\(\int (c e+d e x)^3 (a+b (c+d x)^3)^2 \, dx\) [2857]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {379, 276} \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=\frac {a^2 e^3 (c+d x)^4}{4 d}+\frac {2 a b e^3 (c+d x)^7}{7 d}+\frac {b^2 e^3 (c+d x)^{10}}{10 d} \]

[In]

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

[Out]

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

Rule 276

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

Rule 379

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[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]

Rubi steps \begin{align*} \text {integral}& = \frac {e^3 \text {Subst}\left (\int x^3 \left (a+b x^3\right )^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int \left (a^2 x^3+2 a b x^6+b^2 x^9\right ) \, dx,x,c+d x\right )}{d} \\ & = \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} \\ \end{align*}

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 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=e^3 \left (c^3 \left (a+b c^3\right )^2 x+\frac {3}{2} c^2 \left (a^2+4 a b c^3+3 b^2 c^6\right ) d x^2+c \left (a^2+10 a b c^3+12 b^2 c^6\right ) d^2 x^3+\frac {1}{4} \left (a^2+40 a b c^3+84 b^2 c^6\right ) d^3 x^4+\frac {6}{5} b c^2 \left (5 a+21 b c^3\right ) d^4 x^5+b c \left (2 a+21 b c^3\right ) d^5 x^6+\frac {2}{7} b \left (a+42 b c^3\right ) d^6 x^7+\frac {9}{2} b^2 c^2 d^7 x^8+b^2 c d^8 x^9+\frac {1}{10} b^2 d^9 x^{10}\right ) \]

[In]

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

[Out]

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)

Maple [B] (verified)

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

Time = 3.94 (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	\(\left (12 c^{3} e^{3} b^{2} d^{6}+\frac {2}{7} a b \,d^{6} e^{3}\right ) x^{7}+\left (\frac {126}{5} c^{5} e^{3} b^{2} d^{4}+6 a b \,c^{2} d^{4} e^{3}\right ) x^{5}+\left (21 b^{2} c^{6} d^{3} e^{3}+10 a b \,c^{3} d^{3} e^{3}+\frac {1}{4} a^{2} d^{3} e^{3}\right ) x^{4}+\left (\frac {9}{2} b^{2} c^{8} d \,e^{3}+6 a b \,c^{5} d \,e^{3}+\frac {3}{2} a^{2} c^{2} d \,e^{3}\right ) x^{2}+\left (21 c^{4} e^{3} b^{2} d^{5}+2 a b c \,d^{5} e^{3}\right ) x^{6}+\left (b^{2} c^{9} e^{3}+2 a b \,c^{6} e^{3}+a^{2} c^{3} e^{3}\right ) x +\left (12 b^{2} c^{7} d^{2} e^{3}+10 a b \,c^{4} d^{2} e^{3}+a^{2} c \,d^{2} e^{3}\right ) x^{3}+c \,e^{3} d^{8} b^{2} x^{9}+\frac {d^{9} e^{3} b^{2} x^{10}}{10}+\frac {9 c^{2} e^{3} d^{7} b^{2} x^{8}}{2}\)	\(297\)
risch	\(\frac {1}{10} d^{9} e^{3} b^{2} x^{10}+c \,e^{3} d^{8} b^{2} x^{9}+\frac {9}{2} c^{2} e^{3} d^{7} b^{2} x^{8}+12 e^{3} x^{7} c^{3} b^{2} d^{6}+\frac {2}{7} e^{3} x^{7} a b \,d^{6}+21 e^{3} b^{2} c^{4} d^{5} x^{6}+2 e^{3} a b c \,d^{5} x^{6}+\frac {126}{5} e^{3} x^{5} c^{5} b^{2} d^{4}+6 e^{3} x^{5} a b \,c^{2} d^{4}+21 e^{3} x^{4} b^{2} c^{6} d^{3}+10 e^{3} x^{4} a b \,c^{3} d^{3}+\frac {1}{4} e^{3} a^{2} d^{3} x^{4}+12 e^{3} b^{2} c^{7} d^{2} x^{3}+10 e^{3} a b \,c^{4} d^{2} x^{3}+e^{3} a^{2} c \,d^{2} x^{3}+\frac {9}{2} e^{3} x^{2} b^{2} c^{8} d +6 e^{3} x^{2} a b \,c^{5} d +\frac {3}{2} e^{3} a^{2} c^{2} d \,x^{2}+e^{3} b^{2} c^{9} x +2 e^{3} a b \,c^{6} x +e^{3} a^{2} c^{3} x\)	\(312\)
parallelrisch	\(\frac {1}{10} d^{9} e^{3} b^{2} x^{10}+c \,e^{3} d^{8} b^{2} x^{9}+\frac {9}{2} c^{2} e^{3} d^{7} b^{2} x^{8}+12 e^{3} x^{7} c^{3} b^{2} d^{6}+\frac {2}{7} e^{3} x^{7} a b \,d^{6}+21 e^{3} b^{2} c^{4} d^{5} x^{6}+2 e^{3} a b c \,d^{5} x^{6}+\frac {126}{5} e^{3} x^{5} c^{5} b^{2} d^{4}+6 e^{3} x^{5} a b \,c^{2} d^{4}+21 e^{3} x^{4} b^{2} c^{6} d^{3}+10 e^{3} x^{4} a b \,c^{3} d^{3}+\frac {1}{4} e^{3} a^{2} d^{3} x^{4}+12 e^{3} b^{2} c^{7} d^{2} x^{3}+10 e^{3} a b \,c^{4} d^{2} x^{3}+e^{3} a^{2} c \,d^{2} x^{3}+\frac {9}{2} e^{3} x^{2} b^{2} c^{8} d +6 e^{3} x^{2} a b \,c^{5} d +\frac {3}{2} e^{3} a^{2} c^{2} d \,x^{2}+e^{3} b^{2} c^{9} x +2 e^{3} a b \,c^{6} x +e^{3} a^{2} c^{3} x\)	\(312\)
default	\(\frac {d^{9} e^{3} b^{2} x^{10}}{10}+c \,e^{3} d^{8} b^{2} x^{9}+\frac {9 c^{2} e^{3} d^{7} b^{2} x^{8}}{2}+\frac {\left (64 c^{3} e^{3} b^{2} d^{6}+d^{3} e^{3} \left (2 \left (c^{3} b +a \right ) b \,d^{3}+18 b^{2} c^{3} d^{3}\right )\right ) x^{7}}{7}+\frac {\left (51 c^{4} e^{3} b^{2} d^{5}+3 c \,e^{3} d^{2} \left (2 \left (c^{3} b +a \right ) b \,d^{3}+18 b^{2} c^{3} d^{3}\right )+d^{3} e^{3} \left (6 \left (c^{3} b +a \right ) b c \,d^{2}+9 b^{2} c^{4} d^{2}\right )\right ) x^{6}}{6}+\frac {\left (15 c^{5} e^{3} b^{2} d^{4}+3 c^{2} e^{3} d \left (2 \left (c^{3} b +a \right ) b \,d^{3}+18 b^{2} c^{3} d^{3}\right )+3 c \,e^{3} d^{2} \left (6 \left (c^{3} b +a \right ) b c \,d^{2}+9 b^{2} c^{4} d^{2}\right )+6 d^{4} e^{3} \left (c^{3} b +a \right ) b \,c^{2}\right ) x^{5}}{5}+\frac {\left (c^{3} e^{3} \left (2 \left (c^{3} b +a \right ) b \,d^{3}+18 b^{2} c^{3} d^{3}\right )+3 c^{2} e^{3} d \left (6 \left (c^{3} b +a \right ) b c \,d^{2}+9 b^{2} c^{4} d^{2}\right )+18 c^{3} e^{3} d^{3} \left (c^{3} b +a \right ) b +d^{3} e^{3} \left (c^{3} b +a \right )^{2}\right ) x^{4}}{4}+\frac {\left (c^{3} e^{3} \left (6 \left (c^{3} b +a \right ) b c \,d^{2}+9 b^{2} c^{4} d^{2}\right )+18 c^{4} e^{3} d^{2} \left (c^{3} b +a \right ) b +3 c \,e^{3} d^{2} \left (c^{3} b +a \right )^{2}\right ) x^{3}}{3}+\frac {\left (6 c^{5} e^{3} \left (c^{3} b +a \right ) b d +3 c^{2} e^{3} d \left (c^{3} b +a \right )^{2}\right ) x^{2}}{2}+c^{3} e^{3} \left (c^{3} b +a \right )^{2} x\)	\(536\)

[In]

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

[Out]

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+35
28*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)

Fricas [B] (verification not implemented)

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

Time = 0.24 (sec) , antiderivative size = 240, normalized size of antiderivative = 4.00 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=\frac {1}{10} \, b^{2} d^{9} e^{3} x^{10} + b^{2} c d^{8} e^{3} x^{9} + \frac {9}{2} \, b^{2} c^{2} d^{7} e^{3} x^{8} + \frac {2}{7} \, {\left (42 \, b^{2} c^{3} + a b\right )} d^{6} e^{3} x^{7} + {\left (21 \, b^{2} c^{4} + 2 \, a b c\right )} d^{5} e^{3} x^{6} + \frac {6}{5} \, {\left (21 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{4} e^{3} x^{5} + \frac {1}{4} \, {\left (84 \, b^{2} c^{6} + 40 \, a b c^{3} + a^{2}\right )} d^{3} e^{3} x^{4} + {\left (12 \, b^{2} c^{7} + 10 \, a b c^{4} + a^{2} c\right )} d^{2} e^{3} x^{3} + \frac {3}{2} \, {\left (3 \, b^{2} c^{8} + 4 \, a b c^{5} + a^{2} c^{2}\right )} d e^{3} x^{2} + {\left (b^{2} c^{9} + 2 \, a b c^{6} + a^{2} c^{3}\right )} e^{3} x \]

[In]

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

[Out]

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 = 0.05 (sec) , antiderivative size = 323, normalized size of antiderivative = 5.38 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=\frac {9 b^{2} c^{2} d^{7} e^{3} x^{8}}{2} + b^{2} c d^{8} e^{3} x^{9} + \frac {b^{2} d^{9} e^{3} x^{10}}{10} + x^{7} \cdot \left (\frac {2 a b d^{6} e^{3}}{7} + 12 b^{2} c^{3} d^{6} e^{3}\right ) + x^{6} \cdot \left (2 a b c d^{5} e^{3} + 21 b^{2} c^{4} d^{5} e^{3}\right ) + x^{5} \cdot \left (6 a b c^{2} d^{4} e^{3} + \frac {126 b^{2} c^{5} d^{4} e^{3}}{5}\right ) + x^{4} \left (\frac {a^{2} d^{3} e^{3}}{4} + 10 a b c^{3} d^{3} e^{3} + 21 b^{2} c^{6} d^{3} e^{3}\right ) + x^{3} \left (a^{2} c d^{2} e^{3} + 10 a b c^{4} d^{2} e^{3} + 12 b^{2} c^{7} d^{2} e^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c^{2} d e^{3}}{2} + 6 a b c^{5} d e^{3} + \frac {9 b^{2} c^{8} d e^{3}}{2}\right ) + x \left (a^{2} c^{3} e^{3} + 2 a b c^{6} e^{3} + b^{2} c^{9} e^{3}\right ) \]

[In]

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

[Out]

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**10/10 + 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 = 0.19 (sec) , antiderivative size = 240, normalized size of antiderivative = 4.00 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=\frac {1}{10} \, b^{2} d^{9} e^{3} x^{10} + b^{2} c d^{8} e^{3} x^{9} + \frac {9}{2} \, b^{2} c^{2} d^{7} e^{3} x^{8} + \frac {2}{7} \, {\left (42 \, b^{2} c^{3} + a b\right )} d^{6} e^{3} x^{7} + {\left (21 \, b^{2} c^{4} + 2 \, a b c\right )} d^{5} e^{3} x^{6} + \frac {6}{5} \, {\left (21 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{4} e^{3} x^{5} + \frac {1}{4} \, {\left (84 \, b^{2} c^{6} + 40 \, a b c^{3} + a^{2}\right )} d^{3} e^{3} x^{4} + {\left (12 \, b^{2} c^{7} + 10 \, a b c^{4} + a^{2} c\right )} d^{2} e^{3} x^{3} + \frac {3}{2} \, {\left (3 \, b^{2} c^{8} + 4 \, a b c^{5} + a^{2} c^{2}\right )} d e^{3} x^{2} + {\left (b^{2} c^{9} + 2 \, a b c^{6} + a^{2} c^{3}\right )} e^{3} x \]

[In]

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

[Out]

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 = 0.29 (sec) , antiderivative size = 311, normalized size of antiderivative = 5.18 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=\frac {1}{10} \, b^{2} d^{9} e^{3} x^{10} + b^{2} c d^{8} e^{3} x^{9} + \frac {9}{2} \, b^{2} c^{2} d^{7} e^{3} x^{8} + 12 \, b^{2} c^{3} d^{6} e^{3} x^{7} + 21 \, b^{2} c^{4} d^{5} e^{3} x^{6} + \frac {126}{5} \, b^{2} c^{5} d^{4} e^{3} x^{5} + 21 \, b^{2} c^{6} d^{3} e^{3} x^{4} + \frac {2}{7} \, a b d^{6} e^{3} x^{7} + 12 \, b^{2} c^{7} d^{2} e^{3} x^{3} + 2 \, a b c d^{5} e^{3} x^{6} + \frac {9}{2} \, b^{2} c^{8} d e^{3} x^{2} + 6 \, a b c^{2} d^{4} e^{3} x^{5} + b^{2} c^{9} e^{3} x + 10 \, a b c^{3} d^{3} e^{3} x^{4} + 10 \, a b c^{4} d^{2} e^{3} x^{3} + 6 \, a b c^{5} d e^{3} x^{2} + 2 \, a b c^{6} e^{3} x + \frac {1}{4} \, a^{2} d^{3} e^{3} x^{4} + a^{2} c d^{2} e^{3} x^{3} + \frac {3}{2} \, a^{2} c^{2} d e^{3} x^{2} + a^{2} c^{3} e^{3} x \]

[In]

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

[Out]

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 = 5.53 (sec) , antiderivative size = 221, normalized size of antiderivative = 3.68 \[ \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^2 \, dx=c^3\,e^3\,x\,{\left (b\,c^3+a\right )}^2+\frac {b^2\,d^9\,e^3\,x^{10}}{10}+\frac {d^3\,e^3\,x^4\,\left (a^2+40\,a\,b\,c^3+84\,b^2\,c^6\right )}{4}+\frac {3\,c^2\,d\,e^3\,x^2\,\left (a^2+4\,a\,b\,c^3+3\,b^2\,c^6\right )}{2}+c\,d^2\,e^3\,x^3\,\left (a^2+10\,a\,b\,c^3+12\,b^2\,c^6\right )+\frac {9\,b^2\,c^2\,d^7\,e^3\,x^8}{2}+\frac {2\,b\,d^6\,e^3\,x^7\,\left (42\,b\,c^3+a\right )}{7}+b^2\,c\,d^8\,e^3\,x^9+b\,c\,d^5\,e^3\,x^6\,\left (21\,b\,c^3+2\,a\right )+\frac {6\,b\,c^2\,d^4\,e^3\,x^5\,\left (21\,b\,c^3+5\,a\right )}{5} \]

[In]

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

[Out]

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