\(\int (a+b x)^4 (c+d x)^7 \, dx\) [72]

Optimal result

Integrand size = 15, antiderivative size = 119 \[ \int (a+b x)^4 (c+d x)^7 \, dx=\frac {(b c-a d)^4 (c+d x)^8}{8 d^5}-\frac {4 b (b c-a d)^3 (c+d x)^9}{9 d^5}+\frac {3 b^2 (b c-a d)^2 (c+d x)^{10}}{5 d^5}-\frac {4 b^3 (b c-a d) (c+d x)^{11}}{11 d^5}+\frac {b^4 (c+d x)^{12}}{12 d^5} \] Output:

1/8*(-a*d+b*c)^4*(d*x+c)^8/d^5-4/9*b*(-a*d+b*c)^3*(d*x+c)^9/d^5+3/5*b^2*(- 
a*d+b*c)^2*(d*x+c)^10/d^5-4/11*b^3*(-a*d+b*c)*(d*x+c)^11/d^5+1/12*b^4*(d*x 
+c)^12/d^5
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(473\) vs. \(2(119)=238\).

Time = 0.03 (sec) , antiderivative size = 473, normalized size of antiderivative = 3.97 \[ \int (a+b x)^4 (c+d x)^7 \, dx=a^4 c^7 x+\frac {1}{2} a^3 c^6 (4 b c+7 a d) x^2+\frac {1}{3} a^2 c^5 \left (6 b^2 c^2+28 a b c d+21 a^2 d^2\right ) x^3+\frac {1}{4} a c^4 \left (4 b^3 c^3+42 a b^2 c^2 d+84 a^2 b c d^2+35 a^3 d^3\right ) x^4+\frac {1}{5} c^3 \left (b^4 c^4+28 a b^3 c^3 d+126 a^2 b^2 c^2 d^2+140 a^3 b c d^3+35 a^4 d^4\right ) x^5+\frac {7}{6} c^2 d \left (b^4 c^4+12 a b^3 c^3 d+30 a^2 b^2 c^2 d^2+20 a^3 b c d^3+3 a^4 d^4\right ) x^6+c d^2 \left (3 b^4 c^4+20 a b^3 c^3 d+30 a^2 b^2 c^2 d^2+12 a^3 b c d^3+a^4 d^4\right ) x^7+\frac {1}{8} d^3 \left (35 b^4 c^4+140 a b^3 c^3 d+126 a^2 b^2 c^2 d^2+28 a^3 b c d^3+a^4 d^4\right ) x^8+\frac {1}{9} b d^4 \left (35 b^3 c^3+84 a b^2 c^2 d+42 a^2 b c d^2+4 a^3 d^3\right ) x^9+\frac {1}{10} b^2 d^5 \left (21 b^2 c^2+28 a b c d+6 a^2 d^2\right ) x^{10}+\frac {1}{11} b^3 d^6 (7 b c+4 a d) x^{11}+\frac {1}{12} b^4 d^7 x^{12} \] Input:

Integrate[(a + b*x)^4*(c + d*x)^7,x]
 

Output:

a^4*c^7*x + (a^3*c^6*(4*b*c + 7*a*d)*x^2)/2 + (a^2*c^5*(6*b^2*c^2 + 28*a*b 
*c*d + 21*a^2*d^2)*x^3)/3 + (a*c^4*(4*b^3*c^3 + 42*a*b^2*c^2*d + 84*a^2*b* 
c*d^2 + 35*a^3*d^3)*x^4)/4 + (c^3*(b^4*c^4 + 28*a*b^3*c^3*d + 126*a^2*b^2* 
c^2*d^2 + 140*a^3*b*c*d^3 + 35*a^4*d^4)*x^5)/5 + (7*c^2*d*(b^4*c^4 + 12*a* 
b^3*c^3*d + 30*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 + 3*a^4*d^4)*x^6)/6 + c*d^ 
2*(3*b^4*c^4 + 20*a*b^3*c^3*d + 30*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4* 
d^4)*x^7 + (d^3*(35*b^4*c^4 + 140*a*b^3*c^3*d + 126*a^2*b^2*c^2*d^2 + 28*a 
^3*b*c*d^3 + a^4*d^4)*x^8)/8 + (b*d^4*(35*b^3*c^3 + 84*a*b^2*c^2*d + 42*a^ 
2*b*c*d^2 + 4*a^3*d^3)*x^9)/9 + (b^2*d^5*(21*b^2*c^2 + 28*a*b*c*d + 6*a^2* 
d^2)*x^10)/10 + (b^3*d^6*(7*b*c + 4*a*d)*x^11)/11 + (b^4*d^7*x^12)/12
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {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.

\(\displaystyle \int (a+b x)^4 (c+d x)^7 \, dx\)

\(\Big \downarrow \) 49

\(\displaystyle \int \left (-\frac {4 b^3 (c+d x)^{10} (b c-a d)}{d^4}+\frac {6 b^2 (c+d x)^9 (b c-a d)^2}{d^4}-\frac {4 b (c+d x)^8 (b c-a d)^3}{d^4}+\frac {(c+d x)^7 (a d-b c)^4}{d^4}+\frac {b^4 (c+d x)^{11}}{d^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 b^3 (c+d x)^{11} (b c-a d)}{11 d^5}+\frac {3 b^2 (c+d x)^{10} (b c-a d)^2}{5 d^5}-\frac {4 b (c+d x)^9 (b c-a d)^3}{9 d^5}+\frac {(c+d x)^8 (b c-a d)^4}{8 d^5}+\frac {b^4 (c+d x)^{12}}{12 d^5}\)

Input:

Int[(a + b*x)^4*(c + d*x)^7,x]
 

Output:

((b*c - a*d)^4*(c + d*x)^8)/(8*d^5) - (4*b*(b*c - a*d)^3*(c + d*x)^9)/(9*d 
^5) + (3*b^2*(b*c - a*d)^2*(c + d*x)^10)/(5*d^5) - (4*b^3*(b*c - a*d)*(c + 
 d*x)^11)/(11*d^5) + (b^4*(c + d*x)^12)/(12*d^5)
 

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 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. \(482\) vs. \(2(109)=218\).

Time = 0.09 (sec) , antiderivative size = 483, normalized size of antiderivative = 4.06

method	result	size
norman	\(\frac {b^{4} d^{7} x^{12}}{12}+\left (\frac {4}{11} a \,b^{3} d^{7}+\frac {7}{11} b^{4} c \,d^{6}\right ) x^{11}+\left (\frac {3}{5} a^{2} b^{2} d^{7}+\frac {14}{5} a \,b^{3} c \,d^{6}+\frac {21}{10} b^{4} c^{2} d^{5}\right ) x^{10}+\left (\frac {4}{9} a^{3} b \,d^{7}+\frac {14}{3} a^{2} b^{2} c \,d^{6}+\frac {28}{3} a \,b^{3} c^{2} d^{5}+\frac {35}{9} b^{4} c^{3} d^{4}\right ) x^{9}+\left (\frac {1}{8} a^{4} d^{7}+\frac {7}{2} a^{3} b c \,d^{6}+\frac {63}{4} a^{2} b^{2} c^{2} d^{5}+\frac {35}{2} a \,b^{3} c^{3} d^{4}+\frac {35}{8} b^{4} c^{4} d^{3}\right ) x^{8}+\left (a^{4} c \,d^{6}+12 a^{3} b \,c^{2} d^{5}+30 a^{2} b^{2} c^{3} d^{4}+20 a \,b^{3} c^{4} d^{3}+3 b^{4} c^{5} d^{2}\right ) x^{7}+\left (\frac {7}{2} a^{4} c^{2} d^{5}+\frac {70}{3} a^{3} b \,c^{3} d^{4}+35 a^{2} b^{2} c^{4} d^{3}+14 a \,b^{3} c^{5} d^{2}+\frac {7}{6} b^{4} c^{6} d \right ) x^{6}+\left (7 a^{4} c^{3} d^{4}+28 a^{3} b \,c^{4} d^{3}+\frac {126}{5} a^{2} b^{2} c^{5} d^{2}+\frac {28}{5} a \,b^{3} c^{6} d +\frac {1}{5} b^{4} c^{7}\right ) x^{5}+\left (\frac {35}{4} a^{4} c^{4} d^{3}+21 a^{3} b \,c^{5} d^{2}+\frac {21}{2} a^{2} b^{2} c^{6} d +a \,b^{3} c^{7}\right ) x^{4}+\left (7 a^{4} c^{5} d^{2}+\frac {28}{3} a^{3} b \,c^{6} d +2 a^{2} b^{2} c^{7}\right ) x^{3}+\left (\frac {7}{2} a^{4} c^{6} d +2 a^{3} b \,c^{7}\right ) x^{2}+a^{4} c^{7} x\)	\(483\)
default	\(\frac {b^{4} d^{7} x^{12}}{12}+\frac {\left (4 a \,b^{3} d^{7}+7 b^{4} c \,d^{6}\right ) x^{11}}{11}+\frac {\left (6 a^{2} b^{2} d^{7}+28 a \,b^{3} c \,d^{6}+21 b^{4} c^{2} d^{5}\right ) x^{10}}{10}+\frac {\left (4 a^{3} b \,d^{7}+42 a^{2} b^{2} c \,d^{6}+84 a \,b^{3} c^{2} d^{5}+35 b^{4} c^{3} d^{4}\right ) x^{9}}{9}+\frac {\left (a^{4} d^{7}+28 a^{3} b c \,d^{6}+126 a^{2} b^{2} c^{2} d^{5}+140 a \,b^{3} c^{3} d^{4}+35 b^{4} c^{4} d^{3}\right ) x^{8}}{8}+\frac {\left (7 a^{4} c \,d^{6}+84 a^{3} b \,c^{2} d^{5}+210 a^{2} b^{2} c^{3} d^{4}+140 a \,b^{3} c^{4} d^{3}+21 b^{4} c^{5} d^{2}\right ) x^{7}}{7}+\frac {\left (21 a^{4} c^{2} d^{5}+140 a^{3} b \,c^{3} d^{4}+210 a^{2} b^{2} c^{4} d^{3}+84 a \,b^{3} c^{5} d^{2}+7 b^{4} c^{6} d \right ) x^{6}}{6}+\frac {\left (35 a^{4} c^{3} d^{4}+140 a^{3} b \,c^{4} d^{3}+126 a^{2} b^{2} c^{5} d^{2}+28 a \,b^{3} c^{6} d +b^{4} c^{7}\right ) x^{5}}{5}+\frac {\left (35 a^{4} c^{4} d^{3}+84 a^{3} b \,c^{5} d^{2}+42 a^{2} b^{2} c^{6} d +4 a \,b^{3} c^{7}\right ) x^{4}}{4}+\frac {\left (21 a^{4} c^{5} d^{2}+28 a^{3} b \,c^{6} d +6 a^{2} b^{2} c^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{4} c^{6} d +4 a^{3} b \,c^{7}\right ) x^{2}}{2}+a^{4} c^{7} x\)	\(493\)
gosper	\(\frac {1}{12} b^{4} d^{7} x^{12}+a^{4} c^{7} x +\frac {7}{2} x^{6} a^{4} c^{2} d^{5}+\frac {7}{6} x^{6} b^{4} c^{6} d +\frac {35}{9} x^{9} b^{4} c^{3} d^{4}+\frac {35}{8} x^{8} b^{4} c^{4} d^{3}+\frac {35}{2} x^{8} a \,b^{3} c^{3} d^{4}+\frac {70}{3} x^{6} a^{3} b \,c^{3} d^{4}+35 x^{6} a^{2} b^{2} c^{4} d^{3}+14 x^{6} a \,b^{3} c^{5} d^{2}+28 x^{5} a^{3} b \,c^{4} d^{3}+\frac {126}{5} x^{5} a^{2} b^{2} c^{5} d^{2}+\frac {28}{5} x^{5} a \,b^{3} c^{6} d +21 x^{4} a^{3} b \,c^{5} d^{2}+\frac {21}{2} x^{4} a^{2} b^{2} c^{6} d +\frac {28}{3} x^{3} a^{3} b \,c^{6} d +12 a^{3} b \,c^{2} d^{5} x^{7}+30 a^{2} b^{2} c^{3} d^{4} x^{7}+20 a \,b^{3} c^{4} d^{3} x^{7}+\frac {63}{4} x^{8} a^{2} b^{2} c^{2} d^{5}+\frac {28}{3} x^{9} a \,b^{3} c^{2} d^{5}+\frac {7}{2} x^{8} a^{3} b c \,d^{6}+\frac {14}{5} x^{10} a \,b^{3} c \,d^{6}+\frac {14}{3} x^{9} a^{2} b^{2} c \,d^{6}+\frac {21}{10} x^{10} b^{4} c^{2} d^{5}+\frac {4}{9} x^{9} a^{3} b \,d^{7}+7 x^{5} a^{4} c^{3} d^{4}+\frac {35}{4} x^{4} a^{4} c^{4} d^{3}+x^{4} a \,b^{3} c^{7}+7 x^{3} a^{4} c^{5} d^{2}+2 x^{3} a^{2} b^{2} c^{7}+\frac {7}{2} x^{2} a^{4} c^{6} d +2 x^{2} a^{3} b \,c^{7}+a^{4} c \,d^{6} x^{7}+3 b^{4} c^{5} d^{2} x^{7}+\frac {4}{11} x^{11} a \,b^{3} d^{7}+\frac {7}{11} x^{11} b^{4} c \,d^{6}+\frac {3}{5} x^{10} a^{2} b^{2} d^{7}+\frac {1}{8} x^{8} a^{4} d^{7}+\frac {1}{5} x^{5} b^{4} c^{7}\)	\(547\)
risch	\(\frac {1}{12} b^{4} d^{7} x^{12}+a^{4} c^{7} x +\frac {7}{2} x^{6} a^{4} c^{2} d^{5}+\frac {7}{6} x^{6} b^{4} c^{6} d +\frac {35}{9} x^{9} b^{4} c^{3} d^{4}+\frac {35}{8} x^{8} b^{4} c^{4} d^{3}+\frac {35}{2} x^{8} a \,b^{3} c^{3} d^{4}+\frac {70}{3} x^{6} a^{3} b \,c^{3} d^{4}+35 x^{6} a^{2} b^{2} c^{4} d^{3}+14 x^{6} a \,b^{3} c^{5} d^{2}+28 x^{5} a^{3} b \,c^{4} d^{3}+\frac {126}{5} x^{5} a^{2} b^{2} c^{5} d^{2}+\frac {28}{5} x^{5} a \,b^{3} c^{6} d +21 x^{4} a^{3} b \,c^{5} d^{2}+\frac {21}{2} x^{4} a^{2} b^{2} c^{6} d +\frac {28}{3} x^{3} a^{3} b \,c^{6} d +12 a^{3} b \,c^{2} d^{5} x^{7}+30 a^{2} b^{2} c^{3} d^{4} x^{7}+20 a \,b^{3} c^{4} d^{3} x^{7}+\frac {63}{4} x^{8} a^{2} b^{2} c^{2} d^{5}+\frac {28}{3} x^{9} a \,b^{3} c^{2} d^{5}+\frac {7}{2} x^{8} a^{3} b c \,d^{6}+\frac {14}{5} x^{10} a \,b^{3} c \,d^{6}+\frac {14}{3} x^{9} a^{2} b^{2} c \,d^{6}+\frac {21}{10} x^{10} b^{4} c^{2} d^{5}+\frac {4}{9} x^{9} a^{3} b \,d^{7}+7 x^{5} a^{4} c^{3} d^{4}+\frac {35}{4} x^{4} a^{4} c^{4} d^{3}+x^{4} a \,b^{3} c^{7}+7 x^{3} a^{4} c^{5} d^{2}+2 x^{3} a^{2} b^{2} c^{7}+\frac {7}{2} x^{2} a^{4} c^{6} d +2 x^{2} a^{3} b \,c^{7}+a^{4} c \,d^{6} x^{7}+3 b^{4} c^{5} d^{2} x^{7}+\frac {4}{11} x^{11} a \,b^{3} d^{7}+\frac {7}{11} x^{11} b^{4} c \,d^{6}+\frac {3}{5} x^{10} a^{2} b^{2} d^{7}+\frac {1}{8} x^{8} a^{4} d^{7}+\frac {1}{5} x^{5} b^{4} c^{7}\)	\(547\)
parallelrisch	\(\frac {1}{12} b^{4} d^{7} x^{12}+a^{4} c^{7} x +\frac {7}{2} x^{6} a^{4} c^{2} d^{5}+\frac {7}{6} x^{6} b^{4} c^{6} d +\frac {35}{9} x^{9} b^{4} c^{3} d^{4}+\frac {35}{8} x^{8} b^{4} c^{4} d^{3}+\frac {35}{2} x^{8} a \,b^{3} c^{3} d^{4}+\frac {70}{3} x^{6} a^{3} b \,c^{3} d^{4}+35 x^{6} a^{2} b^{2} c^{4} d^{3}+14 x^{6} a \,b^{3} c^{5} d^{2}+28 x^{5} a^{3} b \,c^{4} d^{3}+\frac {126}{5} x^{5} a^{2} b^{2} c^{5} d^{2}+\frac {28}{5} x^{5} a \,b^{3} c^{6} d +21 x^{4} a^{3} b \,c^{5} d^{2}+\frac {21}{2} x^{4} a^{2} b^{2} c^{6} d +\frac {28}{3} x^{3} a^{3} b \,c^{6} d +12 a^{3} b \,c^{2} d^{5} x^{7}+30 a^{2} b^{2} c^{3} d^{4} x^{7}+20 a \,b^{3} c^{4} d^{3} x^{7}+\frac {63}{4} x^{8} a^{2} b^{2} c^{2} d^{5}+\frac {28}{3} x^{9} a \,b^{3} c^{2} d^{5}+\frac {7}{2} x^{8} a^{3} b c \,d^{6}+\frac {14}{5} x^{10} a \,b^{3} c \,d^{6}+\frac {14}{3} x^{9} a^{2} b^{2} c \,d^{6}+\frac {21}{10} x^{10} b^{4} c^{2} d^{5}+\frac {4}{9} x^{9} a^{3} b \,d^{7}+7 x^{5} a^{4} c^{3} d^{4}+\frac {35}{4} x^{4} a^{4} c^{4} d^{3}+x^{4} a \,b^{3} c^{7}+7 x^{3} a^{4} c^{5} d^{2}+2 x^{3} a^{2} b^{2} c^{7}+\frac {7}{2} x^{2} a^{4} c^{6} d +2 x^{2} a^{3} b \,c^{7}+a^{4} c \,d^{6} x^{7}+3 b^{4} c^{5} d^{2} x^{7}+\frac {4}{11} x^{11} a \,b^{3} d^{7}+\frac {7}{11} x^{11} b^{4} c \,d^{6}+\frac {3}{5} x^{10} a^{2} b^{2} d^{7}+\frac {1}{8} x^{8} a^{4} d^{7}+\frac {1}{5} x^{5} b^{4} c^{7}\)	\(547\)
orering	\(\frac {x \left (330 b^{4} d^{7} x^{11}+1440 a \,b^{3} d^{7} x^{10}+2520 b^{4} c \,d^{6} x^{10}+2376 a^{2} b^{2} d^{7} x^{9}+11088 a \,b^{3} c \,d^{6} x^{9}+8316 b^{4} c^{2} d^{5} x^{9}+1760 a^{3} b \,d^{7} x^{8}+18480 a^{2} b^{2} c \,d^{6} x^{8}+36960 a \,b^{3} c^{2} d^{5} x^{8}+15400 b^{4} c^{3} d^{4} x^{8}+495 a^{4} d^{7} x^{7}+13860 a^{3} b c \,d^{6} x^{7}+62370 a^{2} b^{2} c^{2} d^{5} x^{7}+69300 a \,b^{3} c^{3} d^{4} x^{7}+17325 b^{4} c^{4} d^{3} x^{7}+3960 a^{4} c \,d^{6} x^{6}+47520 a^{3} b \,c^{2} d^{5} x^{6}+118800 a^{2} b^{2} c^{3} d^{4} x^{6}+79200 a \,b^{3} c^{4} d^{3} x^{6}+11880 b^{4} c^{5} d^{2} x^{6}+13860 a^{4} c^{2} d^{5} x^{5}+92400 a^{3} b \,c^{3} d^{4} x^{5}+138600 a^{2} b^{2} c^{4} d^{3} x^{5}+55440 a \,b^{3} c^{5} d^{2} x^{5}+4620 b^{4} c^{6} d \,x^{5}+27720 a^{4} c^{3} d^{4} x^{4}+110880 a^{3} b \,c^{4} d^{3} x^{4}+99792 a^{2} b^{2} c^{5} d^{2} x^{4}+22176 a \,b^{3} c^{6} d \,x^{4}+792 b^{4} c^{7} x^{4}+34650 a^{4} c^{4} d^{3} x^{3}+83160 a^{3} b \,c^{5} d^{2} x^{3}+41580 a^{2} b^{2} c^{6} d \,x^{3}+3960 a \,b^{3} c^{7} x^{3}+27720 a^{4} c^{5} d^{2} x^{2}+36960 a^{3} b \,c^{6} d \,x^{2}+7920 a^{2} b^{2} c^{7} x^{2}+13860 a^{4} c^{6} d x +7920 a^{3} b \,c^{7} x +3960 a^{4} c^{7}\right )}{3960}\)	\(548\)

Input:

int((b*x+a)^4*(d*x+c)^7,x,method=_RETURNVERBOSE)
 

Output:

1/12*b^4*d^7*x^12+(4/11*a*b^3*d^7+7/11*b^4*c*d^6)*x^11+(3/5*a^2*b^2*d^7+14 
/5*a*b^3*c*d^6+21/10*b^4*c^2*d^5)*x^10+(4/9*a^3*b*d^7+14/3*a^2*b^2*c*d^6+2 
8/3*a*b^3*c^2*d^5+35/9*b^4*c^3*d^4)*x^9+(1/8*a^4*d^7+7/2*a^3*b*c*d^6+63/4* 
a^2*b^2*c^2*d^5+35/2*a*b^3*c^3*d^4+35/8*b^4*c^4*d^3)*x^8+(a^4*c*d^6+12*a^3 
*b*c^2*d^5+30*a^2*b^2*c^3*d^4+20*a*b^3*c^4*d^3+3*b^4*c^5*d^2)*x^7+(7/2*a^4 
*c^2*d^5+70/3*a^3*b*c^3*d^4+35*a^2*b^2*c^4*d^3+14*a*b^3*c^5*d^2+7/6*b^4*c^ 
6*d)*x^6+(7*a^4*c^3*d^4+28*a^3*b*c^4*d^3+126/5*a^2*b^2*c^5*d^2+28/5*a*b^3* 
c^6*d+1/5*b^4*c^7)*x^5+(35/4*a^4*c^4*d^3+21*a^3*b*c^5*d^2+21/2*a^2*b^2*c^6 
*d+a*b^3*c^7)*x^4+(7*a^4*c^5*d^2+28/3*a^3*b*c^6*d+2*a^2*b^2*c^7)*x^3+(7/2* 
a^4*c^6*d+2*a^3*b*c^7)*x^2+a^4*c^7*x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (109) = 218\).

Time = 0.09 (sec) , antiderivative size = 489, normalized size of antiderivative = 4.11 \[ \int (a+b x)^4 (c+d x)^7 \, dx=\frac {1}{12} \, b^{4} d^{7} x^{12} + a^{4} c^{7} x + \frac {1}{11} \, {\left (7 \, b^{4} c d^{6} + 4 \, a b^{3} d^{7}\right )} x^{11} + \frac {1}{10} \, {\left (21 \, b^{4} c^{2} d^{5} + 28 \, a b^{3} c d^{6} + 6 \, a^{2} b^{2} d^{7}\right )} x^{10} + \frac {1}{9} \, {\left (35 \, b^{4} c^{3} d^{4} + 84 \, a b^{3} c^{2} d^{5} + 42 \, a^{2} b^{2} c d^{6} + 4 \, a^{3} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (35 \, b^{4} c^{4} d^{3} + 140 \, a b^{3} c^{3} d^{4} + 126 \, a^{2} b^{2} c^{2} d^{5} + 28 \, a^{3} b c d^{6} + a^{4} d^{7}\right )} x^{8} + {\left (3 \, b^{4} c^{5} d^{2} + 20 \, a b^{3} c^{4} d^{3} + 30 \, a^{2} b^{2} c^{3} d^{4} + 12 \, a^{3} b c^{2} d^{5} + a^{4} c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (b^{4} c^{6} d + 12 \, a b^{3} c^{5} d^{2} + 30 \, a^{2} b^{2} c^{4} d^{3} + 20 \, a^{3} b c^{3} d^{4} + 3 \, a^{4} c^{2} d^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{7} + 28 \, a b^{3} c^{6} d + 126 \, a^{2} b^{2} c^{5} d^{2} + 140 \, a^{3} b c^{4} d^{3} + 35 \, a^{4} c^{3} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} c^{7} + 42 \, a^{2} b^{2} c^{6} d + 84 \, a^{3} b c^{5} d^{2} + 35 \, a^{4} c^{4} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{7} + 28 \, a^{3} b c^{6} d + 21 \, a^{4} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c^{7} + 7 \, a^{4} c^{6} d\right )} x^{2} \] Input:

integrate((b*x+a)^4*(d*x+c)^7,x, algorithm="fricas")
 

Output:

1/12*b^4*d^7*x^12 + a^4*c^7*x + 1/11*(7*b^4*c*d^6 + 4*a*b^3*d^7)*x^11 + 1/ 
10*(21*b^4*c^2*d^5 + 28*a*b^3*c*d^6 + 6*a^2*b^2*d^7)*x^10 + 1/9*(35*b^4*c^ 
3*d^4 + 84*a*b^3*c^2*d^5 + 42*a^2*b^2*c*d^6 + 4*a^3*b*d^7)*x^9 + 1/8*(35*b 
^4*c^4*d^3 + 140*a*b^3*c^3*d^4 + 126*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^ 
4*d^7)*x^8 + (3*b^4*c^5*d^2 + 20*a*b^3*c^4*d^3 + 30*a^2*b^2*c^3*d^4 + 12*a 
^3*b*c^2*d^5 + a^4*c*d^6)*x^7 + 7/6*(b^4*c^6*d + 12*a*b^3*c^5*d^2 + 30*a^2 
*b^2*c^4*d^3 + 20*a^3*b*c^3*d^4 + 3*a^4*c^2*d^5)*x^6 + 1/5*(b^4*c^7 + 28*a 
*b^3*c^6*d + 126*a^2*b^2*c^5*d^2 + 140*a^3*b*c^4*d^3 + 35*a^4*c^3*d^4)*x^5 
 + 1/4*(4*a*b^3*c^7 + 42*a^2*b^2*c^6*d + 84*a^3*b*c^5*d^2 + 35*a^4*c^4*d^3 
)*x^4 + 1/3*(6*a^2*b^2*c^7 + 28*a^3*b*c^6*d + 21*a^4*c^5*d^2)*x^3 + 1/2*(4 
*a^3*b*c^7 + 7*a^4*c^6*d)*x^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (107) = 214\).

Time = 0.05 (sec) , antiderivative size = 549, normalized size of antiderivative = 4.61 \[ \int (a+b x)^4 (c+d x)^7 \, dx=a^{4} c^{7} x + \frac {b^{4} d^{7} x^{12}}{12} + x^{11} \cdot \left (\frac {4 a b^{3} d^{7}}{11} + \frac {7 b^{4} c d^{6}}{11}\right ) + x^{10} \cdot \left (\frac {3 a^{2} b^{2} d^{7}}{5} + \frac {14 a b^{3} c d^{6}}{5} + \frac {21 b^{4} c^{2} d^{5}}{10}\right ) + x^{9} \cdot \left (\frac {4 a^{3} b d^{7}}{9} + \frac {14 a^{2} b^{2} c d^{6}}{3} + \frac {28 a b^{3} c^{2} d^{5}}{3} + \frac {35 b^{4} c^{3} d^{4}}{9}\right ) + x^{8} \left (\frac {a^{4} d^{7}}{8} + \frac {7 a^{3} b c d^{6}}{2} + \frac {63 a^{2} b^{2} c^{2} d^{5}}{4} + \frac {35 a b^{3} c^{3} d^{4}}{2} + \frac {35 b^{4} c^{4} d^{3}}{8}\right ) + x^{7} \left (a^{4} c d^{6} + 12 a^{3} b c^{2} d^{5} + 30 a^{2} b^{2} c^{3} d^{4} + 20 a b^{3} c^{4} d^{3} + 3 b^{4} c^{5} d^{2}\right ) + x^{6} \cdot \left (\frac {7 a^{4} c^{2} d^{5}}{2} + \frac {70 a^{3} b c^{3} d^{4}}{3} + 35 a^{2} b^{2} c^{4} d^{3} + 14 a b^{3} c^{5} d^{2} + \frac {7 b^{4} c^{6} d}{6}\right ) + x^{5} \cdot \left (7 a^{4} c^{3} d^{4} + 28 a^{3} b c^{4} d^{3} + \frac {126 a^{2} b^{2} c^{5} d^{2}}{5} + \frac {28 a b^{3} c^{6} d}{5} + \frac {b^{4} c^{7}}{5}\right ) + x^{4} \cdot \left (\frac {35 a^{4} c^{4} d^{3}}{4} + 21 a^{3} b c^{5} d^{2} + \frac {21 a^{2} b^{2} c^{6} d}{2} + a b^{3} c^{7}\right ) + x^{3} \cdot \left (7 a^{4} c^{5} d^{2} + \frac {28 a^{3} b c^{6} d}{3} + 2 a^{2} b^{2} c^{7}\right ) + x^{2} \cdot \left (\frac {7 a^{4} c^{6} d}{2} + 2 a^{3} b c^{7}\right ) \] Input:

integrate((b*x+a)**4*(d*x+c)**7,x)
 

Output:

a**4*c**7*x + b**4*d**7*x**12/12 + x**11*(4*a*b**3*d**7/11 + 7*b**4*c*d**6 
/11) + x**10*(3*a**2*b**2*d**7/5 + 14*a*b**3*c*d**6/5 + 21*b**4*c**2*d**5/ 
10) + x**9*(4*a**3*b*d**7/9 + 14*a**2*b**2*c*d**6/3 + 28*a*b**3*c**2*d**5/ 
3 + 35*b**4*c**3*d**4/9) + x**8*(a**4*d**7/8 + 7*a**3*b*c*d**6/2 + 63*a**2 
*b**2*c**2*d**5/4 + 35*a*b**3*c**3*d**4/2 + 35*b**4*c**4*d**3/8) + x**7*(a 
**4*c*d**6 + 12*a**3*b*c**2*d**5 + 30*a**2*b**2*c**3*d**4 + 20*a*b**3*c**4 
*d**3 + 3*b**4*c**5*d**2) + x**6*(7*a**4*c**2*d**5/2 + 70*a**3*b*c**3*d**4 
/3 + 35*a**2*b**2*c**4*d**3 + 14*a*b**3*c**5*d**2 + 7*b**4*c**6*d/6) + x** 
5*(7*a**4*c**3*d**4 + 28*a**3*b*c**4*d**3 + 126*a**2*b**2*c**5*d**2/5 + 28 
*a*b**3*c**6*d/5 + b**4*c**7/5) + x**4*(35*a**4*c**4*d**3/4 + 21*a**3*b*c* 
*5*d**2 + 21*a**2*b**2*c**6*d/2 + a*b**3*c**7) + x**3*(7*a**4*c**5*d**2 + 
28*a**3*b*c**6*d/3 + 2*a**2*b**2*c**7) + x**2*(7*a**4*c**6*d/2 + 2*a**3*b* 
c**7)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (109) = 218\).

Time = 0.04 (sec) , antiderivative size = 489, normalized size of antiderivative = 4.11 \[ \int (a+b x)^4 (c+d x)^7 \, dx=\frac {1}{12} \, b^{4} d^{7} x^{12} + a^{4} c^{7} x + \frac {1}{11} \, {\left (7 \, b^{4} c d^{6} + 4 \, a b^{3} d^{7}\right )} x^{11} + \frac {1}{10} \, {\left (21 \, b^{4} c^{2} d^{5} + 28 \, a b^{3} c d^{6} + 6 \, a^{2} b^{2} d^{7}\right )} x^{10} + \frac {1}{9} \, {\left (35 \, b^{4} c^{3} d^{4} + 84 \, a b^{3} c^{2} d^{5} + 42 \, a^{2} b^{2} c d^{6} + 4 \, a^{3} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (35 \, b^{4} c^{4} d^{3} + 140 \, a b^{3} c^{3} d^{4} + 126 \, a^{2} b^{2} c^{2} d^{5} + 28 \, a^{3} b c d^{6} + a^{4} d^{7}\right )} x^{8} + {\left (3 \, b^{4} c^{5} d^{2} + 20 \, a b^{3} c^{4} d^{3} + 30 \, a^{2} b^{2} c^{3} d^{4} + 12 \, a^{3} b c^{2} d^{5} + a^{4} c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (b^{4} c^{6} d + 12 \, a b^{3} c^{5} d^{2} + 30 \, a^{2} b^{2} c^{4} d^{3} + 20 \, a^{3} b c^{3} d^{4} + 3 \, a^{4} c^{2} d^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{7} + 28 \, a b^{3} c^{6} d + 126 \, a^{2} b^{2} c^{5} d^{2} + 140 \, a^{3} b c^{4} d^{3} + 35 \, a^{4} c^{3} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} c^{7} + 42 \, a^{2} b^{2} c^{6} d + 84 \, a^{3} b c^{5} d^{2} + 35 \, a^{4} c^{4} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{7} + 28 \, a^{3} b c^{6} d + 21 \, a^{4} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c^{7} + 7 \, a^{4} c^{6} d\right )} x^{2} \] Input:

integrate((b*x+a)^4*(d*x+c)^7,x, algorithm="maxima")
 

Output:

1/12*b^4*d^7*x^12 + a^4*c^7*x + 1/11*(7*b^4*c*d^6 + 4*a*b^3*d^7)*x^11 + 1/ 
10*(21*b^4*c^2*d^5 + 28*a*b^3*c*d^6 + 6*a^2*b^2*d^7)*x^10 + 1/9*(35*b^4*c^ 
3*d^4 + 84*a*b^3*c^2*d^5 + 42*a^2*b^2*c*d^6 + 4*a^3*b*d^7)*x^9 + 1/8*(35*b 
^4*c^4*d^3 + 140*a*b^3*c^3*d^4 + 126*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^ 
4*d^7)*x^8 + (3*b^4*c^5*d^2 + 20*a*b^3*c^4*d^3 + 30*a^2*b^2*c^3*d^4 + 12*a 
^3*b*c^2*d^5 + a^4*c*d^6)*x^7 + 7/6*(b^4*c^6*d + 12*a*b^3*c^5*d^2 + 30*a^2 
*b^2*c^4*d^3 + 20*a^3*b*c^3*d^4 + 3*a^4*c^2*d^5)*x^6 + 1/5*(b^4*c^7 + 28*a 
*b^3*c^6*d + 126*a^2*b^2*c^5*d^2 + 140*a^3*b*c^4*d^3 + 35*a^4*c^3*d^4)*x^5 
 + 1/4*(4*a*b^3*c^7 + 42*a^2*b^2*c^6*d + 84*a^3*b*c^5*d^2 + 35*a^4*c^4*d^3 
)*x^4 + 1/3*(6*a^2*b^2*c^7 + 28*a^3*b*c^6*d + 21*a^4*c^5*d^2)*x^3 + 1/2*(4 
*a^3*b*c^7 + 7*a^4*c^6*d)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (109) = 218\).

Time = 0.12 (sec) , antiderivative size = 546, normalized size of antiderivative = 4.59 \[ \int (a+b x)^4 (c+d x)^7 \, dx=\frac {1}{12} \, b^{4} d^{7} x^{12} + \frac {7}{11} \, b^{4} c d^{6} x^{11} + \frac {4}{11} \, a b^{3} d^{7} x^{11} + \frac {21}{10} \, b^{4} c^{2} d^{5} x^{10} + \frac {14}{5} \, a b^{3} c d^{6} x^{10} + \frac {3}{5} \, a^{2} b^{2} d^{7} x^{10} + \frac {35}{9} \, b^{4} c^{3} d^{4} x^{9} + \frac {28}{3} \, a b^{3} c^{2} d^{5} x^{9} + \frac {14}{3} \, a^{2} b^{2} c d^{6} x^{9} + \frac {4}{9} \, a^{3} b d^{7} x^{9} + \frac {35}{8} \, b^{4} c^{4} d^{3} x^{8} + \frac {35}{2} \, a b^{3} c^{3} d^{4} x^{8} + \frac {63}{4} \, a^{2} b^{2} c^{2} d^{5} x^{8} + \frac {7}{2} \, a^{3} b c d^{6} x^{8} + \frac {1}{8} \, a^{4} d^{7} x^{8} + 3 \, b^{4} c^{5} d^{2} x^{7} + 20 \, a b^{3} c^{4} d^{3} x^{7} + 30 \, a^{2} b^{2} c^{3} d^{4} x^{7} + 12 \, a^{3} b c^{2} d^{5} x^{7} + a^{4} c d^{6} x^{7} + \frac {7}{6} \, b^{4} c^{6} d x^{6} + 14 \, a b^{3} c^{5} d^{2} x^{6} + 35 \, a^{2} b^{2} c^{4} d^{3} x^{6} + \frac {70}{3} \, a^{3} b c^{3} d^{4} x^{6} + \frac {7}{2} \, a^{4} c^{2} d^{5} x^{6} + \frac {1}{5} \, b^{4} c^{7} x^{5} + \frac {28}{5} \, a b^{3} c^{6} d x^{5} + \frac {126}{5} \, a^{2} b^{2} c^{5} d^{2} x^{5} + 28 \, a^{3} b c^{4} d^{3} x^{5} + 7 \, a^{4} c^{3} d^{4} x^{5} + a b^{3} c^{7} x^{4} + \frac {21}{2} \, a^{2} b^{2} c^{6} d x^{4} + 21 \, a^{3} b c^{5} d^{2} x^{4} + \frac {35}{4} \, a^{4} c^{4} d^{3} x^{4} + 2 \, a^{2} b^{2} c^{7} x^{3} + \frac {28}{3} \, a^{3} b c^{6} d x^{3} + 7 \, a^{4} c^{5} d^{2} x^{3} + 2 \, a^{3} b c^{7} x^{2} + \frac {7}{2} \, a^{4} c^{6} d x^{2} + a^{4} c^{7} x \] Input:

integrate((b*x+a)^4*(d*x+c)^7,x, algorithm="giac")
 

Output:

1/12*b^4*d^7*x^12 + 7/11*b^4*c*d^6*x^11 + 4/11*a*b^3*d^7*x^11 + 21/10*b^4* 
c^2*d^5*x^10 + 14/5*a*b^3*c*d^6*x^10 + 3/5*a^2*b^2*d^7*x^10 + 35/9*b^4*c^3 
*d^4*x^9 + 28/3*a*b^3*c^2*d^5*x^9 + 14/3*a^2*b^2*c*d^6*x^9 + 4/9*a^3*b*d^7 
*x^9 + 35/8*b^4*c^4*d^3*x^8 + 35/2*a*b^3*c^3*d^4*x^8 + 63/4*a^2*b^2*c^2*d^ 
5*x^8 + 7/2*a^3*b*c*d^6*x^8 + 1/8*a^4*d^7*x^8 + 3*b^4*c^5*d^2*x^7 + 20*a*b 
^3*c^4*d^3*x^7 + 30*a^2*b^2*c^3*d^4*x^7 + 12*a^3*b*c^2*d^5*x^7 + a^4*c*d^6 
*x^7 + 7/6*b^4*c^6*d*x^6 + 14*a*b^3*c^5*d^2*x^6 + 35*a^2*b^2*c^4*d^3*x^6 + 
 70/3*a^3*b*c^3*d^4*x^6 + 7/2*a^4*c^2*d^5*x^6 + 1/5*b^4*c^7*x^5 + 28/5*a*b 
^3*c^6*d*x^5 + 126/5*a^2*b^2*c^5*d^2*x^5 + 28*a^3*b*c^4*d^3*x^5 + 7*a^4*c^ 
3*d^4*x^5 + a*b^3*c^7*x^4 + 21/2*a^2*b^2*c^6*d*x^4 + 21*a^3*b*c^5*d^2*x^4 
+ 35/4*a^4*c^4*d^3*x^4 + 2*a^2*b^2*c^7*x^3 + 28/3*a^3*b*c^6*d*x^3 + 7*a^4* 
c^5*d^2*x^3 + 2*a^3*b*c^7*x^2 + 7/2*a^4*c^6*d*x^2 + a^4*c^7*x
 

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 470, normalized size of antiderivative = 3.95 \[ \int (a+b x)^4 (c+d x)^7 \, dx=x^5\,\left (7\,a^4\,c^3\,d^4+28\,a^3\,b\,c^4\,d^3+\frac {126\,a^2\,b^2\,c^5\,d^2}{5}+\frac {28\,a\,b^3\,c^6\,d}{5}+\frac {b^4\,c^7}{5}\right )+x^8\,\left (\frac {a^4\,d^7}{8}+\frac {7\,a^3\,b\,c\,d^6}{2}+\frac {63\,a^2\,b^2\,c^2\,d^5}{4}+\frac {35\,a\,b^3\,c^3\,d^4}{2}+\frac {35\,b^4\,c^4\,d^3}{8}\right )+x^4\,\left (\frac {35\,a^4\,c^4\,d^3}{4}+21\,a^3\,b\,c^5\,d^2+\frac {21\,a^2\,b^2\,c^6\,d}{2}+a\,b^3\,c^7\right )+x^9\,\left (\frac {4\,a^3\,b\,d^7}{9}+\frac {14\,a^2\,b^2\,c\,d^6}{3}+\frac {28\,a\,b^3\,c^2\,d^5}{3}+\frac {35\,b^4\,c^3\,d^4}{9}\right )+x^7\,\left (a^4\,c\,d^6+12\,a^3\,b\,c^2\,d^5+30\,a^2\,b^2\,c^3\,d^4+20\,a\,b^3\,c^4\,d^3+3\,b^4\,c^5\,d^2\right )+x^6\,\left (\frac {7\,a^4\,c^2\,d^5}{2}+\frac {70\,a^3\,b\,c^3\,d^4}{3}+35\,a^2\,b^2\,c^4\,d^3+14\,a\,b^3\,c^5\,d^2+\frac {7\,b^4\,c^6\,d}{6}\right )+a^4\,c^7\,x+\frac {b^4\,d^7\,x^{12}}{12}+\frac {a^3\,c^6\,x^2\,\left (7\,a\,d+4\,b\,c\right )}{2}+\frac {b^3\,d^6\,x^{11}\,\left (4\,a\,d+7\,b\,c\right )}{11}+\frac {a^2\,c^5\,x^3\,\left (21\,a^2\,d^2+28\,a\,b\,c\,d+6\,b^2\,c^2\right )}{3}+\frac {b^2\,d^5\,x^{10}\,\left (6\,a^2\,d^2+28\,a\,b\,c\,d+21\,b^2\,c^2\right )}{10} \] Input:

int((a + b*x)^4*(c + d*x)^7,x)
 

Output:

x^5*((b^4*c^7)/5 + 7*a^4*c^3*d^4 + 28*a^3*b*c^4*d^3 + (126*a^2*b^2*c^5*d^2 
)/5 + (28*a*b^3*c^6*d)/5) + x^8*((a^4*d^7)/8 + (35*b^4*c^4*d^3)/8 + (35*a* 
b^3*c^3*d^4)/2 + (63*a^2*b^2*c^2*d^5)/4 + (7*a^3*b*c*d^6)/2) + x^4*(a*b^3* 
c^7 + (35*a^4*c^4*d^3)/4 + (21*a^2*b^2*c^6*d)/2 + 21*a^3*b*c^5*d^2) + x^9* 
((4*a^3*b*d^7)/9 + (35*b^4*c^3*d^4)/9 + (28*a*b^3*c^2*d^5)/3 + (14*a^2*b^2 
*c*d^6)/3) + x^7*(a^4*c*d^6 + 3*b^4*c^5*d^2 + 20*a*b^3*c^4*d^3 + 12*a^3*b* 
c^2*d^5 + 30*a^2*b^2*c^3*d^4) + x^6*((7*b^4*c^6*d)/6 + (7*a^4*c^2*d^5)/2 + 
 14*a*b^3*c^5*d^2 + (70*a^3*b*c^3*d^4)/3 + 35*a^2*b^2*c^4*d^3) + a^4*c^7*x 
 + (b^4*d^7*x^12)/12 + (a^3*c^6*x^2*(7*a*d + 4*b*c))/2 + (b^3*d^6*x^11*(4* 
a*d + 7*b*c))/11 + (a^2*c^5*x^3*(21*a^2*d^2 + 6*b^2*c^2 + 28*a*b*c*d))/3 + 
 (b^2*d^5*x^10*(6*a^2*d^2 + 21*b^2*c^2 + 28*a*b*c*d))/10
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 547, normalized size of antiderivative = 4.60 \[ \int (a+b x)^4 (c+d x)^7 \, dx=\frac {x \left (330 b^{4} d^{7} x^{11}+1440 a \,b^{3} d^{7} x^{10}+2520 b^{4} c \,d^{6} x^{10}+2376 a^{2} b^{2} d^{7} x^{9}+11088 a \,b^{3} c \,d^{6} x^{9}+8316 b^{4} c^{2} d^{5} x^{9}+1760 a^{3} b \,d^{7} x^{8}+18480 a^{2} b^{2} c \,d^{6} x^{8}+36960 a \,b^{3} c^{2} d^{5} x^{8}+15400 b^{4} c^{3} d^{4} x^{8}+495 a^{4} d^{7} x^{7}+13860 a^{3} b c \,d^{6} x^{7}+62370 a^{2} b^{2} c^{2} d^{5} x^{7}+69300 a \,b^{3} c^{3} d^{4} x^{7}+17325 b^{4} c^{4} d^{3} x^{7}+3960 a^{4} c \,d^{6} x^{6}+47520 a^{3} b \,c^{2} d^{5} x^{6}+118800 a^{2} b^{2} c^{3} d^{4} x^{6}+79200 a \,b^{3} c^{4} d^{3} x^{6}+11880 b^{4} c^{5} d^{2} x^{6}+13860 a^{4} c^{2} d^{5} x^{5}+92400 a^{3} b \,c^{3} d^{4} x^{5}+138600 a^{2} b^{2} c^{4} d^{3} x^{5}+55440 a \,b^{3} c^{5} d^{2} x^{5}+4620 b^{4} c^{6} d \,x^{5}+27720 a^{4} c^{3} d^{4} x^{4}+110880 a^{3} b \,c^{4} d^{3} x^{4}+99792 a^{2} b^{2} c^{5} d^{2} x^{4}+22176 a \,b^{3} c^{6} d \,x^{4}+792 b^{4} c^{7} x^{4}+34650 a^{4} c^{4} d^{3} x^{3}+83160 a^{3} b \,c^{5} d^{2} x^{3}+41580 a^{2} b^{2} c^{6} d \,x^{3}+3960 a \,b^{3} c^{7} x^{3}+27720 a^{4} c^{5} d^{2} x^{2}+36960 a^{3} b \,c^{6} d \,x^{2}+7920 a^{2} b^{2} c^{7} x^{2}+13860 a^{4} c^{6} d x +7920 a^{3} b \,c^{7} x +3960 a^{4} c^{7}\right )}{3960} \] Input:

int((b*x+a)^4*(d*x+c)^7,x)
 

Output:

(x*(3960*a**4*c**7 + 13860*a**4*c**6*d*x + 27720*a**4*c**5*d**2*x**2 + 346 
50*a**4*c**4*d**3*x**3 + 27720*a**4*c**3*d**4*x**4 + 13860*a**4*c**2*d**5* 
x**5 + 3960*a**4*c*d**6*x**6 + 495*a**4*d**7*x**7 + 7920*a**3*b*c**7*x + 3 
6960*a**3*b*c**6*d*x**2 + 83160*a**3*b*c**5*d**2*x**3 + 110880*a**3*b*c**4 
*d**3*x**4 + 92400*a**3*b*c**3*d**4*x**5 + 47520*a**3*b*c**2*d**5*x**6 + 1 
3860*a**3*b*c*d**6*x**7 + 1760*a**3*b*d**7*x**8 + 7920*a**2*b**2*c**7*x**2 
 + 41580*a**2*b**2*c**6*d*x**3 + 99792*a**2*b**2*c**5*d**2*x**4 + 138600*a 
**2*b**2*c**4*d**3*x**5 + 118800*a**2*b**2*c**3*d**4*x**6 + 62370*a**2*b** 
2*c**2*d**5*x**7 + 18480*a**2*b**2*c*d**6*x**8 + 2376*a**2*b**2*d**7*x**9 
+ 3960*a*b**3*c**7*x**3 + 22176*a*b**3*c**6*d*x**4 + 55440*a*b**3*c**5*d** 
2*x**5 + 79200*a*b**3*c**4*d**3*x**6 + 69300*a*b**3*c**3*d**4*x**7 + 36960 
*a*b**3*c**2*d**5*x**8 + 11088*a*b**3*c*d**6*x**9 + 1440*a*b**3*d**7*x**10 
 + 792*b**4*c**7*x**4 + 4620*b**4*c**6*d*x**5 + 11880*b**4*c**5*d**2*x**6 
+ 17325*b**4*c**4*d**3*x**7 + 15400*b**4*c**3*d**4*x**8 + 8316*b**4*c**2*d 
**5*x**9 + 2520*b**4*c*d**6*x**10 + 330*b**4*d**7*x**11))/3960