\(\int (a+b x)^3 (c+d x)^{10} \, dx\) [102]

Optimal result

Integrand size = 15, antiderivative size = 92 \[ \int (a+b x)^3 (c+d x)^{10} \, dx=-\frac {(b c-a d)^3 (c+d x)^{11}}{11 d^4}+\frac {b (b c-a d)^2 (c+d x)^{12}}{4 d^4}-\frac {3 b^2 (b c-a d) (c+d x)^{13}}{13 d^4}+\frac {b^3 (c+d x)^{14}}{14 d^4} \] Output:

-1/11*(-a*d+b*c)^3*(d*x+c)^11/d^4+1/4*b*(-a*d+b*c)^2*(d*x+c)^12/d^4-3/13*b 
^2*(-a*d+b*c)*(d*x+c)^13/d^4+1/14*b^3*(d*x+c)^14/d^4
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(511\) vs. \(2(92)=184\).

Time = 0.03 (sec) , antiderivative size = 511, normalized size of antiderivative = 5.55 \[ \int (a+b x)^3 (c+d x)^{10} \, dx=a^3 c^{10} x+\frac {1}{2} a^2 c^9 (3 b c+10 a d) x^2+a c^8 \left (b^2 c^2+10 a b c d+15 a^2 d^2\right ) x^3+\frac {1}{4} c^7 \left (b^3 c^3+30 a b^2 c^2 d+135 a^2 b c d^2+120 a^3 d^3\right ) x^4+c^6 d \left (2 b^3 c^3+27 a b^2 c^2 d+72 a^2 b c d^2+42 a^3 d^3\right ) x^5+\frac {3}{2} c^5 d^2 \left (5 b^3 c^3+40 a b^2 c^2 d+70 a^2 b c d^2+28 a^3 d^3\right ) x^6+\frac {6}{7} c^4 d^3 \left (20 b^3 c^3+105 a b^2 c^2 d+126 a^2 b c d^2+35 a^3 d^3\right ) x^7+\frac {3}{4} c^3 d^4 \left (35 b^3 c^3+126 a b^2 c^2 d+105 a^2 b c d^2+20 a^3 d^3\right ) x^8+c^2 d^5 \left (28 b^3 c^3+70 a b^2 c^2 d+40 a^2 b c d^2+5 a^3 d^3\right ) x^9+\frac {1}{2} c d^6 \left (42 b^3 c^3+72 a b^2 c^2 d+27 a^2 b c d^2+2 a^3 d^3\right ) x^{10}+\frac {1}{11} d^7 \left (120 b^3 c^3+135 a b^2 c^2 d+30 a^2 b c d^2+a^3 d^3\right ) x^{11}+\frac {1}{4} b d^8 \left (15 b^2 c^2+10 a b c d+a^2 d^2\right ) x^{12}+\frac {1}{13} b^2 d^9 (10 b c+3 a d) x^{13}+\frac {1}{14} b^3 d^{10} x^{14} \] Input:

Integrate[(a + b*x)^3*(c + d*x)^10,x]
 

Output:

a^3*c^10*x + (a^2*c^9*(3*b*c + 10*a*d)*x^2)/2 + a*c^8*(b^2*c^2 + 10*a*b*c* 
d + 15*a^2*d^2)*x^3 + (c^7*(b^3*c^3 + 30*a*b^2*c^2*d + 135*a^2*b*c*d^2 + 1 
20*a^3*d^3)*x^4)/4 + c^6*d*(2*b^3*c^3 + 27*a*b^2*c^2*d + 72*a^2*b*c*d^2 + 
42*a^3*d^3)*x^5 + (3*c^5*d^2*(5*b^3*c^3 + 40*a*b^2*c^2*d + 70*a^2*b*c*d^2 
+ 28*a^3*d^3)*x^6)/2 + (6*c^4*d^3*(20*b^3*c^3 + 105*a*b^2*c^2*d + 126*a^2* 
b*c*d^2 + 35*a^3*d^3)*x^7)/7 + (3*c^3*d^4*(35*b^3*c^3 + 126*a*b^2*c^2*d + 
105*a^2*b*c*d^2 + 20*a^3*d^3)*x^8)/4 + c^2*d^5*(28*b^3*c^3 + 70*a*b^2*c^2* 
d + 40*a^2*b*c*d^2 + 5*a^3*d^3)*x^9 + (c*d^6*(42*b^3*c^3 + 72*a*b^2*c^2*d 
+ 27*a^2*b*c*d^2 + 2*a^3*d^3)*x^10)/2 + (d^7*(120*b^3*c^3 + 135*a*b^2*c^2* 
d + 30*a^2*b*c*d^2 + a^3*d^3)*x^11)/11 + (b*d^8*(15*b^2*c^2 + 10*a*b*c*d + 
 a^2*d^2)*x^12)/4 + (b^2*d^9*(10*b*c + 3*a*d)*x^13)/13 + (b^3*d^10*x^14)/1 
4
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 92, 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.

\(\displaystyle \int (a+b x)^3 (c+d x)^{10} \, dx\)

\(\Big \downarrow \) 49

\(\displaystyle \int \left (-\frac {3 b^2 (c+d x)^{12} (b c-a d)}{d^3}+\frac {3 b (c+d x)^{11} (b c-a d)^2}{d^3}+\frac {(c+d x)^{10} (a d-b c)^3}{d^3}+\frac {b^3 (c+d x)^{13}}{d^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 b^2 (c+d x)^{13} (b c-a d)}{13 d^4}+\frac {b (c+d x)^{12} (b c-a d)^2}{4 d^4}-\frac {(c+d x)^{11} (b c-a d)^3}{11 d^4}+\frac {b^3 (c+d x)^{14}}{14 d^4}\)

Input:

Int[(a + b*x)^3*(c + d*x)^10,x]
 

Output:

-1/11*((b*c - a*d)^3*(c + d*x)^11)/d^4 + (b*(b*c - a*d)^2*(c + d*x)^12)/(4 
*d^4) - (3*b^2*(b*c - a*d)*(c + d*x)^13)/(13*d^4) + (b^3*(c + d*x)^14)/(14 
*d^4)
 

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. \(528\) vs. \(2(84)=168\).

Time = 0.09 (sec) , antiderivative size = 529, normalized size of antiderivative = 5.75

method	result	size
norman	\(a^{3} c^{10} x +\left (5 a^{3} c^{9} d +\frac {3}{2} a^{2} b \,c^{10}\right ) x^{2}+\left (15 a^{3} c^{8} d^{2}+10 a^{2} b \,c^{9} d +a \,b^{2} c^{10}\right ) x^{3}+\left (30 a^{3} c^{7} d^{3}+\frac {135}{4} a^{2} b \,c^{8} d^{2}+\frac {15}{2} a \,b^{2} c^{9} d +\frac {1}{4} b^{3} c^{10}\right ) x^{4}+\left (42 a^{3} c^{6} d^{4}+72 a^{2} b \,c^{7} d^{3}+27 a \,b^{2} c^{8} d^{2}+2 b^{3} c^{9} d \right ) x^{5}+\left (42 a^{3} c^{5} d^{5}+105 a^{2} b \,c^{6} d^{4}+60 a \,b^{2} c^{7} d^{3}+\frac {15}{2} b^{3} c^{8} d^{2}\right ) x^{6}+\left (30 a^{3} c^{4} d^{6}+108 a^{2} b \,c^{5} d^{5}+90 a \,b^{2} c^{6} d^{4}+\frac {120}{7} b^{3} c^{7} d^{3}\right ) x^{7}+\left (15 a^{3} c^{3} d^{7}+\frac {315}{4} a^{2} b \,c^{4} d^{6}+\frac {189}{2} a \,b^{2} c^{5} d^{5}+\frac {105}{4} b^{3} c^{6} d^{4}\right ) x^{8}+\left (5 a^{3} c^{2} d^{8}+40 a^{2} b \,c^{3} d^{7}+70 a \,b^{2} c^{4} d^{6}+28 b^{3} c^{5} d^{5}\right ) x^{9}+\left (a^{3} c \,d^{9}+\frac {27}{2} a^{2} b \,c^{2} d^{8}+36 a \,b^{2} c^{3} d^{7}+21 b^{3} c^{4} d^{6}\right ) x^{10}+\left (\frac {1}{11} a^{3} d^{10}+\frac {30}{11} a^{2} b c \,d^{9}+\frac {135}{11} a \,b^{2} c^{2} d^{8}+\frac {120}{11} b^{3} c^{3} d^{7}\right ) x^{11}+\left (\frac {1}{4} a^{2} b \,d^{10}+\frac {5}{2} a \,b^{2} c \,d^{9}+\frac {15}{4} b^{3} c^{2} d^{8}\right ) x^{12}+\left (\frac {3}{13} a \,b^{2} d^{10}+\frac {10}{13} b^{3} c \,d^{9}\right ) x^{13}+\frac {b^{3} d^{10} x^{14}}{14}\)	\(529\)
default	\(\frac {b^{3} d^{10} x^{14}}{14}+\frac {\left (3 a \,b^{2} d^{10}+10 b^{3} c \,d^{9}\right ) x^{13}}{13}+\frac {\left (3 a^{2} b \,d^{10}+30 a \,b^{2} c \,d^{9}+45 b^{3} c^{2} d^{8}\right ) x^{12}}{12}+\frac {\left (a^{3} d^{10}+30 a^{2} b c \,d^{9}+135 a \,b^{2} c^{2} d^{8}+120 b^{3} c^{3} d^{7}\right ) x^{11}}{11}+\frac {\left (10 a^{3} c \,d^{9}+135 a^{2} b \,c^{2} d^{8}+360 a \,b^{2} c^{3} d^{7}+210 b^{3} c^{4} d^{6}\right ) x^{10}}{10}+\frac {\left (45 a^{3} c^{2} d^{8}+360 a^{2} b \,c^{3} d^{7}+630 a \,b^{2} c^{4} d^{6}+252 b^{3} c^{5} d^{5}\right ) x^{9}}{9}+\frac {\left (120 a^{3} c^{3} d^{7}+630 a^{2} b \,c^{4} d^{6}+756 a \,b^{2} c^{5} d^{5}+210 b^{3} c^{6} d^{4}\right ) x^{8}}{8}+\frac {\left (210 a^{3} c^{4} d^{6}+756 a^{2} b \,c^{5} d^{5}+630 a \,b^{2} c^{6} d^{4}+120 b^{3} c^{7} d^{3}\right ) x^{7}}{7}+\frac {\left (252 a^{3} c^{5} d^{5}+630 a^{2} b \,c^{6} d^{4}+360 a \,b^{2} c^{7} d^{3}+45 b^{3} c^{8} d^{2}\right ) x^{6}}{6}+\frac {\left (210 a^{3} c^{6} d^{4}+360 a^{2} b \,c^{7} d^{3}+135 a \,b^{2} c^{8} d^{2}+10 b^{3} c^{9} d \right ) x^{5}}{5}+\frac {\left (120 a^{3} c^{7} d^{3}+135 a^{2} b \,c^{8} d^{2}+30 a \,b^{2} c^{9} d +b^{3} c^{10}\right ) x^{4}}{4}+\frac {\left (45 a^{3} c^{8} d^{2}+30 a^{2} b \,c^{9} d +3 a \,b^{2} c^{10}\right ) x^{3}}{3}+\frac {\left (10 a^{3} c^{9} d +3 a^{2} b \,c^{10}\right ) x^{2}}{2}+a^{3} c^{10} x\)	\(541\)
gosper	\(\frac {189}{2} x^{8} a \,b^{2} c^{5} d^{5}+\frac {27}{2} x^{10} a^{2} b \,c^{2} d^{8}+36 x^{10} a \,b^{2} c^{3} d^{7}+\frac {30}{11} x^{11} a^{2} b c \,d^{9}+\frac {135}{11} x^{11} a \,b^{2} c^{2} d^{8}+\frac {5}{2} x^{12} a \,b^{2} c \,d^{9}+10 a^{2} b \,c^{9} d \,x^{3}+72 a^{2} b \,c^{7} d^{3} x^{5}+27 a \,b^{2} c^{8} d^{2} x^{5}+40 a^{2} b \,c^{3} d^{7} x^{9}+70 a \,b^{2} c^{4} d^{6} x^{9}+90 x^{7} a \,b^{2} c^{6} d^{4}+\frac {315}{4} x^{8} a^{2} b \,c^{4} d^{6}+108 x^{7} a^{2} b \,c^{5} d^{5}+\frac {15}{2} x^{4} a \,b^{2} c^{9} d +105 x^{6} a^{2} b \,c^{6} d^{4}+60 x^{6} a \,b^{2} c^{7} d^{3}+\frac {135}{4} x^{4} a^{2} b \,c^{8} d^{2}+5 x^{2} a^{3} c^{9} d +\frac {3}{2} x^{2} a^{2} b \,c^{10}+30 x^{4} a^{3} c^{7} d^{3}+42 x^{6} a^{3} c^{5} d^{5}+\frac {15}{2} x^{6} b^{3} c^{8} d^{2}+30 x^{7} a^{3} c^{4} d^{6}+\frac {120}{7} x^{7} b^{3} c^{7} d^{3}+15 x^{8} a^{3} c^{3} d^{7}+\frac {120}{11} x^{11} b^{3} c^{3} d^{7}+\frac {1}{4} x^{12} a^{2} b \,d^{10}+\frac {15}{4} x^{12} b^{3} c^{2} d^{8}+\frac {3}{13} x^{13} a \,b^{2} d^{10}+\frac {10}{13} x^{13} b^{3} c \,d^{9}+15 a^{3} c^{8} d^{2} x^{3}+a \,b^{2} c^{10} x^{3}+42 a^{3} c^{6} d^{4} x^{5}+2 b^{3} c^{9} d \,x^{5}+5 a^{3} c^{2} d^{8} x^{9}+28 b^{3} c^{5} d^{5} x^{9}+\frac {105}{4} x^{8} b^{3} c^{6} d^{4}+x^{10} a^{3} c \,d^{9}+21 x^{10} b^{3} c^{4} d^{6}+\frac {1}{14} b^{3} d^{10} x^{14}+\frac {1}{4} x^{4} b^{3} c^{10}+\frac {1}{11} x^{11} a^{3} d^{10}+a^{3} c^{10} x\)	\(595\)
risch	\(\frac {189}{2} x^{8} a \,b^{2} c^{5} d^{5}+\frac {27}{2} x^{10} a^{2} b \,c^{2} d^{8}+36 x^{10} a \,b^{2} c^{3} d^{7}+\frac {30}{11} x^{11} a^{2} b c \,d^{9}+\frac {135}{11} x^{11} a \,b^{2} c^{2} d^{8}+\frac {5}{2} x^{12} a \,b^{2} c \,d^{9}+10 a^{2} b \,c^{9} d \,x^{3}+72 a^{2} b \,c^{7} d^{3} x^{5}+27 a \,b^{2} c^{8} d^{2} x^{5}+40 a^{2} b \,c^{3} d^{7} x^{9}+70 a \,b^{2} c^{4} d^{6} x^{9}+90 x^{7} a \,b^{2} c^{6} d^{4}+\frac {315}{4} x^{8} a^{2} b \,c^{4} d^{6}+108 x^{7} a^{2} b \,c^{5} d^{5}+\frac {15}{2} x^{4} a \,b^{2} c^{9} d +105 x^{6} a^{2} b \,c^{6} d^{4}+60 x^{6} a \,b^{2} c^{7} d^{3}+\frac {135}{4} x^{4} a^{2} b \,c^{8} d^{2}+5 x^{2} a^{3} c^{9} d +\frac {3}{2} x^{2} a^{2} b \,c^{10}+30 x^{4} a^{3} c^{7} d^{3}+42 x^{6} a^{3} c^{5} d^{5}+\frac {15}{2} x^{6} b^{3} c^{8} d^{2}+30 x^{7} a^{3} c^{4} d^{6}+\frac {120}{7} x^{7} b^{3} c^{7} d^{3}+15 x^{8} a^{3} c^{3} d^{7}+\frac {120}{11} x^{11} b^{3} c^{3} d^{7}+\frac {1}{4} x^{12} a^{2} b \,d^{10}+\frac {15}{4} x^{12} b^{3} c^{2} d^{8}+\frac {3}{13} x^{13} a \,b^{2} d^{10}+\frac {10}{13} x^{13} b^{3} c \,d^{9}+15 a^{3} c^{8} d^{2} x^{3}+a \,b^{2} c^{10} x^{3}+42 a^{3} c^{6} d^{4} x^{5}+2 b^{3} c^{9} d \,x^{5}+5 a^{3} c^{2} d^{8} x^{9}+28 b^{3} c^{5} d^{5} x^{9}+\frac {105}{4} x^{8} b^{3} c^{6} d^{4}+x^{10} a^{3} c \,d^{9}+21 x^{10} b^{3} c^{4} d^{6}+\frac {1}{14} b^{3} d^{10} x^{14}+\frac {1}{4} x^{4} b^{3} c^{10}+\frac {1}{11} x^{11} a^{3} d^{10}+a^{3} c^{10} x\)	\(595\)
parallelrisch	\(\frac {189}{2} x^{8} a \,b^{2} c^{5} d^{5}+\frac {27}{2} x^{10} a^{2} b \,c^{2} d^{8}+36 x^{10} a \,b^{2} c^{3} d^{7}+\frac {30}{11} x^{11} a^{2} b c \,d^{9}+\frac {135}{11} x^{11} a \,b^{2} c^{2} d^{8}+\frac {5}{2} x^{12} a \,b^{2} c \,d^{9}+10 a^{2} b \,c^{9} d \,x^{3}+72 a^{2} b \,c^{7} d^{3} x^{5}+27 a \,b^{2} c^{8} d^{2} x^{5}+40 a^{2} b \,c^{3} d^{7} x^{9}+70 a \,b^{2} c^{4} d^{6} x^{9}+90 x^{7} a \,b^{2} c^{6} d^{4}+\frac {315}{4} x^{8} a^{2} b \,c^{4} d^{6}+108 x^{7} a^{2} b \,c^{5} d^{5}+\frac {15}{2} x^{4} a \,b^{2} c^{9} d +105 x^{6} a^{2} b \,c^{6} d^{4}+60 x^{6} a \,b^{2} c^{7} d^{3}+\frac {135}{4} x^{4} a^{2} b \,c^{8} d^{2}+5 x^{2} a^{3} c^{9} d +\frac {3}{2} x^{2} a^{2} b \,c^{10}+30 x^{4} a^{3} c^{7} d^{3}+42 x^{6} a^{3} c^{5} d^{5}+\frac {15}{2} x^{6} b^{3} c^{8} d^{2}+30 x^{7} a^{3} c^{4} d^{6}+\frac {120}{7} x^{7} b^{3} c^{7} d^{3}+15 x^{8} a^{3} c^{3} d^{7}+\frac {120}{11} x^{11} b^{3} c^{3} d^{7}+\frac {1}{4} x^{12} a^{2} b \,d^{10}+\frac {15}{4} x^{12} b^{3} c^{2} d^{8}+\frac {3}{13} x^{13} a \,b^{2} d^{10}+\frac {10}{13} x^{13} b^{3} c \,d^{9}+15 a^{3} c^{8} d^{2} x^{3}+a \,b^{2} c^{10} x^{3}+42 a^{3} c^{6} d^{4} x^{5}+2 b^{3} c^{9} d \,x^{5}+5 a^{3} c^{2} d^{8} x^{9}+28 b^{3} c^{5} d^{5} x^{9}+\frac {105}{4} x^{8} b^{3} c^{6} d^{4}+x^{10} a^{3} c \,d^{9}+21 x^{10} b^{3} c^{4} d^{6}+\frac {1}{14} b^{3} d^{10} x^{14}+\frac {1}{4} x^{4} b^{3} c^{10}+\frac {1}{11} x^{11} a^{3} d^{10}+a^{3} c^{10} x\)	\(595\)
orering	\(\frac {x \left (286 b^{3} d^{10} x^{13}+924 a \,b^{2} d^{10} x^{12}+3080 b^{3} c \,d^{9} x^{12}+1001 a^{2} b \,d^{10} x^{11}+10010 a \,b^{2} c \,d^{9} x^{11}+15015 b^{3} c^{2} d^{8} x^{11}+364 a^{3} d^{10} x^{10}+10920 a^{2} b c \,d^{9} x^{10}+49140 a \,b^{2} c^{2} d^{8} x^{10}+43680 b^{3} c^{3} d^{7} x^{10}+4004 a^{3} c \,d^{9} x^{9}+54054 a^{2} b \,c^{2} d^{8} x^{9}+144144 a \,b^{2} c^{3} d^{7} x^{9}+84084 b^{3} c^{4} d^{6} x^{9}+20020 a^{3} c^{2} d^{8} x^{8}+160160 a^{2} b \,c^{3} d^{7} x^{8}+280280 a \,b^{2} c^{4} d^{6} x^{8}+112112 b^{3} c^{5} d^{5} x^{8}+60060 a^{3} c^{3} d^{7} x^{7}+315315 a^{2} b \,c^{4} d^{6} x^{7}+378378 a \,b^{2} c^{5} d^{5} x^{7}+105105 b^{3} c^{6} d^{4} x^{7}+120120 a^{3} c^{4} d^{6} x^{6}+432432 a^{2} b \,c^{5} d^{5} x^{6}+360360 a \,b^{2} c^{6} d^{4} x^{6}+68640 b^{3} c^{7} d^{3} x^{6}+168168 a^{3} c^{5} d^{5} x^{5}+420420 a^{2} b \,c^{6} d^{4} x^{5}+240240 a \,b^{2} c^{7} d^{3} x^{5}+30030 b^{3} c^{8} d^{2} x^{5}+168168 a^{3} c^{6} d^{4} x^{4}+288288 a^{2} b \,c^{7} d^{3} x^{4}+108108 a \,b^{2} c^{8} d^{2} x^{4}+8008 b^{3} c^{9} d \,x^{4}+120120 a^{3} c^{7} d^{3} x^{3}+135135 a^{2} b \,c^{8} d^{2} x^{3}+30030 a \,b^{2} c^{9} d \,x^{3}+1001 b^{3} c^{10} x^{3}+60060 a^{3} c^{8} d^{2} x^{2}+40040 a^{2} b \,c^{9} d \,x^{2}+4004 a \,b^{2} c^{10} x^{2}+20020 a^{3} c^{9} d x +6006 a^{2} b \,c^{10} x +4004 a^{3} c^{10}\right )}{4004}\)	\(596\)

Input:

int((b*x+a)^3*(d*x+c)^10,x,method=_RETURNVERBOSE)
 

Output:

a^3*c^10*x+(5*a^3*c^9*d+3/2*a^2*b*c^10)*x^2+(15*a^3*c^8*d^2+10*a^2*b*c^9*d 
+a*b^2*c^10)*x^3+(30*a^3*c^7*d^3+135/4*a^2*b*c^8*d^2+15/2*a*b^2*c^9*d+1/4* 
b^3*c^10)*x^4+(42*a^3*c^6*d^4+72*a^2*b*c^7*d^3+27*a*b^2*c^8*d^2+2*b^3*c^9* 
d)*x^5+(42*a^3*c^5*d^5+105*a^2*b*c^6*d^4+60*a*b^2*c^7*d^3+15/2*b^3*c^8*d^2 
)*x^6+(30*a^3*c^4*d^6+108*a^2*b*c^5*d^5+90*a*b^2*c^6*d^4+120/7*b^3*c^7*d^3 
)*x^7+(15*a^3*c^3*d^7+315/4*a^2*b*c^4*d^6+189/2*a*b^2*c^5*d^5+105/4*b^3*c^ 
6*d^4)*x^8+(5*a^3*c^2*d^8+40*a^2*b*c^3*d^7+70*a*b^2*c^4*d^6+28*b^3*c^5*d^5 
)*x^9+(a^3*c*d^9+27/2*a^2*b*c^2*d^8+36*a*b^2*c^3*d^7+21*b^3*c^4*d^6)*x^10+ 
(1/11*a^3*d^10+30/11*a^2*b*c*d^9+135/11*a*b^2*c^2*d^8+120/11*b^3*c^3*d^7)* 
x^11+(1/4*a^2*b*d^10+5/2*a*b^2*c*d^9+15/4*b^3*c^2*d^8)*x^12+(3/13*a*b^2*d^ 
10+10/13*b^3*c*d^9)*x^13+1/14*b^3*d^10*x^14
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (84) = 168\).

Time = 0.08 (sec) , antiderivative size = 535, normalized size of antiderivative = 5.82 \[ \int (a+b x)^3 (c+d x)^{10} \, dx=\frac {1}{14} \, b^{3} d^{10} x^{14} + a^{3} c^{10} x + \frac {1}{13} \, {\left (10 \, b^{3} c d^{9} + 3 \, a b^{2} d^{10}\right )} x^{13} + \frac {1}{4} \, {\left (15 \, b^{3} c^{2} d^{8} + 10 \, a b^{2} c d^{9} + a^{2} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (120 \, b^{3} c^{3} d^{7} + 135 \, a b^{2} c^{2} d^{8} + 30 \, a^{2} b c d^{9} + a^{3} d^{10}\right )} x^{11} + \frac {1}{2} \, {\left (42 \, b^{3} c^{4} d^{6} + 72 \, a b^{2} c^{3} d^{7} + 27 \, a^{2} b c^{2} d^{8} + 2 \, a^{3} c d^{9}\right )} x^{10} + {\left (28 \, b^{3} c^{5} d^{5} + 70 \, a b^{2} c^{4} d^{6} + 40 \, a^{2} b c^{3} d^{7} + 5 \, a^{3} c^{2} d^{8}\right )} x^{9} + \frac {3}{4} \, {\left (35 \, b^{3} c^{6} d^{4} + 126 \, a b^{2} c^{5} d^{5} + 105 \, a^{2} b c^{4} d^{6} + 20 \, a^{3} c^{3} d^{7}\right )} x^{8} + \frac {6}{7} \, {\left (20 \, b^{3} c^{7} d^{3} + 105 \, a b^{2} c^{6} d^{4} + 126 \, a^{2} b c^{5} d^{5} + 35 \, a^{3} c^{4} d^{6}\right )} x^{7} + \frac {3}{2} \, {\left (5 \, b^{3} c^{8} d^{2} + 40 \, a b^{2} c^{7} d^{3} + 70 \, a^{2} b c^{6} d^{4} + 28 \, a^{3} c^{5} d^{5}\right )} x^{6} + {\left (2 \, b^{3} c^{9} d + 27 \, a b^{2} c^{8} d^{2} + 72 \, a^{2} b c^{7} d^{3} + 42 \, a^{3} c^{6} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{10} + 30 \, a b^{2} c^{9} d + 135 \, a^{2} b c^{8} d^{2} + 120 \, a^{3} c^{7} d^{3}\right )} x^{4} + {\left (a b^{2} c^{10} + 10 \, a^{2} b c^{9} d + 15 \, a^{3} c^{8} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{10} + 10 \, a^{3} c^{9} d\right )} x^{2} \] Input:

integrate((b*x+a)^3*(d*x+c)^10,x, algorithm="fricas")
 

Output:

1/14*b^3*d^10*x^14 + a^3*c^10*x + 1/13*(10*b^3*c*d^9 + 3*a*b^2*d^10)*x^13 
+ 1/4*(15*b^3*c^2*d^8 + 10*a*b^2*c*d^9 + a^2*b*d^10)*x^12 + 1/11*(120*b^3* 
c^3*d^7 + 135*a*b^2*c^2*d^8 + 30*a^2*b*c*d^9 + a^3*d^10)*x^11 + 1/2*(42*b^ 
3*c^4*d^6 + 72*a*b^2*c^3*d^7 + 27*a^2*b*c^2*d^8 + 2*a^3*c*d^9)*x^10 + (28* 
b^3*c^5*d^5 + 70*a*b^2*c^4*d^6 + 40*a^2*b*c^3*d^7 + 5*a^3*c^2*d^8)*x^9 + 3 
/4*(35*b^3*c^6*d^4 + 126*a*b^2*c^5*d^5 + 105*a^2*b*c^4*d^6 + 20*a^3*c^3*d^ 
7)*x^8 + 6/7*(20*b^3*c^7*d^3 + 105*a*b^2*c^6*d^4 + 126*a^2*b*c^5*d^5 + 35* 
a^3*c^4*d^6)*x^7 + 3/2*(5*b^3*c^8*d^2 + 40*a*b^2*c^7*d^3 + 70*a^2*b*c^6*d^ 
4 + 28*a^3*c^5*d^5)*x^6 + (2*b^3*c^9*d + 27*a*b^2*c^8*d^2 + 72*a^2*b*c^7*d 
^3 + 42*a^3*c^6*d^4)*x^5 + 1/4*(b^3*c^10 + 30*a*b^2*c^9*d + 135*a^2*b*c^8* 
d^2 + 120*a^3*c^7*d^3)*x^4 + (a*b^2*c^10 + 10*a^2*b*c^9*d + 15*a^3*c^8*d^2 
)*x^3 + 1/2*(3*a^2*b*c^10 + 10*a^3*c^9*d)*x^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (80) = 160\).

Time = 0.05 (sec) , antiderivative size = 586, normalized size of antiderivative = 6.37 \[ \int (a+b x)^3 (c+d x)^{10} \, dx=a^{3} c^{10} x + \frac {b^{3} d^{10} x^{14}}{14} + x^{13} \cdot \left (\frac {3 a b^{2} d^{10}}{13} + \frac {10 b^{3} c d^{9}}{13}\right ) + x^{12} \left (\frac {a^{2} b d^{10}}{4} + \frac {5 a b^{2} c d^{9}}{2} + \frac {15 b^{3} c^{2} d^{8}}{4}\right ) + x^{11} \left (\frac {a^{3} d^{10}}{11} + \frac {30 a^{2} b c d^{9}}{11} + \frac {135 a b^{2} c^{2} d^{8}}{11} + \frac {120 b^{3} c^{3} d^{7}}{11}\right ) + x^{10} \left (a^{3} c d^{9} + \frac {27 a^{2} b c^{2} d^{8}}{2} + 36 a b^{2} c^{3} d^{7} + 21 b^{3} c^{4} d^{6}\right ) + x^{9} \cdot \left (5 a^{3} c^{2} d^{8} + 40 a^{2} b c^{3} d^{7} + 70 a b^{2} c^{4} d^{6} + 28 b^{3} c^{5} d^{5}\right ) + x^{8} \cdot \left (15 a^{3} c^{3} d^{7} + \frac {315 a^{2} b c^{4} d^{6}}{4} + \frac {189 a b^{2} c^{5} d^{5}}{2} + \frac {105 b^{3} c^{6} d^{4}}{4}\right ) + x^{7} \cdot \left (30 a^{3} c^{4} d^{6} + 108 a^{2} b c^{5} d^{5} + 90 a b^{2} c^{6} d^{4} + \frac {120 b^{3} c^{7} d^{3}}{7}\right ) + x^{6} \cdot \left (42 a^{3} c^{5} d^{5} + 105 a^{2} b c^{6} d^{4} + 60 a b^{2} c^{7} d^{3} + \frac {15 b^{3} c^{8} d^{2}}{2}\right ) + x^{5} \cdot \left (42 a^{3} c^{6} d^{4} + 72 a^{2} b c^{7} d^{3} + 27 a b^{2} c^{8} d^{2} + 2 b^{3} c^{9} d\right ) + x^{4} \cdot \left (30 a^{3} c^{7} d^{3} + \frac {135 a^{2} b c^{8} d^{2}}{4} + \frac {15 a b^{2} c^{9} d}{2} + \frac {b^{3} c^{10}}{4}\right ) + x^{3} \cdot \left (15 a^{3} c^{8} d^{2} + 10 a^{2} b c^{9} d + a b^{2} c^{10}\right ) + x^{2} \cdot \left (5 a^{3} c^{9} d + \frac {3 a^{2} b c^{10}}{2}\right ) \] Input:

integrate((b*x+a)**3*(d*x+c)**10,x)
 

Output:

a**3*c**10*x + b**3*d**10*x**14/14 + x**13*(3*a*b**2*d**10/13 + 10*b**3*c* 
d**9/13) + x**12*(a**2*b*d**10/4 + 5*a*b**2*c*d**9/2 + 15*b**3*c**2*d**8/4 
) + x**11*(a**3*d**10/11 + 30*a**2*b*c*d**9/11 + 135*a*b**2*c**2*d**8/11 + 
 120*b**3*c**3*d**7/11) + x**10*(a**3*c*d**9 + 27*a**2*b*c**2*d**8/2 + 36* 
a*b**2*c**3*d**7 + 21*b**3*c**4*d**6) + x**9*(5*a**3*c**2*d**8 + 40*a**2*b 
*c**3*d**7 + 70*a*b**2*c**4*d**6 + 28*b**3*c**5*d**5) + x**8*(15*a**3*c**3 
*d**7 + 315*a**2*b*c**4*d**6/4 + 189*a*b**2*c**5*d**5/2 + 105*b**3*c**6*d* 
*4/4) + x**7*(30*a**3*c**4*d**6 + 108*a**2*b*c**5*d**5 + 90*a*b**2*c**6*d* 
*4 + 120*b**3*c**7*d**3/7) + x**6*(42*a**3*c**5*d**5 + 105*a**2*b*c**6*d** 
4 + 60*a*b**2*c**7*d**3 + 15*b**3*c**8*d**2/2) + x**5*(42*a**3*c**6*d**4 + 
 72*a**2*b*c**7*d**3 + 27*a*b**2*c**8*d**2 + 2*b**3*c**9*d) + x**4*(30*a** 
3*c**7*d**3 + 135*a**2*b*c**8*d**2/4 + 15*a*b**2*c**9*d/2 + b**3*c**10/4) 
+ x**3*(15*a**3*c**8*d**2 + 10*a**2*b*c**9*d + a*b**2*c**10) + x**2*(5*a** 
3*c**9*d + 3*a**2*b*c**10/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (84) = 168\).

Time = 0.04 (sec) , antiderivative size = 535, normalized size of antiderivative = 5.82 \[ \int (a+b x)^3 (c+d x)^{10} \, dx=\frac {1}{14} \, b^{3} d^{10} x^{14} + a^{3} c^{10} x + \frac {1}{13} \, {\left (10 \, b^{3} c d^{9} + 3 \, a b^{2} d^{10}\right )} x^{13} + \frac {1}{4} \, {\left (15 \, b^{3} c^{2} d^{8} + 10 \, a b^{2} c d^{9} + a^{2} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (120 \, b^{3} c^{3} d^{7} + 135 \, a b^{2} c^{2} d^{8} + 30 \, a^{2} b c d^{9} + a^{3} d^{10}\right )} x^{11} + \frac {1}{2} \, {\left (42 \, b^{3} c^{4} d^{6} + 72 \, a b^{2} c^{3} d^{7} + 27 \, a^{2} b c^{2} d^{8} + 2 \, a^{3} c d^{9}\right )} x^{10} + {\left (28 \, b^{3} c^{5} d^{5} + 70 \, a b^{2} c^{4} d^{6} + 40 \, a^{2} b c^{3} d^{7} + 5 \, a^{3} c^{2} d^{8}\right )} x^{9} + \frac {3}{4} \, {\left (35 \, b^{3} c^{6} d^{4} + 126 \, a b^{2} c^{5} d^{5} + 105 \, a^{2} b c^{4} d^{6} + 20 \, a^{3} c^{3} d^{7}\right )} x^{8} + \frac {6}{7} \, {\left (20 \, b^{3} c^{7} d^{3} + 105 \, a b^{2} c^{6} d^{4} + 126 \, a^{2} b c^{5} d^{5} + 35 \, a^{3} c^{4} d^{6}\right )} x^{7} + \frac {3}{2} \, {\left (5 \, b^{3} c^{8} d^{2} + 40 \, a b^{2} c^{7} d^{3} + 70 \, a^{2} b c^{6} d^{4} + 28 \, a^{3} c^{5} d^{5}\right )} x^{6} + {\left (2 \, b^{3} c^{9} d + 27 \, a b^{2} c^{8} d^{2} + 72 \, a^{2} b c^{7} d^{3} + 42 \, a^{3} c^{6} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{10} + 30 \, a b^{2} c^{9} d + 135 \, a^{2} b c^{8} d^{2} + 120 \, a^{3} c^{7} d^{3}\right )} x^{4} + {\left (a b^{2} c^{10} + 10 \, a^{2} b c^{9} d + 15 \, a^{3} c^{8} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{10} + 10 \, a^{3} c^{9} d\right )} x^{2} \] Input:

integrate((b*x+a)^3*(d*x+c)^10,x, algorithm="maxima")
 

Output:

1/14*b^3*d^10*x^14 + a^3*c^10*x + 1/13*(10*b^3*c*d^9 + 3*a*b^2*d^10)*x^13 
+ 1/4*(15*b^3*c^2*d^8 + 10*a*b^2*c*d^9 + a^2*b*d^10)*x^12 + 1/11*(120*b^3* 
c^3*d^7 + 135*a*b^2*c^2*d^8 + 30*a^2*b*c*d^9 + a^3*d^10)*x^11 + 1/2*(42*b^ 
3*c^4*d^6 + 72*a*b^2*c^3*d^7 + 27*a^2*b*c^2*d^8 + 2*a^3*c*d^9)*x^10 + (28* 
b^3*c^5*d^5 + 70*a*b^2*c^4*d^6 + 40*a^2*b*c^3*d^7 + 5*a^3*c^2*d^8)*x^9 + 3 
/4*(35*b^3*c^6*d^4 + 126*a*b^2*c^5*d^5 + 105*a^2*b*c^4*d^6 + 20*a^3*c^3*d^ 
7)*x^8 + 6/7*(20*b^3*c^7*d^3 + 105*a*b^2*c^6*d^4 + 126*a^2*b*c^5*d^5 + 35* 
a^3*c^4*d^6)*x^7 + 3/2*(5*b^3*c^8*d^2 + 40*a*b^2*c^7*d^3 + 70*a^2*b*c^6*d^ 
4 + 28*a^3*c^5*d^5)*x^6 + (2*b^3*c^9*d + 27*a*b^2*c^8*d^2 + 72*a^2*b*c^7*d 
^3 + 42*a^3*c^6*d^4)*x^5 + 1/4*(b^3*c^10 + 30*a*b^2*c^9*d + 135*a^2*b*c^8* 
d^2 + 120*a^3*c^7*d^3)*x^4 + (a*b^2*c^10 + 10*a^2*b*c^9*d + 15*a^3*c^8*d^2 
)*x^3 + 1/2*(3*a^2*b*c^10 + 10*a^3*c^9*d)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (84) = 168\).

Time = 0.12 (sec) , antiderivative size = 594, normalized size of antiderivative = 6.46 \[ \int (a+b x)^3 (c+d x)^{10} \, dx=\frac {1}{14} \, b^{3} d^{10} x^{14} + \frac {10}{13} \, b^{3} c d^{9} x^{13} + \frac {3}{13} \, a b^{2} d^{10} x^{13} + \frac {15}{4} \, b^{3} c^{2} d^{8} x^{12} + \frac {5}{2} \, a b^{2} c d^{9} x^{12} + \frac {1}{4} \, a^{2} b d^{10} x^{12} + \frac {120}{11} \, b^{3} c^{3} d^{7} x^{11} + \frac {135}{11} \, a b^{2} c^{2} d^{8} x^{11} + \frac {30}{11} \, a^{2} b c d^{9} x^{11} + \frac {1}{11} \, a^{3} d^{10} x^{11} + 21 \, b^{3} c^{4} d^{6} x^{10} + 36 \, a b^{2} c^{3} d^{7} x^{10} + \frac {27}{2} \, a^{2} b c^{2} d^{8} x^{10} + a^{3} c d^{9} x^{10} + 28 \, b^{3} c^{5} d^{5} x^{9} + 70 \, a b^{2} c^{4} d^{6} x^{9} + 40 \, a^{2} b c^{3} d^{7} x^{9} + 5 \, a^{3} c^{2} d^{8} x^{9} + \frac {105}{4} \, b^{3} c^{6} d^{4} x^{8} + \frac {189}{2} \, a b^{2} c^{5} d^{5} x^{8} + \frac {315}{4} \, a^{2} b c^{4} d^{6} x^{8} + 15 \, a^{3} c^{3} d^{7} x^{8} + \frac {120}{7} \, b^{3} c^{7} d^{3} x^{7} + 90 \, a b^{2} c^{6} d^{4} x^{7} + 108 \, a^{2} b c^{5} d^{5} x^{7} + 30 \, a^{3} c^{4} d^{6} x^{7} + \frac {15}{2} \, b^{3} c^{8} d^{2} x^{6} + 60 \, a b^{2} c^{7} d^{3} x^{6} + 105 \, a^{2} b c^{6} d^{4} x^{6} + 42 \, a^{3} c^{5} d^{5} x^{6} + 2 \, b^{3} c^{9} d x^{5} + 27 \, a b^{2} c^{8} d^{2} x^{5} + 72 \, a^{2} b c^{7} d^{3} x^{5} + 42 \, a^{3} c^{6} d^{4} x^{5} + \frac {1}{4} \, b^{3} c^{10} x^{4} + \frac {15}{2} \, a b^{2} c^{9} d x^{4} + \frac {135}{4} \, a^{2} b c^{8} d^{2} x^{4} + 30 \, a^{3} c^{7} d^{3} x^{4} + a b^{2} c^{10} x^{3} + 10 \, a^{2} b c^{9} d x^{3} + 15 \, a^{3} c^{8} d^{2} x^{3} + \frac {3}{2} \, a^{2} b c^{10} x^{2} + 5 \, a^{3} c^{9} d x^{2} + a^{3} c^{10} x \] Input:

integrate((b*x+a)^3*(d*x+c)^10,x, algorithm="giac")
 

Output:

1/14*b^3*d^10*x^14 + 10/13*b^3*c*d^9*x^13 + 3/13*a*b^2*d^10*x^13 + 15/4*b^ 
3*c^2*d^8*x^12 + 5/2*a*b^2*c*d^9*x^12 + 1/4*a^2*b*d^10*x^12 + 120/11*b^3*c 
^3*d^7*x^11 + 135/11*a*b^2*c^2*d^8*x^11 + 30/11*a^2*b*c*d^9*x^11 + 1/11*a^ 
3*d^10*x^11 + 21*b^3*c^4*d^6*x^10 + 36*a*b^2*c^3*d^7*x^10 + 27/2*a^2*b*c^2 
*d^8*x^10 + a^3*c*d^9*x^10 + 28*b^3*c^5*d^5*x^9 + 70*a*b^2*c^4*d^6*x^9 + 4 
0*a^2*b*c^3*d^7*x^9 + 5*a^3*c^2*d^8*x^9 + 105/4*b^3*c^6*d^4*x^8 + 189/2*a* 
b^2*c^5*d^5*x^8 + 315/4*a^2*b*c^4*d^6*x^8 + 15*a^3*c^3*d^7*x^8 + 120/7*b^3 
*c^7*d^3*x^7 + 90*a*b^2*c^6*d^4*x^7 + 108*a^2*b*c^5*d^5*x^7 + 30*a^3*c^4*d 
^6*x^7 + 15/2*b^3*c^8*d^2*x^6 + 60*a*b^2*c^7*d^3*x^6 + 105*a^2*b*c^6*d^4*x 
^6 + 42*a^3*c^5*d^5*x^6 + 2*b^3*c^9*d*x^5 + 27*a*b^2*c^8*d^2*x^5 + 72*a^2* 
b*c^7*d^3*x^5 + 42*a^3*c^6*d^4*x^5 + 1/4*b^3*c^10*x^4 + 15/2*a*b^2*c^9*d*x 
^4 + 135/4*a^2*b*c^8*d^2*x^4 + 30*a^3*c^7*d^3*x^4 + a*b^2*c^10*x^3 + 10*a^ 
2*b*c^9*d*x^3 + 15*a^3*c^8*d^2*x^3 + 3/2*a^2*b*c^10*x^2 + 5*a^3*c^9*d*x^2 
+ a^3*c^10*x
 

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 495, normalized size of antiderivative = 5.38 \[ \int (a+b x)^3 (c+d x)^{10} \, dx=x^4\,\left (30\,a^3\,c^7\,d^3+\frac {135\,a^2\,b\,c^8\,d^2}{4}+\frac {15\,a\,b^2\,c^9\,d}{2}+\frac {b^3\,c^{10}}{4}\right )+x^{11}\,\left (\frac {a^3\,d^{10}}{11}+\frac {30\,a^2\,b\,c\,d^9}{11}+\frac {135\,a\,b^2\,c^2\,d^8}{11}+\frac {120\,b^3\,c^3\,d^7}{11}\right )+a^3\,c^{10}\,x+\frac {b^3\,d^{10}\,x^{14}}{14}+\frac {3\,c^5\,d^2\,x^6\,\left (28\,a^3\,d^3+70\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}{2}+c^2\,d^5\,x^9\,\left (5\,a^3\,d^3+40\,a^2\,b\,c\,d^2+70\,a\,b^2\,c^2\,d+28\,b^3\,c^3\right )+\frac {6\,c^4\,d^3\,x^7\,\left (35\,a^3\,d^3+126\,a^2\,b\,c\,d^2+105\,a\,b^2\,c^2\,d+20\,b^3\,c^3\right )}{7}+\frac {3\,c^3\,d^4\,x^8\,\left (20\,a^3\,d^3+105\,a^2\,b\,c\,d^2+126\,a\,b^2\,c^2\,d+35\,b^3\,c^3\right )}{4}+\frac {a^2\,c^9\,x^2\,\left (10\,a\,d+3\,b\,c\right )}{2}+\frac {b^2\,d^9\,x^{13}\,\left (3\,a\,d+10\,b\,c\right )}{13}+a\,c^8\,x^3\,\left (15\,a^2\,d^2+10\,a\,b\,c\,d+b^2\,c^2\right )+\frac {b\,d^8\,x^{12}\,\left (a^2\,d^2+10\,a\,b\,c\,d+15\,b^2\,c^2\right )}{4}+c^6\,d\,x^5\,\left (42\,a^3\,d^3+72\,a^2\,b\,c\,d^2+27\,a\,b^2\,c^2\,d+2\,b^3\,c^3\right )+\frac {c\,d^6\,x^{10}\,\left (2\,a^3\,d^3+27\,a^2\,b\,c\,d^2+72\,a\,b^2\,c^2\,d+42\,b^3\,c^3\right )}{2} \] Input:

int((a + b*x)^3*(c + d*x)^10,x)
 

Output:

x^4*((b^3*c^10)/4 + 30*a^3*c^7*d^3 + (135*a^2*b*c^8*d^2)/4 + (15*a*b^2*c^9 
*d)/2) + x^11*((a^3*d^10)/11 + (120*b^3*c^3*d^7)/11 + (135*a*b^2*c^2*d^8)/ 
11 + (30*a^2*b*c*d^9)/11) + a^3*c^10*x + (b^3*d^10*x^14)/14 + (3*c^5*d^2*x 
^6*(28*a^3*d^3 + 5*b^3*c^3 + 40*a*b^2*c^2*d + 70*a^2*b*c*d^2))/2 + c^2*d^5 
*x^9*(5*a^3*d^3 + 28*b^3*c^3 + 70*a*b^2*c^2*d + 40*a^2*b*c*d^2) + (6*c^4*d 
^3*x^7*(35*a^3*d^3 + 20*b^3*c^3 + 105*a*b^2*c^2*d + 126*a^2*b*c*d^2))/7 + 
(3*c^3*d^4*x^8*(20*a^3*d^3 + 35*b^3*c^3 + 126*a*b^2*c^2*d + 105*a^2*b*c*d^ 
2))/4 + (a^2*c^9*x^2*(10*a*d + 3*b*c))/2 + (b^2*d^9*x^13*(3*a*d + 10*b*c)) 
/13 + a*c^8*x^3*(15*a^2*d^2 + b^2*c^2 + 10*a*b*c*d) + (b*d^8*x^12*(a^2*d^2 
 + 15*b^2*c^2 + 10*a*b*c*d))/4 + c^6*d*x^5*(42*a^3*d^3 + 2*b^3*c^3 + 27*a* 
b^2*c^2*d + 72*a^2*b*c*d^2) + (c*d^6*x^10*(2*a^3*d^3 + 42*b^3*c^3 + 72*a*b 
^2*c^2*d + 27*a^2*b*c*d^2))/2
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 595, normalized size of antiderivative = 6.47 \[ \int (a+b x)^3 (c+d x)^{10} \, dx=\frac {x \left (286 b^{3} d^{10} x^{13}+924 a \,b^{2} d^{10} x^{12}+3080 b^{3} c \,d^{9} x^{12}+1001 a^{2} b \,d^{10} x^{11}+10010 a \,b^{2} c \,d^{9} x^{11}+15015 b^{3} c^{2} d^{8} x^{11}+364 a^{3} d^{10} x^{10}+10920 a^{2} b c \,d^{9} x^{10}+49140 a \,b^{2} c^{2} d^{8} x^{10}+43680 b^{3} c^{3} d^{7} x^{10}+4004 a^{3} c \,d^{9} x^{9}+54054 a^{2} b \,c^{2} d^{8} x^{9}+144144 a \,b^{2} c^{3} d^{7} x^{9}+84084 b^{3} c^{4} d^{6} x^{9}+20020 a^{3} c^{2} d^{8} x^{8}+160160 a^{2} b \,c^{3} d^{7} x^{8}+280280 a \,b^{2} c^{4} d^{6} x^{8}+112112 b^{3} c^{5} d^{5} x^{8}+60060 a^{3} c^{3} d^{7} x^{7}+315315 a^{2} b \,c^{4} d^{6} x^{7}+378378 a \,b^{2} c^{5} d^{5} x^{7}+105105 b^{3} c^{6} d^{4} x^{7}+120120 a^{3} c^{4} d^{6} x^{6}+432432 a^{2} b \,c^{5} d^{5} x^{6}+360360 a \,b^{2} c^{6} d^{4} x^{6}+68640 b^{3} c^{7} d^{3} x^{6}+168168 a^{3} c^{5} d^{5} x^{5}+420420 a^{2} b \,c^{6} d^{4} x^{5}+240240 a \,b^{2} c^{7} d^{3} x^{5}+30030 b^{3} c^{8} d^{2} x^{5}+168168 a^{3} c^{6} d^{4} x^{4}+288288 a^{2} b \,c^{7} d^{3} x^{4}+108108 a \,b^{2} c^{8} d^{2} x^{4}+8008 b^{3} c^{9} d \,x^{4}+120120 a^{3} c^{7} d^{3} x^{3}+135135 a^{2} b \,c^{8} d^{2} x^{3}+30030 a \,b^{2} c^{9} d \,x^{3}+1001 b^{3} c^{10} x^{3}+60060 a^{3} c^{8} d^{2} x^{2}+40040 a^{2} b \,c^{9} d \,x^{2}+4004 a \,b^{2} c^{10} x^{2}+20020 a^{3} c^{9} d x +6006 a^{2} b \,c^{10} x +4004 a^{3} c^{10}\right )}{4004} \] Input:

int((b*x+a)^3*(d*x+c)^10,x)
 

Output:

(x*(4004*a**3*c**10 + 20020*a**3*c**9*d*x + 60060*a**3*c**8*d**2*x**2 + 12 
0120*a**3*c**7*d**3*x**3 + 168168*a**3*c**6*d**4*x**4 + 168168*a**3*c**5*d 
**5*x**5 + 120120*a**3*c**4*d**6*x**6 + 60060*a**3*c**3*d**7*x**7 + 20020* 
a**3*c**2*d**8*x**8 + 4004*a**3*c*d**9*x**9 + 364*a**3*d**10*x**10 + 6006* 
a**2*b*c**10*x + 40040*a**2*b*c**9*d*x**2 + 135135*a**2*b*c**8*d**2*x**3 + 
 288288*a**2*b*c**7*d**3*x**4 + 420420*a**2*b*c**6*d**4*x**5 + 432432*a**2 
*b*c**5*d**5*x**6 + 315315*a**2*b*c**4*d**6*x**7 + 160160*a**2*b*c**3*d**7 
*x**8 + 54054*a**2*b*c**2*d**8*x**9 + 10920*a**2*b*c*d**9*x**10 + 1001*a** 
2*b*d**10*x**11 + 4004*a*b**2*c**10*x**2 + 30030*a*b**2*c**9*d*x**3 + 1081 
08*a*b**2*c**8*d**2*x**4 + 240240*a*b**2*c**7*d**3*x**5 + 360360*a*b**2*c* 
*6*d**4*x**6 + 378378*a*b**2*c**5*d**5*x**7 + 280280*a*b**2*c**4*d**6*x**8 
 + 144144*a*b**2*c**3*d**7*x**9 + 49140*a*b**2*c**2*d**8*x**10 + 10010*a*b 
**2*c*d**9*x**11 + 924*a*b**2*d**10*x**12 + 1001*b**3*c**10*x**3 + 8008*b* 
*3*c**9*d*x**4 + 30030*b**3*c**8*d**2*x**5 + 68640*b**3*c**7*d**3*x**6 + 1 
05105*b**3*c**6*d**4*x**7 + 112112*b**3*c**5*d**5*x**8 + 84084*b**3*c**4*d 
**6*x**9 + 43680*b**3*c**3*d**7*x**10 + 15015*b**3*c**2*d**8*x**11 + 3080* 
b**3*c*d**9*x**12 + 286*b**3*d**10*x**13))/4004