{2} c \,d^{2} x^{2}+210 a^{2} c^{2} d x +140 a^{2} c^{3}\right )}{140} \] Input:

int((d*e*x+c*e)^3*(a+b*(d*x+c)^3)^2,x)
 

Output:

(e**3*x*(140*a**2*c**3 + 210*a**2*c**2*d*x + 140*a**2*c*d**2*x**2 + 35*a** 
2*d**3*x**3 + 280*a*b*c**6 + 840*a*b*c**5*d*x + 1400*a*b*c**4*d**2*x**2 + 
1400*a*b*c**3*d**3*x**3 + 840*a*b*c**2*d**4*x**4 + 280*a*b*c*d**5*x**5 + 4 
0*a*b*d**6*x**6 + 140*b**2*c**9 + 630*b**2*c**8*d*x + 1680*b**2*c**7*d**2* 
x**2 + 2940*b**2*c**6*d**3*x**3 + 3528*b**2*c**5*d**4*x**4 + 2940*b**2*c** 
4*d**5*x**5 + 1680*b**2*c**3*d**6*x**6 + 630*b**2*c**2*d**7*x**7 + 140*b** 
2*c*d**8*x**8 + 14*b**2*d**9*x**9))/140