Integrand size = 46, antiderivative size = 92 \[ \int (c+d x)^3 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=\frac {(b c-a d)^3 (a+b x)^5}{5 b^4}+\frac {d (b c-a d)^2 (a+b x)^6}{2 b^4}+\frac {3 d^2 (b c-a d) (a+b x)^7}{7 b^4}+\frac {d^3 (a+b x)^8}{8 b^4} \] Output:
1/5*(-a*d+b*c)^3*(b*x+a)^5/b^4+1/2*d*(-a*d+b*c)^2*(b*x+a)^6/b^4+3/7*d^2*(- a*d+b*c)*(b*x+a)^7/b^4+1/8*d^3*(b*x+a)^8/b^4
Leaf count is larger than twice the leaf count of optimal. \(217\) vs. \(2(92)=184\).
Time = 0.01 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.36 \[ \int (c+d x)^3 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=a^4 c^3 x+\frac {1}{2} a^3 c^2 (4 b c+3 a d) x^2+a^2 c \left (2 b^2 c^2+4 a b c d+a^2 d^2\right ) x^3+\frac {1}{4} a \left (4 b^3 c^3+18 a b^2 c^2 d+12 a^2 b c d^2+a^3 d^3\right ) x^4+\frac {1}{5} b \left (b^3 c^3+12 a b^2 c^2 d+18 a^2 b c d^2+4 a^3 d^3\right ) x^5+\frac {1}{2} b^2 d \left (b^2 c^2+4 a b c d+2 a^2 d^2\right ) x^6+\frac {1}{7} b^3 d^2 (3 b c+4 a d) x^7+\frac {1}{8} b^4 d^3 x^8 \] Input:
Integrate[(c + d*x)^3*(a^4 + 4*a^3*b*x + 6*a^2*b^2*x^2 + 4*a*b^3*x^3 + b^4 *x^4),x]
Output:
a^4*c^3*x + (a^3*c^2*(4*b*c + 3*a*d)*x^2)/2 + a^2*c*(2*b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^3 + (a*(4*b^3*c^3 + 18*a*b^2*c^2*d + 12*a^2*b*c*d^2 + a^3*d^ 3)*x^4)/4 + (b*(b^3*c^3 + 12*a*b^2*c^2*d + 18*a^2*b*c*d^2 + 4*a^3*d^3)*x^5 )/5 + (b^2*d*(b^2*c^2 + 4*a*b*c*d + 2*a^2*d^2)*x^6)/2 + (b^3*d^2*(3*b*c + 4*a*d)*x^7)/7 + (b^4*d^3*x^8)/8
Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2006, 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.
| \(\displaystyle \int \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) (c+d x)^3 \, dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int (a+b x)^4 (c+d x)^3dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {3 d^2 (a+b x)^6 (b c-a d)}{b^3}+\frac {3 d (a+b x)^5 (b c-a d)^2}{b^3}+\frac {(a+b x)^4 (b c-a d)^3}{b^3}+\frac {d^3 (a+b x)^7}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 d^2 (a+b x)^7 (b c-a d)}{7 b^4}+\frac {d (a+b x)^6 (b c-a d)^2}{2 b^4}+\frac {(a+b x)^5 (b c-a d)^3}{5 b^4}+\frac {d^3 (a+b x)^8}{8 b^4}\) |
Input:
Int[(c + d*x)^3*(a^4 + 4*a^3*b*x + 6*a^2*b^2*x^2 + 4*a*b^3*x^3 + b^4*x^4), x]
Output:
((b*c - a*d)^3*(a + b*x)^5)/(5*b^4) + (d*(b*c - a*d)^2*(a + b*x)^6)/(2*b^4 ) + (3*d^2*(b*c - a*d)*(a + b*x)^7)/(7*b^4) + (d^3*(a + b*x)^8)/(8*b^4)
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[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(221\) vs. \(2(84)=168\).
Time = 0.08 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.41
| method | result | size |
| norman | \(\frac {b^{4} d^{3} x^{8}}{8}+\left (\frac {4}{7} a \,b^{3} d^{3}+\frac {3}{7} b^{4} c \,d^{2}\right ) x^{7}+\left (a^{2} b^{2} d^{3}+2 a \,b^{3} c \,d^{2}+\frac {1}{2} b^{4} c^{2} d \right ) x^{6}+\left (\frac {4}{5} a^{3} b \,d^{3}+\frac {18}{5} a^{2} b^{2} c \,d^{2}+\frac {12}{5} a \,b^{3} c^{2} d +\frac {1}{5} b^{4} c^{3}\right ) x^{5}+\left (\frac {1}{4} a^{4} d^{3}+3 a^{3} b c \,d^{2}+\frac {9}{2} a^{2} b^{2} c^{2} d +a \,b^{3} c^{3}\right ) x^{4}+\left (a^{4} c \,d^{2}+4 a^{3} b \,c^{2} d +2 a^{2} b^{2} c^{3}\right ) x^{3}+\left (\frac {3}{2} a^{4} c^{2} d +2 a^{3} b \,c^{3}\right ) x^{2}+a^{4} c^{3} x\) | \(222\) |
| default | \(\frac {b^{4} d^{3} x^{8}}{8}+\frac {\left (4 a \,b^{3} d^{3}+3 b^{4} c \,d^{2}\right ) x^{7}}{7}+\frac {\left (6 a^{2} b^{2} d^{3}+12 a \,b^{3} c \,d^{2}+3 b^{4} c^{2} d \right ) x^{6}}{6}+\frac {\left (4 a^{3} b \,d^{3}+18 a^{2} b^{2} c \,d^{2}+12 a \,b^{3} c^{2} d +b^{4} c^{3}\right ) x^{5}}{5}+\frac {\left (a^{4} d^{3}+12 a^{3} b c \,d^{2}+18 a^{2} b^{2} c^{2} d +4 a \,b^{3} c^{3}\right ) x^{4}}{4}+\frac {\left (3 a^{4} c \,d^{2}+12 a^{3} b \,c^{2} d +6 a^{2} b^{2} c^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{4} c^{2} d +4 a^{3} b \,c^{3}\right ) x^{2}}{2}+a^{4} c^{3} x\) | \(229\) |
| risch | \(\frac {1}{8} b^{4} d^{3} x^{8}+\frac {4}{7} x^{7} a \,b^{3} d^{3}+\frac {3}{7} x^{7} b^{4} c \,d^{2}+x^{6} a^{2} b^{2} d^{3}+2 x^{6} a \,b^{3} c \,d^{2}+\frac {1}{2} x^{6} b^{4} c^{2} d +\frac {4}{5} x^{5} a^{3} b \,d^{3}+\frac {18}{5} x^{5} a^{2} b^{2} c \,d^{2}+\frac {12}{5} x^{5} a \,b^{3} c^{2} d +\frac {1}{5} x^{5} b^{4} c^{3}+\frac {1}{4} x^{4} a^{4} d^{3}+3 x^{4} a^{3} b c \,d^{2}+\frac {9}{2} x^{4} a^{2} b^{2} c^{2} d +x^{4} a \,b^{3} c^{3}+a^{4} c \,d^{2} x^{3}+4 a^{3} b \,c^{2} d \,x^{3}+2 a^{2} b^{2} c^{3} x^{3}+\frac {3}{2} x^{2} a^{4} c^{2} d +2 x^{2} a^{3} b \,c^{3}+a^{4} c^{3} x\) | \(246\) |
| parallelrisch | \(\frac {1}{8} b^{4} d^{3} x^{8}+\frac {4}{7} x^{7} a \,b^{3} d^{3}+\frac {3}{7} x^{7} b^{4} c \,d^{2}+x^{6} a^{2} b^{2} d^{3}+2 x^{6} a \,b^{3} c \,d^{2}+\frac {1}{2} x^{6} b^{4} c^{2} d +\frac {4}{5} x^{5} a^{3} b \,d^{3}+\frac {18}{5} x^{5} a^{2} b^{2} c \,d^{2}+\frac {12}{5} x^{5} a \,b^{3} c^{2} d +\frac {1}{5} x^{5} b^{4} c^{3}+\frac {1}{4} x^{4} a^{4} d^{3}+3 x^{4} a^{3} b c \,d^{2}+\frac {9}{2} x^{4} a^{2} b^{2} c^{2} d +x^{4} a \,b^{3} c^{3}+a^{4} c \,d^{2} x^{3}+4 a^{3} b \,c^{2} d \,x^{3}+2 a^{2} b^{2} c^{3} x^{3}+\frac {3}{2} x^{2} a^{4} c^{2} d +2 x^{2} a^{3} b \,c^{3}+a^{4} c^{3} x\) | \(246\) |
| gosper | \(\frac {x \left (35 b^{4} d^{3} x^{7}+160 x^{6} a \,b^{3} d^{3}+120 x^{6} b^{4} c \,d^{2}+280 a^{2} b^{2} d^{3} x^{5}+560 a \,b^{3} c \,d^{2} x^{5}+140 b^{4} c^{2} d \,x^{5}+224 x^{4} a^{3} b \,d^{3}+1008 a^{2} b^{2} c \,d^{2} x^{4}+672 x^{4} a \,b^{3} c^{2} d +56 x^{4} b^{4} c^{3}+70 x^{3} a^{4} d^{3}+840 x^{3} a^{3} b c \,d^{2}+1260 x^{3} a^{2} b^{2} c^{2} d +280 x^{3} a \,b^{3} c^{3}+280 a^{4} c \,d^{2} x^{2}+1120 a^{3} b \,c^{2} d \,x^{2}+560 a^{2} b^{2} c^{3} x^{2}+420 x \,a^{4} c^{2} d +560 x \,a^{3} b \,c^{3}+280 a^{4} c^{3}\right )}{280}\) | \(248\) |
| orering | \(\frac {x \left (35 b^{4} d^{3} x^{7}+160 x^{6} a \,b^{3} d^{3}+120 x^{6} b^{4} c \,d^{2}+280 a^{2} b^{2} d^{3} x^{5}+560 a \,b^{3} c \,d^{2} x^{5}+140 b^{4} c^{2} d \,x^{5}+224 x^{4} a^{3} b \,d^{3}+1008 a^{2} b^{2} c \,d^{2} x^{4}+672 x^{4} a \,b^{3} c^{2} d +56 x^{4} b^{4} c^{3}+70 x^{3} a^{4} d^{3}+840 x^{3} a^{3} b c \,d^{2}+1260 x^{3} a^{2} b^{2} c^{2} d +280 x^{3} a \,b^{3} c^{3}+280 a^{4} c \,d^{2} x^{2}+1120 a^{3} b \,c^{2} d \,x^{2}+560 a^{2} b^{2} c^{3} x^{2}+420 x \,a^{4} c^{2} d +560 x \,a^{3} b \,c^{3}+280 a^{4} c^{3}\right ) \left (b^{4} x^{4}+4 a \,x^{3} b^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )}{280 \left (b x +a \right )^{4}}\) | \(293\) |
Input:
int((d*x+c)^3*(b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4),x,method=_ RETURNVERBOSE)
Output:
1/8*b^4*d^3*x^8+(4/7*a*b^3*d^3+3/7*b^4*c*d^2)*x^7+(a^2*b^2*d^3+2*a*b^3*c*d ^2+1/2*b^4*c^2*d)*x^6+(4/5*a^3*b*d^3+18/5*a^2*b^2*c*d^2+12/5*a*b^3*c^2*d+1 /5*b^4*c^3)*x^5+(1/4*a^4*d^3+3*a^3*b*c*d^2+9/2*a^2*b^2*c^2*d+a*b^3*c^3)*x^ 4+(a^4*c*d^2+4*a^3*b*c^2*d+2*a^2*b^2*c^3)*x^3+(3/2*a^4*c^2*d+2*a^3*b*c^3)* x^2+a^4*c^3*x
Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (84) = 168\).
Time = 0.07 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.45 \[ \int (c+d x)^3 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=\frac {1}{8} \, b^{4} d^{3} x^{8} + a^{4} c^{3} x + \frac {1}{7} \, {\left (3 \, b^{4} c d^{2} + 4 \, a b^{3} d^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b^{4} c^{2} d + 4 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{3} + 12 \, a b^{3} c^{2} d + 18 \, a^{2} b^{2} c d^{2} + 4 \, a^{3} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} c^{3} + 18 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} c^{3} + 4 \, a^{3} b c^{2} d + a^{4} c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c^{3} + 3 \, a^{4} c^{2} d\right )} x^{2} \] Input:
integrate((d*x+c)^3*(b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4),x, a lgorithm="fricas")
Output:
1/8*b^4*d^3*x^8 + a^4*c^3*x + 1/7*(3*b^4*c*d^2 + 4*a*b^3*d^3)*x^7 + 1/2*(b ^4*c^2*d + 4*a*b^3*c*d^2 + 2*a^2*b^2*d^3)*x^6 + 1/5*(b^4*c^3 + 12*a*b^3*c^ 2*d + 18*a^2*b^2*c*d^2 + 4*a^3*b*d^3)*x^5 + 1/4*(4*a*b^3*c^3 + 18*a^2*b^2* c^2*d + 12*a^3*b*c*d^2 + a^4*d^3)*x^4 + (2*a^2*b^2*c^3 + 4*a^3*b*c^2*d + a ^4*c*d^2)*x^3 + 1/2*(4*a^3*b*c^3 + 3*a^4*c^2*d)*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 \[ \int (c+d x)^3 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=a^{4} c^{3} x + \frac {b^{4} d^{3} x^{8}}{8} + x^{7} \cdot \left (\frac {4 a b^{3} d^{3}}{7} + \frac {3 b^{4} c d^{2}}{7}\right ) + x^{6} \left (a^{2} b^{2} d^{3} + 2 a b^{3} c d^{2} + \frac {b^{4} c^{2} d}{2}\right ) + x^{5} \cdot \left (\frac {4 a^{3} b d^{3}}{5} + \frac {18 a^{2} b^{2} c d^{2}}{5} + \frac {12 a b^{3} c^{2} d}{5} + \frac {b^{4} c^{3}}{5}\right ) + x^{4} \left (\frac {a^{4} d^{3}}{4} + 3 a^{3} b c d^{2} + \frac {9 a^{2} b^{2} c^{2} d}{2} + a b^{3} c^{3}\right ) + x^{3} \left (a^{4} c d^{2} + 4 a^{3} b c^{2} d + 2 a^{2} b^{2} c^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{4} c^{2} d}{2} + 2 a^{3} b c^{3}\right ) \] Input:
integrate((d*x+c)**3*(b**4*x**4+4*a*b**3*x**3+6*a**2*b**2*x**2+4*a**3*b*x+ a**4),x)
Output:
a**4*c**3*x + b**4*d**3*x**8/8 + x**7*(4*a*b**3*d**3/7 + 3*b**4*c*d**2/7) + x**6*(a**2*b**2*d**3 + 2*a*b**3*c*d**2 + b**4*c**2*d/2) + x**5*(4*a**3*b *d**3/5 + 18*a**2*b**2*c*d**2/5 + 12*a*b**3*c**2*d/5 + b**4*c**3/5) + x**4 *(a**4*d**3/4 + 3*a**3*b*c*d**2 + 9*a**2*b**2*c**2*d/2 + a*b**3*c**3) + x* *3*(a**4*c*d**2 + 4*a**3*b*c**2*d + 2*a**2*b**2*c**3) + x**2*(3*a**4*c**2* d/2 + 2*a**3*b*c**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 \[ \int (c+d x)^3 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=\frac {1}{8} \, b^{4} d^{3} x^{8} + a^{4} c^{3} x + \frac {1}{7} \, {\left (3 \, b^{4} c d^{2} + 4 \, a b^{3} d^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b^{4} c^{2} d + 4 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{3} + 12 \, a b^{3} c^{2} d + 18 \, a^{2} b^{2} c d^{2} + 4 \, a^{3} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} c^{3} + 18 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} c^{3} + 4 \, a^{3} b c^{2} d + a^{4} c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c^{3} + 3 \, a^{4} c^{2} d\right )} x^{2} \] Input:
integrate((d*x+c)^3*(b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4),x, a lgorithm="maxima")
Output:
1/8*b^4*d^3*x^8 + a^4*c^3*x + 1/7*(3*b^4*c*d^2 + 4*a*b^3*d^3)*x^7 + 1/2*(b ^4*c^2*d + 4*a*b^3*c*d^2 + 2*a^2*b^2*d^3)*x^6 + 1/5*(b^4*c^3 + 12*a*b^3*c^ 2*d + 18*a^2*b^2*c*d^2 + 4*a^3*b*d^3)*x^5 + 1/4*(4*a*b^3*c^3 + 18*a^2*b^2* c^2*d + 12*a^3*b*c*d^2 + a^4*d^3)*x^4 + (2*a^2*b^2*c^3 + 4*a^3*b*c^2*d + a ^4*c*d^2)*x^3 + 1/2*(4*a^3*b*c^3 + 3*a^4*c^2*d)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (84) = 168\).
Time = 0.13 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.66 \[ \int (c+d x)^3 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=\frac {1}{8} \, b^{4} d^{3} x^{8} + \frac {3}{7} \, b^{4} c d^{2} x^{7} + \frac {4}{7} \, a b^{3} d^{3} x^{7} + \frac {1}{2} \, b^{4} c^{2} d x^{6} + 2 \, a b^{3} c d^{2} x^{6} + a^{2} b^{2} d^{3} x^{6} + \frac {1}{5} \, b^{4} c^{3} x^{5} + \frac {12}{5} \, a b^{3} c^{2} d x^{5} + \frac {18}{5} \, a^{2} b^{2} c d^{2} x^{5} + \frac {4}{5} \, a^{3} b d^{3} x^{5} + a b^{3} c^{3} x^{4} + \frac {9}{2} \, a^{2} b^{2} c^{2} d x^{4} + 3 \, a^{3} b c d^{2} x^{4} + \frac {1}{4} \, a^{4} d^{3} x^{4} + 2 \, a^{2} b^{2} c^{3} x^{3} + 4 \, a^{3} b c^{2} d x^{3} + a^{4} c d^{2} x^{3} + 2 \, a^{3} b c^{3} x^{2} + \frac {3}{2} \, a^{4} c^{2} d x^{2} + a^{4} c^{3} x \] Input:
integrate((d*x+c)^3*(b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4),x, a lgorithm="giac")
Output:
1/8*b^4*d^3*x^8 + 3/7*b^4*c*d^2*x^7 + 4/7*a*b^3*d^3*x^7 + 1/2*b^4*c^2*d*x^ 6 + 2*a*b^3*c*d^2*x^6 + a^2*b^2*d^3*x^6 + 1/5*b^4*c^3*x^5 + 12/5*a*b^3*c^2 *d*x^5 + 18/5*a^2*b^2*c*d^2*x^5 + 4/5*a^3*b*d^3*x^5 + a*b^3*c^3*x^4 + 9/2* a^2*b^2*c^2*d*x^4 + 3*a^3*b*c*d^2*x^4 + 1/4*a^4*d^3*x^4 + 2*a^2*b^2*c^3*x^ 3 + 4*a^3*b*c^2*d*x^3 + a^4*c*d^2*x^3 + 2*a^3*b*c^3*x^2 + 3/2*a^4*c^2*d*x^ 2 + a^4*c^3*x
Time = 0.16 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.26 \[ \int (c+d x)^3 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=x^4\,\left (\frac {a^4\,d^3}{4}+3\,a^3\,b\,c\,d^2+\frac {9\,a^2\,b^2\,c^2\,d}{2}+a\,b^3\,c^3\right )+x^5\,\left (\frac {4\,a^3\,b\,d^3}{5}+\frac {18\,a^2\,b^2\,c\,d^2}{5}+\frac {12\,a\,b^3\,c^2\,d}{5}+\frac {b^4\,c^3}{5}\right )+a^4\,c^3\,x+\frac {b^4\,d^3\,x^8}{8}+\frac {a^3\,c^2\,x^2\,\left (3\,a\,d+4\,b\,c\right )}{2}+\frac {b^3\,d^2\,x^7\,\left (4\,a\,d+3\,b\,c\right )}{7}+a^2\,c\,x^3\,\left (a^2\,d^2+4\,a\,b\,c\,d+2\,b^2\,c^2\right )+\frac {b^2\,d\,x^6\,\left (2\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{2} \] Input:
int((c + d*x)^3*(a^4 + b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x), x)
Output:
x^4*((a^4*d^3)/4 + a*b^3*c^3 + (9*a^2*b^2*c^2*d)/2 + 3*a^3*b*c*d^2) + x^5* ((b^4*c^3)/5 + (4*a^3*b*d^3)/5 + (18*a^2*b^2*c*d^2)/5 + (12*a*b^3*c^2*d)/5 ) + a^4*c^3*x + (b^4*d^3*x^8)/8 + (a^3*c^2*x^2*(3*a*d + 4*b*c))/2 + (b^3*d ^2*x^7*(4*a*d + 3*b*c))/7 + a^2*c*x^3*(a^2*d^2 + 2*b^2*c^2 + 4*a*b*c*d) + (b^2*d*x^6*(2*a^2*d^2 + b^2*c^2 + 4*a*b*c*d))/2
Time = 0.17 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.68 \[ \int (c+d x)^3 \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=\frac {x \left (35 b^{4} d^{3} x^{7}+160 a \,b^{3} d^{3} x^{6}+120 b^{4} c \,d^{2} x^{6}+280 a^{2} b^{2} d^{3} x^{5}+560 a \,b^{3} c \,d^{2} x^{5}+140 b^{4} c^{2} d \,x^{5}+224 a^{3} b \,d^{3} x^{4}+1008 a^{2} b^{2} c \,d^{2} x^{4}+672 a \,b^{3} c^{2} d \,x^{4}+56 b^{4} c^{3} x^{4}+70 a^{4} d^{3} x^{3}+840 a^{3} b c \,d^{2} x^{3}+1260 a^{2} b^{2} c^{2} d \,x^{3}+280 a \,b^{3} c^{3} x^{3}+280 a^{4} c \,d^{2} x^{2}+1120 a^{3} b \,c^{2} d \,x^{2}+560 a^{2} b^{2} c^{3} x^{2}+420 a^{4} c^{2} d x +560 a^{3} b \,c^{3} x +280 a^{4} c^{3}\right )}{280} \] Input:
int((d*x+c)^3*(b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4),x)
Output:
(x*(280*a**4*c**3 + 420*a**4*c**2*d*x + 280*a**4*c*d**2*x**2 + 70*a**4*d** 3*x**3 + 560*a**3*b*c**3*x + 1120*a**3*b*c**2*d*x**2 + 840*a**3*b*c*d**2*x **3 + 224*a**3*b*d**3*x**4 + 560*a**2*b**2*c**3*x**2 + 1260*a**2*b**2*c**2 *d*x**3 + 1008*a**2*b**2*c*d**2*x**4 + 280*a**2*b**2*d**3*x**5 + 280*a*b** 3*c**3*x**3 + 672*a*b**3*c**2*d*x**4 + 560*a*b**3*c*d**2*x**5 + 160*a*b**3 *d**3*x**6 + 56*b**4*c**3*x**4 + 140*b**4*c**2*d*x**5 + 120*b**4*c*d**2*x* *6 + 35*b**4*d**3*x**7))/280