\(\int (c+d x)^3 (a+b (c+d x)^2)^3 \, dx\) [42]

Optimal result

Integrand size = 21, antiderivative size = 48 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^3 \, dx=-\frac {a \left (a+b (c+d x)^2\right )^4}{8 b^2 d}+\frac {\left (a+b (c+d x)^2\right )^5}{10 b^2 d} \] Output:

-1/8*a*(a+b*(d*x+c)^2)^4/b^2/d+1/10*(a+b*(d*x+c)^2)^5/b^2/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(249\) vs. \(2(48)=96\).

Time = 0.02 (sec) , antiderivative size = 249, normalized size of antiderivative = 5.19 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^3 \, dx=c^3 \left (a+b c^2\right )^3 x+\frac {3}{2} c^2 \left (a+b c^2\right )^2 \left (a+3 b c^2\right ) d x^2+c \left (a^3+10 a^2 b c^2+21 a b^2 c^4+12 b^3 c^6\right ) d^2 x^3+\frac {1}{4} \left (a^3+30 a^2 b c^2+105 a b^2 c^4+84 b^3 c^6\right ) d^3 x^4+\frac {3}{5} b c \left (5 a^2+35 a b c^2+42 b^2 c^4\right ) d^4 x^5+\frac {1}{2} b \left (a^2+21 a b c^2+42 b^2 c^4\right ) d^5 x^6+3 b^2 c \left (a+4 b c^2\right ) d^6 x^7+\frac {3}{8} b^2 \left (a+12 b c^2\right ) d^7 x^8+b^3 c d^8 x^9+\frac {1}{10} b^3 d^9 x^{10} \] Input:

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

Output:

c^3*(a + b*c^2)^3*x + (3*c^2*(a + b*c^2)^2*(a + 3*b*c^2)*d*x^2)/2 + c*(a^3 
 + 10*a^2*b*c^2 + 21*a*b^2*c^4 + 12*b^3*c^6)*d^2*x^3 + ((a^3 + 30*a^2*b*c^ 
2 + 105*a*b^2*c^4 + 84*b^3*c^6)*d^3*x^4)/4 + (3*b*c*(5*a^2 + 35*a*b*c^2 + 
42*b^2*c^4)*d^4*x^5)/5 + (b*(a^2 + 21*a*b*c^2 + 42*b^2*c^4)*d^5*x^6)/2 + 3 
*b^2*c*(a + 4*b*c^2)*d^6*x^7 + (3*b^2*(a + 12*b*c^2)*d^7*x^8)/8 + b^3*c*d^ 
8*x^9 + (b^3*d^9*x^10)/10
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {895, 243, 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.

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

\(\Big \downarrow \) 895

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 49

\(\displaystyle \frac {\int \left (\frac {\left (b (c+d x)^2+a\right )^4}{b}-\frac {a \left (b (c+d x)^2+a\right )^3}{b}\right )d(c+d x)^2}{2 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\left (a+b (c+d x)^2\right )^5}{5 b^2}-\frac {a \left (a+b (c+d x)^2\right )^4}{4 b^2}}{2 d}\)

Input:

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

Output:

(-1/4*(a*(a + b*(c + d*x)^2)^4)/b^2 + (a + b*(c + d*x)^2)^5/(5*b^2))/(2*d)
 

Defintions of rubi rules used

rule 49

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]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 [A] (verified)

Time = 1.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {\frac {a \left (a +b \left (x d +c \right )^{2}\right )^{4}}{4}-\frac {\left (a +b \left (x d +c \right )^{2}\right )^{5}}{5}}{2 d \,b^{2}}\)	\(41\)
norman	\(\frac {d^{9} b^{3} x^{10}}{10}+c \,d^{8} b^{3} x^{9}+\left (\frac {9}{2} c^{2} d^{7} b^{3}+\frac {3}{8} a \,b^{2} d^{7}\right ) x^{8}+\left (12 c^{3} b^{3} d^{6}+3 a \,b^{2} c \,d^{6}\right ) x^{7}+\left (21 c^{4} b^{3} d^{5}+\frac {21}{2} a \,b^{2} c^{2} d^{5}+\frac {1}{2} a^{2} b \,d^{5}\right ) x^{6}+\left (\frac {126}{5} b^{3} c^{5} d^{4}+21 a \,b^{2} c^{3} d^{4}+3 a^{2} b c \,d^{4}\right ) x^{5}+\left (21 b^{3} c^{6} d^{3}+\frac {105}{4} a \,b^{2} c^{4} d^{3}+\frac {15}{2} a^{2} b \,c^{2} d^{3}+\frac {1}{4} a^{3} d^{3}\right ) x^{4}+\left (12 b^{3} c^{7} d^{2}+21 a \,b^{2} c^{5} d^{2}+10 a^{2} b \,c^{3} d^{2}+a^{3} c \,d^{2}\right ) x^{3}+\left (\frac {9}{2} b^{3} c^{8} d +\frac {21}{2} a \,b^{2} c^{6} d +\frac {15}{2} a^{2} b \,c^{4} d +\frac {3}{2} a^{3} c^{2} d \right ) x^{2}+\left (b^{3} c^{9}+3 a \,b^{2} c^{7}+3 a^{2} b \,c^{5}+c^{3} a^{3}\right ) x\)	\(324\)
gosper	\(\frac {x \left (4 d^{9} b^{3} x^{9}+40 c \,d^{8} b^{3} x^{8}+180 x^{7} c^{2} d^{7} b^{3}+480 b^{3} c^{3} d^{6} x^{6}+15 x^{7} a \,b^{2} d^{7}+840 x^{5} c^{4} b^{3} d^{5}+120 a \,b^{2} c \,d^{6} x^{6}+1008 x^{4} b^{3} c^{5} d^{4}+420 x^{5} a \,b^{2} c^{2} d^{5}+840 x^{3} b^{3} c^{6} d^{3}+840 x^{4} a \,b^{2} c^{3} d^{4}+480 b^{3} c^{7} d^{2} x^{2}+20 a^{2} b \,d^{5} x^{5}+1050 x^{3} a \,b^{2} c^{4} d^{3}+180 x \,b^{3} c^{8} d +120 x^{4} a^{2} b c \,d^{4}+840 a \,b^{2} c^{5} d^{2} x^{2}+40 b^{3} c^{9}+300 x^{3} a^{2} b \,c^{2} d^{3}+420 x a \,b^{2} c^{6} d +400 a^{2} b \,c^{3} d^{2} x^{2}+120 a \,b^{2} c^{7}+10 x^{3} a^{3} d^{3}+300 x \,a^{2} b \,c^{4} d +40 a^{3} c \,d^{2} x^{2}+120 a^{2} b \,c^{5}+60 a^{3} c^{2} d x +40 c^{3} a^{3}\right )}{40}\)	\(351\)
parallelrisch	\(3 b \,a^{2} c^{5} x +\frac {3}{2} d \,a^{3} c^{2} x^{2}+\frac {15}{2} d b \,a^{2} c^{4} x^{2}+d^{2} a^{3} c \,x^{3}+a^{3} c^{3} x +\frac {21}{2} d^{5} b^{2} a \,c^{2} x^{6}+21 d^{4} b^{2} a \,c^{3} x^{5}+\frac {105}{4} d^{3} b^{2} a \,c^{4} x^{4}+3 d^{6} b^{2} a c \,x^{7}+c \,d^{8} b^{3} x^{9}+21 d^{2} b^{2} a \,c^{5} x^{3}+\frac {15}{2} d^{3} b \,a^{2} c^{2} x^{4}+\frac {21}{2} d \,b^{2} a \,c^{6} x^{2}+10 d^{2} b \,a^{2} c^{3} x^{3}+3 b^{2} a \,c^{7} x +3 a^{2} b c \,d^{4} x^{5}+\frac {3}{8} d^{7} b^{2} a \,x^{8}+\frac {1}{2} d^{5} b \,a^{2} x^{6}+\frac {9}{2} d^{7} b^{3} c^{2} x^{8}+12 d^{6} b^{3} c^{3} x^{7}+21 d^{5} b^{3} c^{4} x^{6}+\frac {126}{5} d^{4} b^{3} c^{5} x^{5}+21 d^{3} b^{3} c^{6} x^{4}+12 d^{2} b^{3} c^{7} x^{3}+\frac {9}{2} d \,b^{3} c^{8} x^{2}+b^{3} c^{9} x +\frac {1}{4} a^{3} d^{3} x^{4}+\frac {1}{10} d^{9} b^{3} x^{10}\)	\(356\)
orering	\(\frac {x \left (4 d^{9} b^{3} x^{9}+40 c \,d^{8} b^{3} x^{8}+180 x^{7} c^{2} d^{7} b^{3}+480 b^{3} c^{3} d^{6} x^{6}+15 x^{7} a \,b^{2} d^{7}+840 x^{5} c^{4} b^{3} d^{5}+120 a \,b^{2} c \,d^{6} x^{6}+1008 x^{4} b^{3} c^{5} d^{4}+420 x^{5} a \,b^{2} c^{2} d^{5}+840 x^{3} b^{3} c^{6} d^{3}+840 x^{4} a \,b^{2} c^{3} d^{4}+480 b^{3} c^{7} d^{2} x^{2}+20 a^{2} b \,d^{5} x^{5}+1050 x^{3} a \,b^{2} c^{4} d^{3}+180 x \,b^{3} c^{8} d +120 x^{4} a^{2} b c \,d^{4}+840 a \,b^{2} c^{5} d^{2} x^{2}+40 b^{3} c^{9}+300 x^{3} a^{2} b \,c^{2} d^{3}+420 x a \,b^{2} c^{6} d +400 a^{2} b \,c^{3} d^{2} x^{2}+120 a \,b^{2} c^{7}+10 x^{3} a^{3} d^{3}+300 x \,a^{2} b \,c^{4} d +40 a^{3} c \,d^{2} x^{2}+120 a^{2} b \,c^{5}+60 a^{3} c^{2} d x +40 c^{3} a^{3}\right ) \left (a +b \left (x d +c \right )^{2}\right )^{3}}{40 \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{3}}\)	\(387\)
risch	\(3 b \,a^{2} c^{5} x +\frac {3 d \,a^{3} c^{2} x^{2}}{2}+\frac {15 d b \,a^{2} c^{4} x^{2}}{2}+d^{2} a^{3} c \,x^{3}+a^{3} c^{3} x +\frac {21 d^{5} b^{2} a \,c^{2} x^{6}}{2}+21 d^{4} b^{2} a \,c^{3} x^{5}+\frac {105 d^{3} b^{2} a \,c^{4} x^{4}}{4}+3 d^{6} b^{2} a c \,x^{7}+c \,d^{8} b^{3} x^{9}+21 d^{2} b^{2} a \,c^{5} x^{3}+\frac {15 d^{3} b \,a^{2} c^{2} x^{4}}{2}+\frac {21 d \,b^{2} a \,c^{6} x^{2}}{2}+10 d^{2} b \,a^{2} c^{3} x^{3}+3 b^{2} a \,c^{7} x +3 a^{2} b c \,d^{4} x^{5}+\frac {3 d^{7} b^{2} a \,x^{8}}{8}+\frac {d^{5} b \,a^{2} x^{6}}{2}+\frac {9 d^{7} b^{3} c^{2} x^{8}}{2}+12 d^{6} b^{3} c^{3} x^{7}+21 d^{5} b^{3} c^{4} x^{6}+\frac {126 d^{4} b^{3} c^{5} x^{5}}{5}+21 d^{3} b^{3} c^{6} x^{4}+12 d^{2} b^{3} c^{7} x^{3}+\frac {9 d \,b^{3} c^{8} x^{2}}{2}+b^{3} c^{9} x +\frac {3 b^{2} a \,c^{8}}{8 d}+\frac {b \,a^{2} c^{6}}{2 d}+\frac {a^{3} c^{4}}{4 d}+\frac {a^{3} d^{3} x^{4}}{4}-\frac {a^{5}}{40 d \,b^{2}}+\frac {b^{3} c^{10}}{10 d}+\frac {d^{9} b^{3} x^{10}}{10}\)	\(413\)

Input:

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

Output:

-1/2/d/b^2*(1/4*a*(a+b*(d*x+c)^2)^4-1/5*(a+b*(d*x+c)^2)^5)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (44) = 88\).

Time = 0.08 (sec) , antiderivative size = 284, normalized size of antiderivative = 5.92 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^3 \, dx=\frac {1}{10} \, b^{3} d^{9} x^{10} + b^{3} c d^{8} x^{9} + \frac {3}{8} \, {\left (12 \, b^{3} c^{2} + a b^{2}\right )} d^{7} x^{8} + 3 \, {\left (4 \, b^{3} c^{3} + a b^{2} c\right )} d^{6} x^{7} + \frac {1}{2} \, {\left (42 \, b^{3} c^{4} + 21 \, a b^{2} c^{2} + a^{2} b\right )} d^{5} x^{6} + \frac {3}{5} \, {\left (42 \, b^{3} c^{5} + 35 \, a b^{2} c^{3} + 5 \, a^{2} b c\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (84 \, b^{3} c^{6} + 105 \, a b^{2} c^{4} + 30 \, a^{2} b c^{2} + a^{3}\right )} d^{3} x^{4} + {\left (12 \, b^{3} c^{7} + 21 \, a b^{2} c^{5} + 10 \, a^{2} b c^{3} + a^{3} c\right )} d^{2} x^{3} + \frac {3}{2} \, {\left (3 \, b^{3} c^{8} + 7 \, a b^{2} c^{6} + 5 \, a^{2} b c^{4} + a^{3} c^{2}\right )} d x^{2} + {\left (b^{3} c^{9} + 3 \, a b^{2} c^{7} + 3 \, a^{2} b c^{5} + a^{3} c^{3}\right )} x \] Input:

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

Output:

1/10*b^3*d^9*x^10 + b^3*c*d^8*x^9 + 3/8*(12*b^3*c^2 + a*b^2)*d^7*x^8 + 3*( 
4*b^3*c^3 + a*b^2*c)*d^6*x^7 + 1/2*(42*b^3*c^4 + 21*a*b^2*c^2 + a^2*b)*d^5 
*x^6 + 3/5*(42*b^3*c^5 + 35*a*b^2*c^3 + 5*a^2*b*c)*d^4*x^5 + 1/4*(84*b^3*c 
^6 + 105*a*b^2*c^4 + 30*a^2*b*c^2 + a^3)*d^3*x^4 + (12*b^3*c^7 + 21*a*b^2* 
c^5 + 10*a^2*b*c^3 + a^3*c)*d^2*x^3 + 3/2*(3*b^3*c^8 + 7*a*b^2*c^6 + 5*a^2 
*b*c^4 + a^3*c^2)*d*x^2 + (b^3*c^9 + 3*a*b^2*c^7 + 3*a^2*b*c^5 + a^3*c^3)* 
x
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (37) = 74\).

Time = 0.09 (sec) , antiderivative size = 357, normalized size of antiderivative = 7.44 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^3 \, dx=b^{3} c d^{8} x^{9} + \frac {b^{3} d^{9} x^{10}}{10} + x^{8} \cdot \left (\frac {3 a b^{2} d^{7}}{8} + \frac {9 b^{3} c^{2} d^{7}}{2}\right ) + x^{7} \cdot \left (3 a b^{2} c d^{6} + 12 b^{3} c^{3} d^{6}\right ) + x^{6} \left (\frac {a^{2} b d^{5}}{2} + \frac {21 a b^{2} c^{2} d^{5}}{2} + 21 b^{3} c^{4} d^{5}\right ) + x^{5} \cdot \left (3 a^{2} b c d^{4} + 21 a b^{2} c^{3} d^{4} + \frac {126 b^{3} c^{5} d^{4}}{5}\right ) + x^{4} \left (\frac {a^{3} d^{3}}{4} + \frac {15 a^{2} b c^{2} d^{3}}{2} + \frac {105 a b^{2} c^{4} d^{3}}{4} + 21 b^{3} c^{6} d^{3}\right ) + x^{3} \left (a^{3} c d^{2} + 10 a^{2} b c^{3} d^{2} + 21 a b^{2} c^{5} d^{2} + 12 b^{3} c^{7} d^{2}\right ) + x^{2} \cdot \left (\frac {3 a^{3} c^{2} d}{2} + \frac {15 a^{2} b c^{4} d}{2} + \frac {21 a b^{2} c^{6} d}{2} + \frac {9 b^{3} c^{8} d}{2}\right ) + x \left (a^{3} c^{3} + 3 a^{2} b c^{5} + 3 a b^{2} c^{7} + b^{3} c^{9}\right ) \] Input:

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

Output:

b**3*c*d**8*x**9 + b**3*d**9*x**10/10 + x**8*(3*a*b**2*d**7/8 + 9*b**3*c** 
2*d**7/2) + x**7*(3*a*b**2*c*d**6 + 12*b**3*c**3*d**6) + x**6*(a**2*b*d**5 
/2 + 21*a*b**2*c**2*d**5/2 + 21*b**3*c**4*d**5) + x**5*(3*a**2*b*c*d**4 + 
21*a*b**2*c**3*d**4 + 126*b**3*c**5*d**4/5) + x**4*(a**3*d**3/4 + 15*a**2* 
b*c**2*d**3/2 + 105*a*b**2*c**4*d**3/4 + 21*b**3*c**6*d**3) + x**3*(a**3*c 
*d**2 + 10*a**2*b*c**3*d**2 + 21*a*b**2*c**5*d**2 + 12*b**3*c**7*d**2) + x 
**2*(3*a**3*c**2*d/2 + 15*a**2*b*c**4*d/2 + 21*a*b**2*c**6*d/2 + 9*b**3*c* 
*8*d/2) + x*(a**3*c**3 + 3*a**2*b*c**5 + 3*a*b**2*c**7 + b**3*c**9)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (44) = 88\).

Time = 0.04 (sec) , antiderivative size = 284, normalized size of antiderivative = 5.92 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^3 \, dx=\frac {1}{10} \, b^{3} d^{9} x^{10} + b^{3} c d^{8} x^{9} + \frac {3}{8} \, {\left (12 \, b^{3} c^{2} + a b^{2}\right )} d^{7} x^{8} + 3 \, {\left (4 \, b^{3} c^{3} + a b^{2} c\right )} d^{6} x^{7} + \frac {1}{2} \, {\left (42 \, b^{3} c^{4} + 21 \, a b^{2} c^{2} + a^{2} b\right )} d^{5} x^{6} + \frac {3}{5} \, {\left (42 \, b^{3} c^{5} + 35 \, a b^{2} c^{3} + 5 \, a^{2} b c\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (84 \, b^{3} c^{6} + 105 \, a b^{2} c^{4} + 30 \, a^{2} b c^{2} + a^{3}\right )} d^{3} x^{4} + {\left (12 \, b^{3} c^{7} + 21 \, a b^{2} c^{5} + 10 \, a^{2} b c^{3} + a^{3} c\right )} d^{2} x^{3} + \frac {3}{2} \, {\left (3 \, b^{3} c^{8} + 7 \, a b^{2} c^{6} + 5 \, a^{2} b c^{4} + a^{3} c^{2}\right )} d x^{2} + {\left (b^{3} c^{9} + 3 \, a b^{2} c^{7} + 3 \, a^{2} b c^{5} + a^{3} c^{3}\right )} x \] Input:

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

Output:

1/10*b^3*d^9*x^10 + b^3*c*d^8*x^9 + 3/8*(12*b^3*c^2 + a*b^2)*d^7*x^8 + 3*( 
4*b^3*c^3 + a*b^2*c)*d^6*x^7 + 1/2*(42*b^3*c^4 + 21*a*b^2*c^2 + a^2*b)*d^5 
*x^6 + 3/5*(42*b^3*c^5 + 35*a*b^2*c^3 + 5*a^2*b*c)*d^4*x^5 + 1/4*(84*b^3*c 
^6 + 105*a*b^2*c^4 + 30*a^2*b*c^2 + a^3)*d^3*x^4 + (12*b^3*c^7 + 21*a*b^2* 
c^5 + 10*a^2*b*c^3 + a^3*c)*d^2*x^3 + 3/2*(3*b^3*c^8 + 7*a*b^2*c^6 + 5*a^2 
*b*c^4 + a^3*c^2)*d*x^2 + (b^3*c^9 + 3*a*b^2*c^7 + 3*a^2*b*c^5 + a^3*c^3)* 
x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (44) = 88\).

Time = 0.12 (sec) , antiderivative size = 288, normalized size of antiderivative = 6.00 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^3 \, dx=\frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} b^{3} c^{8} + {\left (d x^{2} + 2 \, c x\right )}^{2} b^{3} c^{6} d + {\left (d x^{2} + 2 \, c x\right )}^{3} b^{3} c^{4} d^{2} + \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )}^{4} b^{3} c^{2} d^{3} + \frac {1}{10} \, {\left (d x^{2} + 2 \, c x\right )}^{5} b^{3} d^{4} + \frac {3}{2} \, {\left (d x^{2} + 2 \, c x\right )} a b^{2} c^{6} + \frac {9}{4} \, {\left (d x^{2} + 2 \, c x\right )}^{2} a b^{2} c^{4} d + \frac {3}{2} \, {\left (d x^{2} + 2 \, c x\right )}^{3} a b^{2} c^{2} d^{2} + \frac {3}{8} \, {\left (d x^{2} + 2 \, c x\right )}^{4} a b^{2} d^{3} + \frac {3}{2} \, {\left (d x^{2} + 2 \, c x\right )} a^{2} b c^{4} + \frac {3}{2} \, {\left (d x^{2} + 2 \, c x\right )}^{2} a^{2} b c^{2} d + \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )}^{3} a^{2} b d^{2} + \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} a^{3} c^{2} + \frac {1}{4} \, {\left (d x^{2} + 2 \, c x\right )}^{2} a^{3} d \] Input:

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

Output:

1/2*(d*x^2 + 2*c*x)*b^3*c^8 + (d*x^2 + 2*c*x)^2*b^3*c^6*d + (d*x^2 + 2*c*x 
)^3*b^3*c^4*d^2 + 1/2*(d*x^2 + 2*c*x)^4*b^3*c^2*d^3 + 1/10*(d*x^2 + 2*c*x) 
^5*b^3*d^4 + 3/2*(d*x^2 + 2*c*x)*a*b^2*c^6 + 9/4*(d*x^2 + 2*c*x)^2*a*b^2*c 
^4*d + 3/2*(d*x^2 + 2*c*x)^3*a*b^2*c^2*d^2 + 3/8*(d*x^2 + 2*c*x)^4*a*b^2*d 
^3 + 3/2*(d*x^2 + 2*c*x)*a^2*b*c^4 + 3/2*(d*x^2 + 2*c*x)^2*a^2*b*c^2*d + 1 
/2*(d*x^2 + 2*c*x)^3*a^2*b*d^2 + 1/2*(d*x^2 + 2*c*x)*a^3*c^2 + 1/4*(d*x^2 
+ 2*c*x)^2*a^3*d
 

Mupad [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 237, normalized size of antiderivative = 4.94 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^3 \, dx=c^3\,x\,{\left (b\,c^2+a\right )}^3+\frac {d^3\,x^4\,\left (a^3+30\,a^2\,b\,c^2+105\,a\,b^2\,c^4+84\,b^3\,c^6\right )}{4}+\frac {b^3\,d^9\,x^{10}}{10}+\frac {3\,b^2\,d^7\,x^8\,\left (12\,b\,c^2+a\right )}{8}+b^3\,c\,d^8\,x^9+\frac {b\,d^5\,x^6\,\left (a^2+21\,a\,b\,c^2+42\,b^2\,c^4\right )}{2}+c\,d^2\,x^3\,\left (a^3+10\,a^2\,b\,c^2+21\,a\,b^2\,c^4+12\,b^3\,c^6\right )+\frac {3\,b\,c\,d^4\,x^5\,\left (5\,a^2+35\,a\,b\,c^2+42\,b^2\,c^4\right )}{5}+\frac {3\,c^2\,d\,x^2\,{\left (b\,c^2+a\right )}^2\,\left (3\,b\,c^2+a\right )}{2}+3\,b^2\,c\,d^6\,x^7\,\left (4\,b\,c^2+a\right ) \] Input:

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

Output:

c^3*x*(a + b*c^2)^3 + (d^3*x^4*(a^3 + 84*b^3*c^6 + 30*a^2*b*c^2 + 105*a*b^ 
2*c^4))/4 + (b^3*d^9*x^10)/10 + (3*b^2*d^7*x^8*(a + 12*b*c^2))/8 + b^3*c*d 
^8*x^9 + (b*d^5*x^6*(a^2 + 42*b^2*c^4 + 21*a*b*c^2))/2 + c*d^2*x^3*(a^3 + 
12*b^3*c^6 + 10*a^2*b*c^2 + 21*a*b^2*c^4) + (3*b*c*d^4*x^5*(5*a^2 + 42*b^2 
*c^4 + 35*a*b*c^2))/5 + (3*c^2*d*x^2*(a + b*c^2)^2*(a + 3*b*c^2))/2 + 3*b^ 
2*c*d^6*x^7*(a + 4*b*c^2)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 350, normalized size of antiderivative = 7.29 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^3 \, dx=\frac {x \left (4 b^{3} d^{9} x^{9}+40 b^{3} c \,d^{8} x^{8}+180 b^{3} c^{2} d^{7} x^{7}+480 b^{3} c^{3} d^{6} x^{6}+15 a \,b^{2} d^{7} x^{7}+840 b^{3} c^{4} d^{5} x^{5}+120 a \,b^{2} c \,d^{6} x^{6}+1008 b^{3} c^{5} d^{4} x^{4}+420 a \,b^{2} c^{2} d^{5} x^{5}+840 b^{3} c^{6} d^{3} x^{3}+840 a \,b^{2} c^{3} d^{4} x^{4}+480 b^{3} c^{7} d^{2} x^{2}+20 a^{2} b \,d^{5} x^{5}+1050 a \,b^{2} c^{4} d^{3} x^{3}+180 b^{3} c^{8} d x +120 a^{2} b c \,d^{4} x^{4}+840 a \,b^{2} c^{5} d^{2} x^{2}+40 b^{3} c^{9}+300 a^{2} b \,c^{2} d^{3} x^{3}+420 a \,b^{2} c^{6} d x +400 a^{2} b \,c^{3} d^{2} x^{2}+120 a \,b^{2} c^{7}+10 a^{3} d^{3} x^{3}+300 a^{2} b \,c^{4} d x +40 a^{3} c \,d^{2} x^{2}+120 a^{2} b \,c^{5}+60 a^{3} c^{2} d x +40 a^{3} c^{3}\right )}{40} \] Input:

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

Output:

(x*(40*a**3*c**3 + 60*a**3*c**2*d*x + 40*a**3*c*d**2*x**2 + 10*a**3*d**3*x 
**3 + 120*a**2*b*c**5 + 300*a**2*b*c**4*d*x + 400*a**2*b*c**3*d**2*x**2 + 
300*a**2*b*c**2*d**3*x**3 + 120*a**2*b*c*d**4*x**4 + 20*a**2*b*d**5*x**5 + 
 120*a*b**2*c**7 + 420*a*b**2*c**6*d*x + 840*a*b**2*c**5*d**2*x**2 + 1050* 
a*b**2*c**4*d**3*x**3 + 840*a*b**2*c**3*d**4*x**4 + 420*a*b**2*c**2*d**5*x 
**5 + 120*a*b**2*c*d**6*x**6 + 15*a*b**2*d**7*x**7 + 40*b**3*c**9 + 180*b* 
*3*c**8*d*x + 480*b**3*c**7*d**2*x**2 + 840*b**3*c**6*d**3*x**3 + 1008*b** 
3*c**5*d**4*x**4 + 840*b**3*c**4*d**5*x**5 + 480*b**3*c**3*d**6*x**6 + 180 
*b**3*c**2*d**7*x**7 + 40*b**3*c*d**8*x**8 + 4*b**3*d**9*x**9))/40