\(\int (a+b x)^4 (c+d x)^{10} \, dx\) [101]

Optimal result

Integrand size = 15, antiderivative size = 119 \[ \int (a+b x)^4 (c+d x)^{10} \, dx=\frac {(b c-a d)^4 (c+d x)^{11}}{11 d^5}-\frac {b (b c-a d)^3 (c+d x)^{12}}{3 d^5}+\frac {6 b^2 (b c-a d)^2 (c+d x)^{13}}{13 d^5}-\frac {2 b^3 (b c-a d) (c+d x)^{14}}{7 d^5}+\frac {b^4 (c+d x)^{15}}{15 d^5} \] Output:

1/11*(-a*d+b*c)^4*(d*x+c)^11/d^5-1/3*b*(-a*d+b*c)^3*(d*x+c)^12/d^5+6/13*b^ 
2*(-a*d+b*c)^2*(d*x+c)^13/d^5-2/7*b^3*(-a*d+b*c)*(d*x+c)^14/d^5+1/15*b^4*( 
d*x+c)^15/d^5
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(660\) vs. \(2(119)=238\).

Time = 0.04 (sec) , antiderivative size = 660, normalized size of antiderivative = 5.55 \[ \int (a+b x)^4 (c+d x)^{10} \, dx=a^4 c^{10} x+a^3 c^9 (2 b c+5 a d) x^2+\frac {1}{3} a^2 c^8 \left (6 b^2 c^2+40 a b c d+45 a^2 d^2\right ) x^3+a c^7 \left (b^3 c^3+15 a b^2 c^2 d+45 a^2 b c d^2+30 a^3 d^3\right ) x^4+\frac {1}{5} c^6 \left (b^4 c^4+40 a b^3 c^3 d+270 a^2 b^2 c^2 d^2+480 a^3 b c d^3+210 a^4 d^4\right ) x^5+\frac {1}{3} c^5 d \left (5 b^4 c^4+90 a b^3 c^3 d+360 a^2 b^2 c^2 d^2+420 a^3 b c d^3+126 a^4 d^4\right ) x^6+\frac {3}{7} c^4 d^2 \left (15 b^4 c^4+160 a b^3 c^3 d+420 a^2 b^2 c^2 d^2+336 a^3 b c d^3+70 a^4 d^4\right ) x^7+3 c^3 d^3 \left (5 b^4 c^4+35 a b^3 c^3 d+63 a^2 b^2 c^2 d^2+35 a^3 b c d^3+5 a^4 d^4\right ) x^8+\frac {1}{3} c^2 d^4 \left (70 b^4 c^4+336 a b^3 c^3 d+420 a^2 b^2 c^2 d^2+160 a^3 b c d^3+15 a^4 d^4\right ) x^9+\frac {1}{5} c d^5 \left (126 b^4 c^4+420 a b^3 c^3 d+360 a^2 b^2 c^2 d^2+90 a^3 b c d^3+5 a^4 d^4\right ) x^{10}+\frac {1}{11} d^6 \left (210 b^4 c^4+480 a b^3 c^3 d+270 a^2 b^2 c^2 d^2+40 a^3 b c d^3+a^4 d^4\right ) x^{11}+\frac {1}{3} b d^7 \left (30 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2+a^3 d^3\right ) x^{12}+\frac {1}{13} b^2 d^8 \left (45 b^2 c^2+40 a b c d+6 a^2 d^2\right ) x^{13}+\frac {1}{7} b^3 d^9 (5 b c+2 a d) x^{14}+\frac {1}{15} b^4 d^{10} x^{15} \] Input:

Integrate[(a + b*x)^4*(c + d*x)^10,x]
 

Output:

a^4*c^10*x + a^3*c^9*(2*b*c + 5*a*d)*x^2 + (a^2*c^8*(6*b^2*c^2 + 40*a*b*c* 
d + 45*a^2*d^2)*x^3)/3 + a*c^7*(b^3*c^3 + 15*a*b^2*c^2*d + 45*a^2*b*c*d^2 
+ 30*a^3*d^3)*x^4 + (c^6*(b^4*c^4 + 40*a*b^3*c^3*d + 270*a^2*b^2*c^2*d^2 + 
 480*a^3*b*c*d^3 + 210*a^4*d^4)*x^5)/5 + (c^5*d*(5*b^4*c^4 + 90*a*b^3*c^3* 
d + 360*a^2*b^2*c^2*d^2 + 420*a^3*b*c*d^3 + 126*a^4*d^4)*x^6)/3 + (3*c^4*d 
^2*(15*b^4*c^4 + 160*a*b^3*c^3*d + 420*a^2*b^2*c^2*d^2 + 336*a^3*b*c*d^3 + 
 70*a^4*d^4)*x^7)/7 + 3*c^3*d^3*(5*b^4*c^4 + 35*a*b^3*c^3*d + 63*a^2*b^2*c 
^2*d^2 + 35*a^3*b*c*d^3 + 5*a^4*d^4)*x^8 + (c^2*d^4*(70*b^4*c^4 + 336*a*b^ 
3*c^3*d + 420*a^2*b^2*c^2*d^2 + 160*a^3*b*c*d^3 + 15*a^4*d^4)*x^9)/3 + (c* 
d^5*(126*b^4*c^4 + 420*a*b^3*c^3*d + 360*a^2*b^2*c^2*d^2 + 90*a^3*b*c*d^3 
+ 5*a^4*d^4)*x^10)/5 + (d^6*(210*b^4*c^4 + 480*a*b^3*c^3*d + 270*a^2*b^2*c 
^2*d^2 + 40*a^3*b*c*d^3 + a^4*d^4)*x^11)/11 + (b*d^7*(30*b^3*c^3 + 45*a*b^ 
2*c^2*d + 15*a^2*b*c*d^2 + a^3*d^3)*x^12)/3 + (b^2*d^8*(45*b^2*c^2 + 40*a* 
b*c*d + 6*a^2*d^2)*x^13)/13 + (b^3*d^9*(5*b*c + 2*a*d)*x^14)/7 + (b^4*d^10 
*x^15)/15
 

Rubi [A] (verified)

Time = 0.52 (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.

\(\displaystyle \int (a+b x)^4 (c+d x)^{10} \, dx\)

\(\Big \downarrow \) 49

\(\displaystyle \int \left (-\frac {4 b^3 (c+d x)^{13} (b c-a d)}{d^4}+\frac {6 b^2 (c+d x)^{12} (b c-a d)^2}{d^4}-\frac {4 b (c+d x)^{11} (b c-a d)^3}{d^4}+\frac {(c+d x)^{10} (a d-b c)^4}{d^4}+\frac {b^4 (c+d x)^{14}}{d^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 b^3 (c+d x)^{14} (b c-a d)}{7 d^5}+\frac {6 b^2 (c+d x)^{13} (b c-a d)^2}{13 d^5}-\frac {b (c+d x)^{12} (b c-a d)^3}{3 d^5}+\frac {(c+d x)^{11} (b c-a d)^4}{11 d^5}+\frac {b^4 (c+d x)^{15}}{15 d^5}\)

Input:

Int[(a + b*x)^4*(c + d*x)^10,x]
 

Output:

((b*c - a*d)^4*(c + d*x)^11)/(11*d^5) - (b*(b*c - a*d)^3*(c + d*x)^12)/(3* 
d^5) + (6*b^2*(b*c - a*d)^2*(c + d*x)^13)/(13*d^5) - (2*b^3*(b*c - a*d)*(c 
 + d*x)^14)/(7*d^5) + (b^4*(c + d*x)^15)/(15*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. \(677\) vs. \(2(109)=218\).

Time = 0.10 (sec) , antiderivative size = 678, normalized size of antiderivative = 5.70

method	result	size
norman	\(a^{4} c^{10} x +\left (5 a^{4} c^{9} d +2 a^{3} b \,c^{10}\right ) x^{2}+\left (15 a^{4} c^{8} d^{2}+\frac {40}{3} a^{3} b \,c^{9} d +2 a^{2} b^{2} c^{10}\right ) x^{3}+\left (30 a^{4} c^{7} d^{3}+45 a^{3} b \,c^{8} d^{2}+15 a^{2} b^{2} c^{9} d +a \,b^{3} c^{10}\right ) x^{4}+\left (42 a^{4} c^{6} d^{4}+96 a^{3} b \,c^{7} d^{3}+54 a^{2} b^{2} c^{8} d^{2}+8 a \,b^{3} c^{9} d +\frac {1}{5} b^{4} c^{10}\right ) x^{5}+\left (42 a^{4} c^{5} d^{5}+140 a^{3} b \,c^{6} d^{4}+120 a^{2} b^{2} c^{7} d^{3}+30 a \,b^{3} c^{8} d^{2}+\frac {5}{3} b^{4} c^{9} d \right ) x^{6}+\left (30 a^{4} c^{4} d^{6}+144 a^{3} b \,c^{5} d^{5}+180 a^{2} b^{2} c^{6} d^{4}+\frac {480}{7} a \,b^{3} c^{7} d^{3}+\frac {45}{7} b^{4} c^{8} d^{2}\right ) x^{7}+\left (15 a^{4} c^{3} d^{7}+105 a^{3} b \,c^{4} d^{6}+189 a^{2} b^{2} c^{5} d^{5}+105 a \,b^{3} c^{6} d^{4}+15 b^{4} c^{7} d^{3}\right ) x^{8}+\left (5 a^{4} c^{2} d^{8}+\frac {160}{3} a^{3} b \,c^{3} d^{7}+140 a^{2} b^{2} c^{4} d^{6}+112 a \,b^{3} c^{5} d^{5}+\frac {70}{3} b^{4} c^{6} d^{4}\right ) x^{9}+\left (a^{4} c \,d^{9}+18 a^{3} b \,c^{2} d^{8}+72 a^{2} b^{2} c^{3} d^{7}+84 a \,b^{3} c^{4} d^{6}+\frac {126}{5} b^{4} c^{5} d^{5}\right ) x^{10}+\left (\frac {1}{11} a^{4} d^{10}+\frac {40}{11} a^{3} b c \,d^{9}+\frac {270}{11} a^{2} b^{2} c^{2} d^{8}+\frac {480}{11} a \,b^{3} c^{3} d^{7}+\frac {210}{11} b^{4} c^{4} d^{6}\right ) x^{11}+\left (\frac {1}{3} a^{3} b \,d^{10}+5 a^{2} b^{2} c \,d^{9}+15 a \,b^{3} c^{2} d^{8}+10 b^{4} c^{3} d^{7}\right ) x^{12}+\left (\frac {6}{13} a^{2} b^{2} d^{10}+\frac {40}{13} a \,b^{3} c \,d^{9}+\frac {45}{13} b^{4} c^{2} d^{8}\right ) x^{13}+\left (\frac {2}{7} a \,b^{3} d^{10}+\frac {5}{7} b^{4} c \,d^{9}\right ) x^{14}+\frac {b^{4} d^{10} x^{15}}{15}\)	\(678\)
default	\(\frac {b^{4} d^{10} x^{15}}{15}+\frac {\left (4 a \,b^{3} d^{10}+10 b^{4} c \,d^{9}\right ) x^{14}}{14}+\frac {\left (6 a^{2} b^{2} d^{10}+40 a \,b^{3} c \,d^{9}+45 b^{4} c^{2} d^{8}\right ) x^{13}}{13}+\frac {\left (4 a^{3} b \,d^{10}+60 a^{2} b^{2} c \,d^{9}+180 a \,b^{3} c^{2} d^{8}+120 b^{4} c^{3} d^{7}\right ) x^{12}}{12}+\frac {\left (a^{4} d^{10}+40 a^{3} b c \,d^{9}+270 a^{2} b^{2} c^{2} d^{8}+480 a \,b^{3} c^{3} d^{7}+210 b^{4} c^{4} d^{6}\right ) x^{11}}{11}+\frac {\left (10 a^{4} c \,d^{9}+180 a^{3} b \,c^{2} d^{8}+720 a^{2} b^{2} c^{3} d^{7}+840 a \,b^{3} c^{4} d^{6}+252 b^{4} c^{5} d^{5}\right ) x^{10}}{10}+\frac {\left (45 a^{4} c^{2} d^{8}+480 a^{3} b \,c^{3} d^{7}+1260 a^{2} b^{2} c^{4} d^{6}+1008 a \,b^{3} c^{5} d^{5}+210 b^{4} c^{6} d^{4}\right ) x^{9}}{9}+\frac {\left (120 a^{4} c^{3} d^{7}+840 a^{3} b \,c^{4} d^{6}+1512 a^{2} b^{2} c^{5} d^{5}+840 a \,b^{3} c^{6} d^{4}+120 b^{4} c^{7} d^{3}\right ) x^{8}}{8}+\frac {\left (210 a^{4} c^{4} d^{6}+1008 a^{3} b \,c^{5} d^{5}+1260 a^{2} b^{2} c^{6} d^{4}+480 a \,b^{3} c^{7} d^{3}+45 b^{4} c^{8} d^{2}\right ) x^{7}}{7}+\frac {\left (252 a^{4} c^{5} d^{5}+840 a^{3} b \,c^{6} d^{4}+720 a^{2} b^{2} c^{7} d^{3}+180 a \,b^{3} c^{8} d^{2}+10 b^{4} c^{9} d \right ) x^{6}}{6}+\frac {\left (210 a^{4} c^{6} d^{4}+480 a^{3} b \,c^{7} d^{3}+270 a^{2} b^{2} c^{8} d^{2}+40 a \,b^{3} c^{9} d +b^{4} c^{10}\right ) x^{5}}{5}+\frac {\left (120 a^{4} c^{7} d^{3}+180 a^{3} b \,c^{8} d^{2}+60 a^{2} b^{2} c^{9} d +4 a \,b^{3} c^{10}\right ) x^{4}}{4}+\frac {\left (45 a^{4} c^{8} d^{2}+40 a^{3} b \,c^{9} d +6 a^{2} b^{2} c^{10}\right ) x^{3}}{3}+\frac {\left (10 a^{4} c^{9} d +4 a^{3} b \,c^{10}\right ) x^{2}}{2}+a^{4} c^{10} x\)	\(691\)
gosper	\(\frac {40}{11} x^{11} a^{3} b c \,d^{9}+\frac {270}{11} x^{11} a^{2} b^{2} c^{2} d^{8}+\frac {480}{11} x^{11} a \,b^{3} c^{3} d^{7}+5 x^{12} a^{2} b^{2} c \,d^{9}+\frac {160}{3} x^{9} a^{3} b \,c^{3} d^{7}+140 x^{9} a^{2} b^{2} c^{4} d^{6}+112 x^{9} a \,b^{3} c^{5} d^{5}+18 x^{10} a^{3} b \,c^{2} d^{8}+140 x^{6} a^{3} b \,c^{6} d^{4}+120 x^{6} a^{2} b^{2} c^{7} d^{3}+30 x^{6} a \,b^{3} c^{8} d^{2}+144 x^{7} a^{3} b \,c^{5} d^{5}+180 x^{7} a^{2} b^{2} c^{6} d^{4}+\frac {480}{7} x^{7} a \,b^{3} c^{7} d^{3}+\frac {40}{3} x^{3} a^{3} b \,c^{9} d +96 x^{5} a^{3} b \,c^{7} d^{3}+54 x^{5} a^{2} b^{2} c^{8} d^{2}+8 x^{5} a \,b^{3} c^{9} d +15 x^{12} a \,b^{3} c^{2} d^{8}+\frac {40}{13} x^{13} a \,b^{3} c \,d^{9}+45 a^{3} b \,c^{8} d^{2} x^{4}+15 a^{2} b^{2} c^{9} d \,x^{4}+105 a^{3} b \,c^{4} d^{6} x^{8}+189 a^{2} b^{2} c^{5} d^{5} x^{8}+105 a \,b^{3} c^{6} d^{4} x^{8}+42 x^{5} a^{4} c^{6} d^{4}+42 x^{6} a^{4} c^{5} d^{5}+84 x^{10} a \,b^{3} c^{4} d^{6}+\frac {5}{3} x^{6} b^{4} c^{9} d +30 x^{7} a^{4} c^{4} d^{6}+\frac {45}{7} x^{7} b^{4} c^{8} d^{2}+5 x^{9} a^{4} c^{2} d^{8}+\frac {70}{3} x^{9} b^{4} c^{6} d^{4}+x^{10} a^{4} c \,d^{9}+\frac {126}{5} x^{10} b^{4} c^{5} d^{5}+\frac {210}{11} x^{11} b^{4} c^{4} d^{6}+\frac {1}{3} x^{12} a^{3} b \,d^{10}+10 x^{12} b^{4} c^{3} d^{7}+\frac {6}{13} x^{13} a^{2} b^{2} d^{10}+\frac {45}{13} x^{13} b^{4} c^{2} d^{8}+\frac {2}{7} x^{14} a \,b^{3} d^{10}+\frac {5}{7} x^{14} b^{4} c \,d^{9}+5 a^{4} c^{9} d \,x^{2}+2 a^{3} b \,c^{10} x^{2}+30 a^{4} c^{7} d^{3} x^{4}+a \,b^{3} c^{10} x^{4}+15 a^{4} c^{3} d^{7} x^{8}+15 b^{4} c^{7} d^{3} x^{8}+15 x^{3} a^{4} c^{8} d^{2}+2 x^{3} a^{2} b^{2} c^{10}+\frac {1}{15} b^{4} d^{10} x^{15}+\frac {1}{5} x^{5} b^{4} c^{10}+\frac {1}{11} x^{11} a^{4} d^{10}+a^{4} c^{10} x +72 x^{10} a^{2} b^{2} c^{3} d^{7}\)	\(772\)
risch	\(\frac {40}{11} x^{11} a^{3} b c \,d^{9}+\frac {270}{11} x^{11} a^{2} b^{2} c^{2} d^{8}+\frac {480}{11} x^{11} a \,b^{3} c^{3} d^{7}+5 x^{12} a^{2} b^{2} c \,d^{9}+\frac {160}{3} x^{9} a^{3} b \,c^{3} d^{7}+140 x^{9} a^{2} b^{2} c^{4} d^{6}+112 x^{9} a \,b^{3} c^{5} d^{5}+18 x^{10} a^{3} b \,c^{2} d^{8}+140 x^{6} a^{3} b \,c^{6} d^{4}+120 x^{6} a^{2} b^{2} c^{7} d^{3}+30 x^{6} a \,b^{3} c^{8} d^{2}+144 x^{7} a^{3} b \,c^{5} d^{5}+180 x^{7} a^{2} b^{2} c^{6} d^{4}+\frac {480}{7} x^{7} a \,b^{3} c^{7} d^{3}+\frac {40}{3} x^{3} a^{3} b \,c^{9} d +96 x^{5} a^{3} b \,c^{7} d^{3}+54 x^{5} a^{2} b^{2} c^{8} d^{2}+8 x^{5} a \,b^{3} c^{9} d +15 x^{12} a \,b^{3} c^{2} d^{8}+\frac {40}{13} x^{13} a \,b^{3} c \,d^{9}+45 a^{3} b \,c^{8} d^{2} x^{4}+15 a^{2} b^{2} c^{9} d \,x^{4}+105 a^{3} b \,c^{4} d^{6} x^{8}+189 a^{2} b^{2} c^{5} d^{5} x^{8}+105 a \,b^{3} c^{6} d^{4} x^{8}+42 x^{5} a^{4} c^{6} d^{4}+42 x^{6} a^{4} c^{5} d^{5}+84 x^{10} a \,b^{3} c^{4} d^{6}+\frac {5}{3} x^{6} b^{4} c^{9} d +30 x^{7} a^{4} c^{4} d^{6}+\frac {45}{7} x^{7} b^{4} c^{8} d^{2}+5 x^{9} a^{4} c^{2} d^{8}+\frac {70}{3} x^{9} b^{4} c^{6} d^{4}+x^{10} a^{4} c \,d^{9}+\frac {126}{5} x^{10} b^{4} c^{5} d^{5}+\frac {210}{11} x^{11} b^{4} c^{4} d^{6}+\frac {1}{3} x^{12} a^{3} b \,d^{10}+10 x^{12} b^{4} c^{3} d^{7}+\frac {6}{13} x^{13} a^{2} b^{2} d^{10}+\frac {45}{13} x^{13} b^{4} c^{2} d^{8}+\frac {2}{7} x^{14} a \,b^{3} d^{10}+\frac {5}{7} x^{14} b^{4} c \,d^{9}+5 a^{4} c^{9} d \,x^{2}+2 a^{3} b \,c^{10} x^{2}+30 a^{4} c^{7} d^{3} x^{4}+a \,b^{3} c^{10} x^{4}+15 a^{4} c^{3} d^{7} x^{8}+15 b^{4} c^{7} d^{3} x^{8}+15 x^{3} a^{4} c^{8} d^{2}+2 x^{3} a^{2} b^{2} c^{10}+\frac {1}{15} b^{4} d^{10} x^{15}+\frac {1}{5} x^{5} b^{4} c^{10}+\frac {1}{11} x^{11} a^{4} d^{10}+a^{4} c^{10} x +72 x^{10} a^{2} b^{2} c^{3} d^{7}\)	\(772\)
parallelrisch	\(\frac {40}{11} x^{11} a^{3} b c \,d^{9}+\frac {270}{11} x^{11} a^{2} b^{2} c^{2} d^{8}+\frac {480}{11} x^{11} a \,b^{3} c^{3} d^{7}+5 x^{12} a^{2} b^{2} c \,d^{9}+\frac {160}{3} x^{9} a^{3} b \,c^{3} d^{7}+140 x^{9} a^{2} b^{2} c^{4} d^{6}+112 x^{9} a \,b^{3} c^{5} d^{5}+18 x^{10} a^{3} b \,c^{2} d^{8}+140 x^{6} a^{3} b \,c^{6} d^{4}+120 x^{6} a^{2} b^{2} c^{7} d^{3}+30 x^{6} a \,b^{3} c^{8} d^{2}+144 x^{7} a^{3} b \,c^{5} d^{5}+180 x^{7} a^{2} b^{2} c^{6} d^{4}+\frac {480}{7} x^{7} a \,b^{3} c^{7} d^{3}+\frac {40}{3} x^{3} a^{3} b \,c^{9} d +96 x^{5} a^{3} b \,c^{7} d^{3}+54 x^{5} a^{2} b^{2} c^{8} d^{2}+8 x^{5} a \,b^{3} c^{9} d +15 x^{12} a \,b^{3} c^{2} d^{8}+\frac {40}{13} x^{13} a \,b^{3} c \,d^{9}+45 a^{3} b \,c^{8} d^{2} x^{4}+15 a^{2} b^{2} c^{9} d \,x^{4}+105 a^{3} b \,c^{4} d^{6} x^{8}+189 a^{2} b^{2} c^{5} d^{5} x^{8}+105 a \,b^{3} c^{6} d^{4} x^{8}+42 x^{5} a^{4} c^{6} d^{4}+42 x^{6} a^{4} c^{5} d^{5}+84 x^{10} a \,b^{3} c^{4} d^{6}+\frac {5}{3} x^{6} b^{4} c^{9} d +30 x^{7} a^{4} c^{4} d^{6}+\frac {45}{7} x^{7} b^{4} c^{8} d^{2}+5 x^{9} a^{4} c^{2} d^{8}+\frac {70}{3} x^{9} b^{4} c^{6} d^{4}+x^{10} a^{4} c \,d^{9}+\frac {126}{5} x^{10} b^{4} c^{5} d^{5}+\frac {210}{11} x^{11} b^{4} c^{4} d^{6}+\frac {1}{3} x^{12} a^{3} b \,d^{10}+10 x^{12} b^{4} c^{3} d^{7}+\frac {6}{13} x^{13} a^{2} b^{2} d^{10}+\frac {45}{13} x^{13} b^{4} c^{2} d^{8}+\frac {2}{7} x^{14} a \,b^{3} d^{10}+\frac {5}{7} x^{14} b^{4} c \,d^{9}+5 a^{4} c^{9} d \,x^{2}+2 a^{3} b \,c^{10} x^{2}+30 a^{4} c^{7} d^{3} x^{4}+a \,b^{3} c^{10} x^{4}+15 a^{4} c^{3} d^{7} x^{8}+15 b^{4} c^{7} d^{3} x^{8}+15 x^{3} a^{4} c^{8} d^{2}+2 x^{3} a^{2} b^{2} c^{10}+\frac {1}{15} b^{4} d^{10} x^{15}+\frac {1}{5} x^{5} b^{4} c^{10}+\frac {1}{11} x^{11} a^{4} d^{10}+a^{4} c^{10} x +72 x^{10} a^{2} b^{2} c^{3} d^{7}\)	\(772\)
orering	\(\frac {x \left (1001 b^{4} d^{10} x^{14}+4290 a \,b^{3} d^{10} x^{13}+10725 b^{4} c \,d^{9} x^{13}+6930 a^{2} b^{2} d^{10} x^{12}+46200 a \,b^{3} c \,d^{9} x^{12}+51975 b^{4} c^{2} d^{8} x^{12}+5005 a^{3} b \,d^{10} x^{11}+75075 a^{2} b^{2} c \,d^{9} x^{11}+225225 a \,b^{3} c^{2} d^{8} x^{11}+150150 b^{4} c^{3} d^{7} x^{11}+1365 a^{4} d^{10} x^{10}+54600 a^{3} b c \,d^{9} x^{10}+368550 a^{2} b^{2} c^{2} d^{8} x^{10}+655200 a \,b^{3} c^{3} d^{7} x^{10}+286650 b^{4} c^{4} d^{6} x^{10}+15015 a^{4} c \,d^{9} x^{9}+270270 a^{3} b \,c^{2} d^{8} x^{9}+1081080 a^{2} b^{2} c^{3} d^{7} x^{9}+1261260 a \,b^{3} c^{4} d^{6} x^{9}+378378 b^{4} c^{5} d^{5} x^{9}+75075 a^{4} c^{2} d^{8} x^{8}+800800 a^{3} b \,c^{3} d^{7} x^{8}+2102100 a^{2} b^{2} c^{4} d^{6} x^{8}+1681680 a \,b^{3} c^{5} d^{5} x^{8}+350350 b^{4} c^{6} d^{4} x^{8}+225225 a^{4} c^{3} d^{7} x^{7}+1576575 a^{3} b \,c^{4} d^{6} x^{7}+2837835 a^{2} b^{2} c^{5} d^{5} x^{7}+1576575 a \,b^{3} c^{6} d^{4} x^{7}+225225 b^{4} c^{7} d^{3} x^{7}+450450 a^{4} c^{4} d^{6} x^{6}+2162160 a^{3} b \,c^{5} d^{5} x^{6}+2702700 a^{2} b^{2} c^{6} d^{4} x^{6}+1029600 a \,b^{3} c^{7} d^{3} x^{6}+96525 b^{4} c^{8} d^{2} x^{6}+630630 a^{4} c^{5} d^{5} x^{5}+2102100 a^{3} b \,c^{6} d^{4} x^{5}+1801800 a^{2} b^{2} c^{7} d^{3} x^{5}+450450 a \,b^{3} c^{8} d^{2} x^{5}+25025 b^{4} c^{9} d \,x^{5}+630630 a^{4} c^{6} d^{4} x^{4}+1441440 a^{3} b \,c^{7} d^{3} x^{4}+810810 a^{2} b^{2} c^{8} d^{2} x^{4}+120120 a \,b^{3} c^{9} d \,x^{4}+3003 b^{4} c^{10} x^{4}+450450 a^{4} c^{7} d^{3} x^{3}+675675 a^{3} b \,c^{8} d^{2} x^{3}+225225 a^{2} b^{2} c^{9} d \,x^{3}+15015 a \,b^{3} c^{10} x^{3}+225225 a^{4} c^{8} d^{2} x^{2}+200200 a^{3} b \,c^{9} d \,x^{2}+30030 a^{2} b^{2} c^{10} x^{2}+75075 a^{4} c^{9} d x +30030 a^{3} b \,c^{10} x +15015 a^{4} c^{10}\right )}{15015}\)	\(773\)

Input:

int((b*x+a)^4*(d*x+c)^10,x,method=_RETURNVERBOSE)
 

Output:

a^4*c^10*x+(5*a^4*c^9*d+2*a^3*b*c^10)*x^2+(15*a^4*c^8*d^2+40/3*a^3*b*c^9*d 
+2*a^2*b^2*c^10)*x^3+(30*a^4*c^7*d^3+45*a^3*b*c^8*d^2+15*a^2*b^2*c^9*d+a*b 
^3*c^10)*x^4+(42*a^4*c^6*d^4+96*a^3*b*c^7*d^3+54*a^2*b^2*c^8*d^2+8*a*b^3*c 
^9*d+1/5*b^4*c^10)*x^5+(42*a^4*c^5*d^5+140*a^3*b*c^6*d^4+120*a^2*b^2*c^7*d 
^3+30*a*b^3*c^8*d^2+5/3*b^4*c^9*d)*x^6+(30*a^4*c^4*d^6+144*a^3*b*c^5*d^5+1 
80*a^2*b^2*c^6*d^4+480/7*a*b^3*c^7*d^3+45/7*b^4*c^8*d^2)*x^7+(15*a^4*c^3*d 
^7+105*a^3*b*c^4*d^6+189*a^2*b^2*c^5*d^5+105*a*b^3*c^6*d^4+15*b^4*c^7*d^3) 
*x^8+(5*a^4*c^2*d^8+160/3*a^3*b*c^3*d^7+140*a^2*b^2*c^4*d^6+112*a*b^3*c^5* 
d^5+70/3*b^4*c^6*d^4)*x^9+(a^4*c*d^9+18*a^3*b*c^2*d^8+72*a^2*b^2*c^3*d^7+8 
4*a*b^3*c^4*d^6+126/5*b^4*c^5*d^5)*x^10+(1/11*a^4*d^10+40/11*a^3*b*c*d^9+2 
70/11*a^2*b^2*c^2*d^8+480/11*a*b^3*c^3*d^7+210/11*b^4*c^4*d^6)*x^11+(1/3*a 
^3*b*d^10+5*a^2*b^2*c*d^9+15*a*b^3*c^2*d^8+10*b^4*c^3*d^7)*x^12+(6/13*a^2* 
b^2*d^10+40/13*a*b^3*c*d^9+45/13*b^4*c^2*d^8)*x^13+(2/7*a*b^3*d^10+5/7*b^4 
*c*d^9)*x^14+1/15*b^4*d^10*x^15
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (109) = 218\).

Time = 0.07 (sec) , antiderivative size = 686, normalized size of antiderivative = 5.76 \[ \int (a+b x)^4 (c+d x)^{10} \, dx=\frac {1}{15} \, b^{4} d^{10} x^{15} + a^{4} c^{10} x + \frac {1}{7} \, {\left (5 \, b^{4} c d^{9} + 2 \, a b^{3} d^{10}\right )} x^{14} + \frac {1}{13} \, {\left (45 \, b^{4} c^{2} d^{8} + 40 \, a b^{3} c d^{9} + 6 \, a^{2} b^{2} d^{10}\right )} x^{13} + \frac {1}{3} \, {\left (30 \, b^{4} c^{3} d^{7} + 45 \, a b^{3} c^{2} d^{8} + 15 \, a^{2} b^{2} c d^{9} + a^{3} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (210 \, b^{4} c^{4} d^{6} + 480 \, a b^{3} c^{3} d^{7} + 270 \, a^{2} b^{2} c^{2} d^{8} + 40 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} x^{11} + \frac {1}{5} \, {\left (126 \, b^{4} c^{5} d^{5} + 420 \, a b^{3} c^{4} d^{6} + 360 \, a^{2} b^{2} c^{3} d^{7} + 90 \, a^{3} b c^{2} d^{8} + 5 \, a^{4} c d^{9}\right )} x^{10} + \frac {1}{3} \, {\left (70 \, b^{4} c^{6} d^{4} + 336 \, a b^{3} c^{5} d^{5} + 420 \, a^{2} b^{2} c^{4} d^{6} + 160 \, a^{3} b c^{3} d^{7} + 15 \, a^{4} c^{2} d^{8}\right )} x^{9} + 3 \, {\left (5 \, b^{4} c^{7} d^{3} + 35 \, a b^{3} c^{6} d^{4} + 63 \, a^{2} b^{2} c^{5} d^{5} + 35 \, a^{3} b c^{4} d^{6} + 5 \, a^{4} c^{3} d^{7}\right )} x^{8} + \frac {3}{7} \, {\left (15 \, b^{4} c^{8} d^{2} + 160 \, a b^{3} c^{7} d^{3} + 420 \, a^{2} b^{2} c^{6} d^{4} + 336 \, a^{3} b c^{5} d^{5} + 70 \, a^{4} c^{4} d^{6}\right )} x^{7} + \frac {1}{3} \, {\left (5 \, b^{4} c^{9} d + 90 \, a b^{3} c^{8} d^{2} + 360 \, a^{2} b^{2} c^{7} d^{3} + 420 \, a^{3} b c^{6} d^{4} + 126 \, a^{4} c^{5} d^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{10} + 40 \, a b^{3} c^{9} d + 270 \, a^{2} b^{2} c^{8} d^{2} + 480 \, a^{3} b c^{7} d^{3} + 210 \, a^{4} c^{6} d^{4}\right )} x^{5} + {\left (a b^{3} c^{10} + 15 \, a^{2} b^{2} c^{9} d + 45 \, a^{3} b c^{8} d^{2} + 30 \, a^{4} c^{7} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{10} + 40 \, a^{3} b c^{9} d + 45 \, a^{4} c^{8} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b c^{10} + 5 \, a^{4} c^{9} d\right )} x^{2} \] Input:

integrate((b*x+a)^4*(d*x+c)^10,x, algorithm="fricas")
 

Output:

1/15*b^4*d^10*x^15 + a^4*c^10*x + 1/7*(5*b^4*c*d^9 + 2*a*b^3*d^10)*x^14 + 
1/13*(45*b^4*c^2*d^8 + 40*a*b^3*c*d^9 + 6*a^2*b^2*d^10)*x^13 + 1/3*(30*b^4 
*c^3*d^7 + 45*a*b^3*c^2*d^8 + 15*a^2*b^2*c*d^9 + a^3*b*d^10)*x^12 + 1/11*( 
210*b^4*c^4*d^6 + 480*a*b^3*c^3*d^7 + 270*a^2*b^2*c^2*d^8 + 40*a^3*b*c*d^9 
 + a^4*d^10)*x^11 + 1/5*(126*b^4*c^5*d^5 + 420*a*b^3*c^4*d^6 + 360*a^2*b^2 
*c^3*d^7 + 90*a^3*b*c^2*d^8 + 5*a^4*c*d^9)*x^10 + 1/3*(70*b^4*c^6*d^4 + 33 
6*a*b^3*c^5*d^5 + 420*a^2*b^2*c^4*d^6 + 160*a^3*b*c^3*d^7 + 15*a^4*c^2*d^8 
)*x^9 + 3*(5*b^4*c^7*d^3 + 35*a*b^3*c^6*d^4 + 63*a^2*b^2*c^5*d^5 + 35*a^3* 
b*c^4*d^6 + 5*a^4*c^3*d^7)*x^8 + 3/7*(15*b^4*c^8*d^2 + 160*a*b^3*c^7*d^3 + 
 420*a^2*b^2*c^6*d^4 + 336*a^3*b*c^5*d^5 + 70*a^4*c^4*d^6)*x^7 + 1/3*(5*b^ 
4*c^9*d + 90*a*b^3*c^8*d^2 + 360*a^2*b^2*c^7*d^3 + 420*a^3*b*c^6*d^4 + 126 
*a^4*c^5*d^5)*x^6 + 1/5*(b^4*c^10 + 40*a*b^3*c^9*d + 270*a^2*b^2*c^8*d^2 + 
 480*a^3*b*c^7*d^3 + 210*a^4*c^6*d^4)*x^5 + (a*b^3*c^10 + 15*a^2*b^2*c^9*d 
 + 45*a^3*b*c^8*d^2 + 30*a^4*c^7*d^3)*x^4 + 1/3*(6*a^2*b^2*c^10 + 40*a^3*b 
*c^9*d + 45*a^4*c^8*d^2)*x^3 + (2*a^3*b*c^10 + 5*a^4*c^9*d)*x^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 748 vs. \(2 (105) = 210\).

Time = 0.06 (sec) , antiderivative size = 748, normalized size of antiderivative = 6.29 \[ \int (a+b x)^4 (c+d x)^{10} \, dx=a^{4} c^{10} x + \frac {b^{4} d^{10} x^{15}}{15} + x^{14} \cdot \left (\frac {2 a b^{3} d^{10}}{7} + \frac {5 b^{4} c d^{9}}{7}\right ) + x^{13} \cdot \left (\frac {6 a^{2} b^{2} d^{10}}{13} + \frac {40 a b^{3} c d^{9}}{13} + \frac {45 b^{4} c^{2} d^{8}}{13}\right ) + x^{12} \left (\frac {a^{3} b d^{10}}{3} + 5 a^{2} b^{2} c d^{9} + 15 a b^{3} c^{2} d^{8} + 10 b^{4} c^{3} d^{7}\right ) + x^{11} \left (\frac {a^{4} d^{10}}{11} + \frac {40 a^{3} b c d^{9}}{11} + \frac {270 a^{2} b^{2} c^{2} d^{8}}{11} + \frac {480 a b^{3} c^{3} d^{7}}{11} + \frac {210 b^{4} c^{4} d^{6}}{11}\right ) + x^{10} \left (a^{4} c d^{9} + 18 a^{3} b c^{2} d^{8} + 72 a^{2} b^{2} c^{3} d^{7} + 84 a b^{3} c^{4} d^{6} + \frac {126 b^{4} c^{5} d^{5}}{5}\right ) + x^{9} \cdot \left (5 a^{4} c^{2} d^{8} + \frac {160 a^{3} b c^{3} d^{7}}{3} + 140 a^{2} b^{2} c^{4} d^{6} + 112 a b^{3} c^{5} d^{5} + \frac {70 b^{4} c^{6} d^{4}}{3}\right ) + x^{8} \cdot \left (15 a^{4} c^{3} d^{7} + 105 a^{3} b c^{4} d^{6} + 189 a^{2} b^{2} c^{5} d^{5} + 105 a b^{3} c^{6} d^{4} + 15 b^{4} c^{7} d^{3}\right ) + x^{7} \cdot \left (30 a^{4} c^{4} d^{6} + 144 a^{3} b c^{5} d^{5} + 180 a^{2} b^{2} c^{6} d^{4} + \frac {480 a b^{3} c^{7} d^{3}}{7} + \frac {45 b^{4} c^{8} d^{2}}{7}\right ) + x^{6} \cdot \left (42 a^{4} c^{5} d^{5} + 140 a^{3} b c^{6} d^{4} + 120 a^{2} b^{2} c^{7} d^{3} + 30 a b^{3} c^{8} d^{2} + \frac {5 b^{4} c^{9} d}{3}\right ) + x^{5} \cdot \left (42 a^{4} c^{6} d^{4} + 96 a^{3} b c^{7} d^{3} + 54 a^{2} b^{2} c^{8} d^{2} + 8 a b^{3} c^{9} d + \frac {b^{4} c^{10}}{5}\right ) + x^{4} \cdot \left (30 a^{4} c^{7} d^{3} + 45 a^{3} b c^{8} d^{2} + 15 a^{2} b^{2} c^{9} d + a b^{3} c^{10}\right ) + x^{3} \cdot \left (15 a^{4} c^{8} d^{2} + \frac {40 a^{3} b c^{9} d}{3} + 2 a^{2} b^{2} c^{10}\right ) + x^{2} \cdot \left (5 a^{4} c^{9} d + 2 a^{3} b c^{10}\right ) \] Input:

integrate((b*x+a)**4*(d*x+c)**10,x)
 

Output:

a**4*c**10*x + b**4*d**10*x**15/15 + x**14*(2*a*b**3*d**10/7 + 5*b**4*c*d* 
*9/7) + x**13*(6*a**2*b**2*d**10/13 + 40*a*b**3*c*d**9/13 + 45*b**4*c**2*d 
**8/13) + x**12*(a**3*b*d**10/3 + 5*a**2*b**2*c*d**9 + 15*a*b**3*c**2*d**8 
 + 10*b**4*c**3*d**7) + x**11*(a**4*d**10/11 + 40*a**3*b*c*d**9/11 + 270*a 
**2*b**2*c**2*d**8/11 + 480*a*b**3*c**3*d**7/11 + 210*b**4*c**4*d**6/11) + 
 x**10*(a**4*c*d**9 + 18*a**3*b*c**2*d**8 + 72*a**2*b**2*c**3*d**7 + 84*a* 
b**3*c**4*d**6 + 126*b**4*c**5*d**5/5) + x**9*(5*a**4*c**2*d**8 + 160*a**3 
*b*c**3*d**7/3 + 140*a**2*b**2*c**4*d**6 + 112*a*b**3*c**5*d**5 + 70*b**4* 
c**6*d**4/3) + x**8*(15*a**4*c**3*d**7 + 105*a**3*b*c**4*d**6 + 189*a**2*b 
**2*c**5*d**5 + 105*a*b**3*c**6*d**4 + 15*b**4*c**7*d**3) + x**7*(30*a**4* 
c**4*d**6 + 144*a**3*b*c**5*d**5 + 180*a**2*b**2*c**6*d**4 + 480*a*b**3*c* 
*7*d**3/7 + 45*b**4*c**8*d**2/7) + x**6*(42*a**4*c**5*d**5 + 140*a**3*b*c* 
*6*d**4 + 120*a**2*b**2*c**7*d**3 + 30*a*b**3*c**8*d**2 + 5*b**4*c**9*d/3) 
 + x**5*(42*a**4*c**6*d**4 + 96*a**3*b*c**7*d**3 + 54*a**2*b**2*c**8*d**2 
+ 8*a*b**3*c**9*d + b**4*c**10/5) + x**4*(30*a**4*c**7*d**3 + 45*a**3*b*c* 
*8*d**2 + 15*a**2*b**2*c**9*d + a*b**3*c**10) + x**3*(15*a**4*c**8*d**2 + 
40*a**3*b*c**9*d/3 + 2*a**2*b**2*c**10) + x**2*(5*a**4*c**9*d + 2*a**3*b*c 
**10)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (109) = 218\).

Time = 0.03 (sec) , antiderivative size = 686, normalized size of antiderivative = 5.76 \[ \int (a+b x)^4 (c+d x)^{10} \, dx=\frac {1}{15} \, b^{4} d^{10} x^{15} + a^{4} c^{10} x + \frac {1}{7} \, {\left (5 \, b^{4} c d^{9} + 2 \, a b^{3} d^{10}\right )} x^{14} + \frac {1}{13} \, {\left (45 \, b^{4} c^{2} d^{8} + 40 \, a b^{3} c d^{9} + 6 \, a^{2} b^{2} d^{10}\right )} x^{13} + \frac {1}{3} \, {\left (30 \, b^{4} c^{3} d^{7} + 45 \, a b^{3} c^{2} d^{8} + 15 \, a^{2} b^{2} c d^{9} + a^{3} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (210 \, b^{4} c^{4} d^{6} + 480 \, a b^{3} c^{3} d^{7} + 270 \, a^{2} b^{2} c^{2} d^{8} + 40 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} x^{11} + \frac {1}{5} \, {\left (126 \, b^{4} c^{5} d^{5} + 420 \, a b^{3} c^{4} d^{6} + 360 \, a^{2} b^{2} c^{3} d^{7} + 90 \, a^{3} b c^{2} d^{8} + 5 \, a^{4} c d^{9}\right )} x^{10} + \frac {1}{3} \, {\left (70 \, b^{4} c^{6} d^{4} + 336 \, a b^{3} c^{5} d^{5} + 420 \, a^{2} b^{2} c^{4} d^{6} + 160 \, a^{3} b c^{3} d^{7} + 15 \, a^{4} c^{2} d^{8}\right )} x^{9} + 3 \, {\left (5 \, b^{4} c^{7} d^{3} + 35 \, a b^{3} c^{6} d^{4} + 63 \, a^{2} b^{2} c^{5} d^{5} + 35 \, a^{3} b c^{4} d^{6} + 5 \, a^{4} c^{3} d^{7}\right )} x^{8} + \frac {3}{7} \, {\left (15 \, b^{4} c^{8} d^{2} + 160 \, a b^{3} c^{7} d^{3} + 420 \, a^{2} b^{2} c^{6} d^{4} + 336 \, a^{3} b c^{5} d^{5} + 70 \, a^{4} c^{4} d^{6}\right )} x^{7} + \frac {1}{3} \, {\left (5 \, b^{4} c^{9} d + 90 \, a b^{3} c^{8} d^{2} + 360 \, a^{2} b^{2} c^{7} d^{3} + 420 \, a^{3} b c^{6} d^{4} + 126 \, a^{4} c^{5} d^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{10} + 40 \, a b^{3} c^{9} d + 270 \, a^{2} b^{2} c^{8} d^{2} + 480 \, a^{3} b c^{7} d^{3} + 210 \, a^{4} c^{6} d^{4}\right )} x^{5} + {\left (a b^{3} c^{10} + 15 \, a^{2} b^{2} c^{9} d + 45 \, a^{3} b c^{8} d^{2} + 30 \, a^{4} c^{7} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{10} + 40 \, a^{3} b c^{9} d + 45 \, a^{4} c^{8} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b c^{10} + 5 \, a^{4} c^{9} d\right )} x^{2} \] Input:

integrate((b*x+a)^4*(d*x+c)^10,x, algorithm="maxima")
 

Output:

1/15*b^4*d^10*x^15 + a^4*c^10*x + 1/7*(5*b^4*c*d^9 + 2*a*b^3*d^10)*x^14 + 
1/13*(45*b^4*c^2*d^8 + 40*a*b^3*c*d^9 + 6*a^2*b^2*d^10)*x^13 + 1/3*(30*b^4 
*c^3*d^7 + 45*a*b^3*c^2*d^8 + 15*a^2*b^2*c*d^9 + a^3*b*d^10)*x^12 + 1/11*( 
210*b^4*c^4*d^6 + 480*a*b^3*c^3*d^7 + 270*a^2*b^2*c^2*d^8 + 40*a^3*b*c*d^9 
 + a^4*d^10)*x^11 + 1/5*(126*b^4*c^5*d^5 + 420*a*b^3*c^4*d^6 + 360*a^2*b^2 
*c^3*d^7 + 90*a^3*b*c^2*d^8 + 5*a^4*c*d^9)*x^10 + 1/3*(70*b^4*c^6*d^4 + 33 
6*a*b^3*c^5*d^5 + 420*a^2*b^2*c^4*d^6 + 160*a^3*b*c^3*d^7 + 15*a^4*c^2*d^8 
)*x^9 + 3*(5*b^4*c^7*d^3 + 35*a*b^3*c^6*d^4 + 63*a^2*b^2*c^5*d^5 + 35*a^3* 
b*c^4*d^6 + 5*a^4*c^3*d^7)*x^8 + 3/7*(15*b^4*c^8*d^2 + 160*a*b^3*c^7*d^3 + 
 420*a^2*b^2*c^6*d^4 + 336*a^3*b*c^5*d^5 + 70*a^4*c^4*d^6)*x^7 + 1/3*(5*b^ 
4*c^9*d + 90*a*b^3*c^8*d^2 + 360*a^2*b^2*c^7*d^3 + 420*a^3*b*c^6*d^4 + 126 
*a^4*c^5*d^5)*x^6 + 1/5*(b^4*c^10 + 40*a*b^3*c^9*d + 270*a^2*b^2*c^8*d^2 + 
 480*a^3*b*c^7*d^3 + 210*a^4*c^6*d^4)*x^5 + (a*b^3*c^10 + 15*a^2*b^2*c^9*d 
 + 45*a^3*b*c^8*d^2 + 30*a^4*c^7*d^3)*x^4 + 1/3*(6*a^2*b^2*c^10 + 40*a^3*b 
*c^9*d + 45*a^4*c^8*d^2)*x^3 + (2*a^3*b*c^10 + 5*a^4*c^9*d)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 771 vs. \(2 (109) = 218\).

Time = 0.12 (sec) , antiderivative size = 771, normalized size of antiderivative = 6.48 \[ \int (a+b x)^4 (c+d x)^{10} \, dx=\frac {1}{15} \, b^{4} d^{10} x^{15} + \frac {5}{7} \, b^{4} c d^{9} x^{14} + \frac {2}{7} \, a b^{3} d^{10} x^{14} + \frac {45}{13} \, b^{4} c^{2} d^{8} x^{13} + \frac {40}{13} \, a b^{3} c d^{9} x^{13} + \frac {6}{13} \, a^{2} b^{2} d^{10} x^{13} + 10 \, b^{4} c^{3} d^{7} x^{12} + 15 \, a b^{3} c^{2} d^{8} x^{12} + 5 \, a^{2} b^{2} c d^{9} x^{12} + \frac {1}{3} \, a^{3} b d^{10} x^{12} + \frac {210}{11} \, b^{4} c^{4} d^{6} x^{11} + \frac {480}{11} \, a b^{3} c^{3} d^{7} x^{11} + \frac {270}{11} \, a^{2} b^{2} c^{2} d^{8} x^{11} + \frac {40}{11} \, a^{3} b c d^{9} x^{11} + \frac {1}{11} \, a^{4} d^{10} x^{11} + \frac {126}{5} \, b^{4} c^{5} d^{5} x^{10} + 84 \, a b^{3} c^{4} d^{6} x^{10} + 72 \, a^{2} b^{2} c^{3} d^{7} x^{10} + 18 \, a^{3} b c^{2} d^{8} x^{10} + a^{4} c d^{9} x^{10} + \frac {70}{3} \, b^{4} c^{6} d^{4} x^{9} + 112 \, a b^{3} c^{5} d^{5} x^{9} + 140 \, a^{2} b^{2} c^{4} d^{6} x^{9} + \frac {160}{3} \, a^{3} b c^{3} d^{7} x^{9} + 5 \, a^{4} c^{2} d^{8} x^{9} + 15 \, b^{4} c^{7} d^{3} x^{8} + 105 \, a b^{3} c^{6} d^{4} x^{8} + 189 \, a^{2} b^{2} c^{5} d^{5} x^{8} + 105 \, a^{3} b c^{4} d^{6} x^{8} + 15 \, a^{4} c^{3} d^{7} x^{8} + \frac {45}{7} \, b^{4} c^{8} d^{2} x^{7} + \frac {480}{7} \, a b^{3} c^{7} d^{3} x^{7} + 180 \, a^{2} b^{2} c^{6} d^{4} x^{7} + 144 \, a^{3} b c^{5} d^{5} x^{7} + 30 \, a^{4} c^{4} d^{6} x^{7} + \frac {5}{3} \, b^{4} c^{9} d x^{6} + 30 \, a b^{3} c^{8} d^{2} x^{6} + 120 \, a^{2} b^{2} c^{7} d^{3} x^{6} + 140 \, a^{3} b c^{6} d^{4} x^{6} + 42 \, a^{4} c^{5} d^{5} x^{6} + \frac {1}{5} \, b^{4} c^{10} x^{5} + 8 \, a b^{3} c^{9} d x^{5} + 54 \, a^{2} b^{2} c^{8} d^{2} x^{5} + 96 \, a^{3} b c^{7} d^{3} x^{5} + 42 \, a^{4} c^{6} d^{4} x^{5} + a b^{3} c^{10} x^{4} + 15 \, a^{2} b^{2} c^{9} d x^{4} + 45 \, a^{3} b c^{8} d^{2} x^{4} + 30 \, a^{4} c^{7} d^{3} x^{4} + 2 \, a^{2} b^{2} c^{10} x^{3} + \frac {40}{3} \, a^{3} b c^{9} d x^{3} + 15 \, a^{4} c^{8} d^{2} x^{3} + 2 \, a^{3} b c^{10} x^{2} + 5 \, a^{4} c^{9} d x^{2} + a^{4} c^{10} x \] Input:

integrate((b*x+a)^4*(d*x+c)^10,x, algorithm="giac")
 

Output:

1/15*b^4*d^10*x^15 + 5/7*b^4*c*d^9*x^14 + 2/7*a*b^3*d^10*x^14 + 45/13*b^4* 
c^2*d^8*x^13 + 40/13*a*b^3*c*d^9*x^13 + 6/13*a^2*b^2*d^10*x^13 + 10*b^4*c^ 
3*d^7*x^12 + 15*a*b^3*c^2*d^8*x^12 + 5*a^2*b^2*c*d^9*x^12 + 1/3*a^3*b*d^10 
*x^12 + 210/11*b^4*c^4*d^6*x^11 + 480/11*a*b^3*c^3*d^7*x^11 + 270/11*a^2*b 
^2*c^2*d^8*x^11 + 40/11*a^3*b*c*d^9*x^11 + 1/11*a^4*d^10*x^11 + 126/5*b^4* 
c^5*d^5*x^10 + 84*a*b^3*c^4*d^6*x^10 + 72*a^2*b^2*c^3*d^7*x^10 + 18*a^3*b* 
c^2*d^8*x^10 + a^4*c*d^9*x^10 + 70/3*b^4*c^6*d^4*x^9 + 112*a*b^3*c^5*d^5*x 
^9 + 140*a^2*b^2*c^4*d^6*x^9 + 160/3*a^3*b*c^3*d^7*x^9 + 5*a^4*c^2*d^8*x^9 
 + 15*b^4*c^7*d^3*x^8 + 105*a*b^3*c^6*d^4*x^8 + 189*a^2*b^2*c^5*d^5*x^8 + 
105*a^3*b*c^4*d^6*x^8 + 15*a^4*c^3*d^7*x^8 + 45/7*b^4*c^8*d^2*x^7 + 480/7* 
a*b^3*c^7*d^3*x^7 + 180*a^2*b^2*c^6*d^4*x^7 + 144*a^3*b*c^5*d^5*x^7 + 30*a 
^4*c^4*d^6*x^7 + 5/3*b^4*c^9*d*x^6 + 30*a*b^3*c^8*d^2*x^6 + 120*a^2*b^2*c^ 
7*d^3*x^6 + 140*a^3*b*c^6*d^4*x^6 + 42*a^4*c^5*d^5*x^6 + 1/5*b^4*c^10*x^5 
+ 8*a*b^3*c^9*d*x^5 + 54*a^2*b^2*c^8*d^2*x^5 + 96*a^3*b*c^7*d^3*x^5 + 42*a 
^4*c^6*d^4*x^5 + a*b^3*c^10*x^4 + 15*a^2*b^2*c^9*d*x^4 + 45*a^3*b*c^8*d^2* 
x^4 + 30*a^4*c^7*d^3*x^4 + 2*a^2*b^2*c^10*x^3 + 40/3*a^3*b*c^9*d*x^3 + 15* 
a^4*c^8*d^2*x^3 + 2*a^3*b*c^10*x^2 + 5*a^4*c^9*d*x^2 + a^4*c^10*x
 

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 664, normalized size of antiderivative = 5.58 \[ \int (a+b x)^4 (c+d x)^{10} \, dx=x^5\,\left (42\,a^4\,c^6\,d^4+96\,a^3\,b\,c^7\,d^3+54\,a^2\,b^2\,c^8\,d^2+8\,a\,b^3\,c^9\,d+\frac {b^4\,c^{10}}{5}\right )+x^{11}\,\left (\frac {a^4\,d^{10}}{11}+\frac {40\,a^3\,b\,c\,d^9}{11}+\frac {270\,a^2\,b^2\,c^2\,d^8}{11}+\frac {480\,a\,b^3\,c^3\,d^7}{11}+\frac {210\,b^4\,c^4\,d^6}{11}\right )+x^8\,\left (15\,a^4\,c^3\,d^7+105\,a^3\,b\,c^4\,d^6+189\,a^2\,b^2\,c^5\,d^5+105\,a\,b^3\,c^6\,d^4+15\,b^4\,c^7\,d^3\right )+x^9\,\left (5\,a^4\,c^2\,d^8+\frac {160\,a^3\,b\,c^3\,d^7}{3}+140\,a^2\,b^2\,c^4\,d^6+112\,a\,b^3\,c^5\,d^5+\frac {70\,b^4\,c^6\,d^4}{3}\right )+x^7\,\left (30\,a^4\,c^4\,d^6+144\,a^3\,b\,c^5\,d^5+180\,a^2\,b^2\,c^6\,d^4+\frac {480\,a\,b^3\,c^7\,d^3}{7}+\frac {45\,b^4\,c^8\,d^2}{7}\right )+x^4\,\left (30\,a^4\,c^7\,d^3+45\,a^3\,b\,c^8\,d^2+15\,a^2\,b^2\,c^9\,d+a\,b^3\,c^{10}\right )+x^{12}\,\left (\frac {a^3\,b\,d^{10}}{3}+5\,a^2\,b^2\,c\,d^9+15\,a\,b^3\,c^2\,d^8+10\,b^4\,c^3\,d^7\right )+x^{10}\,\left (a^4\,c\,d^9+18\,a^3\,b\,c^2\,d^8+72\,a^2\,b^2\,c^3\,d^7+84\,a\,b^3\,c^4\,d^6+\frac {126\,b^4\,c^5\,d^5}{5}\right )+x^6\,\left (42\,a^4\,c^5\,d^5+140\,a^3\,b\,c^6\,d^4+120\,a^2\,b^2\,c^7\,d^3+30\,a\,b^3\,c^8\,d^2+\frac {5\,b^4\,c^9\,d}{3}\right )+a^4\,c^{10}\,x+\frac {b^4\,d^{10}\,x^{15}}{15}+a^3\,c^9\,x^2\,\left (5\,a\,d+2\,b\,c\right )+\frac {b^3\,d^9\,x^{14}\,\left (2\,a\,d+5\,b\,c\right )}{7}+\frac {a^2\,c^8\,x^3\,\left (45\,a^2\,d^2+40\,a\,b\,c\,d+6\,b^2\,c^2\right )}{3}+\frac {b^2\,d^8\,x^{13}\,\left (6\,a^2\,d^2+40\,a\,b\,c\,d+45\,b^2\,c^2\right )}{13} \] Input:

int((a + b*x)^4*(c + d*x)^10,x)
 

Output:

x^5*((b^4*c^10)/5 + 42*a^4*c^6*d^4 + 96*a^3*b*c^7*d^3 + 54*a^2*b^2*c^8*d^2 
 + 8*a*b^3*c^9*d) + x^11*((a^4*d^10)/11 + (210*b^4*c^4*d^6)/11 + (480*a*b^ 
3*c^3*d^7)/11 + (270*a^2*b^2*c^2*d^8)/11 + (40*a^3*b*c*d^9)/11) + x^8*(15* 
a^4*c^3*d^7 + 15*b^4*c^7*d^3 + 105*a*b^3*c^6*d^4 + 105*a^3*b*c^4*d^6 + 189 
*a^2*b^2*c^5*d^5) + x^9*(5*a^4*c^2*d^8 + (70*b^4*c^6*d^4)/3 + 112*a*b^3*c^ 
5*d^5 + (160*a^3*b*c^3*d^7)/3 + 140*a^2*b^2*c^4*d^6) + x^7*(30*a^4*c^4*d^6 
 + (45*b^4*c^8*d^2)/7 + (480*a*b^3*c^7*d^3)/7 + 144*a^3*b*c^5*d^5 + 180*a^ 
2*b^2*c^6*d^4) + x^4*(a*b^3*c^10 + 30*a^4*c^7*d^3 + 15*a^2*b^2*c^9*d + 45* 
a^3*b*c^8*d^2) + x^12*((a^3*b*d^10)/3 + 10*b^4*c^3*d^7 + 15*a*b^3*c^2*d^8 
+ 5*a^2*b^2*c*d^9) + x^10*(a^4*c*d^9 + (126*b^4*c^5*d^5)/5 + 84*a*b^3*c^4* 
d^6 + 18*a^3*b*c^2*d^8 + 72*a^2*b^2*c^3*d^7) + x^6*((5*b^4*c^9*d)/3 + 42*a 
^4*c^5*d^5 + 30*a*b^3*c^8*d^2 + 140*a^3*b*c^6*d^4 + 120*a^2*b^2*c^7*d^3) + 
 a^4*c^10*x + (b^4*d^10*x^15)/15 + a^3*c^9*x^2*(5*a*d + 2*b*c) + (b^3*d^9* 
x^14*(2*a*d + 5*b*c))/7 + (a^2*c^8*x^3*(45*a^2*d^2 + 6*b^2*c^2 + 40*a*b*c* 
d))/3 + (b^2*d^8*x^13*(6*a^2*d^2 + 45*b^2*c^2 + 40*a*b*c*d))/13
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 772, normalized size of antiderivative = 6.49 \[ \int (a+b x)^4 (c+d x)^{10} \, dx=\frac {x \left (1001 b^{4} d^{10} x^{14}+4290 a \,b^{3} d^{10} x^{13}+10725 b^{4} c \,d^{9} x^{13}+6930 a^{2} b^{2} d^{10} x^{12}+46200 a \,b^{3} c \,d^{9} x^{12}+51975 b^{4} c^{2} d^{8} x^{12}+5005 a^{3} b \,d^{10} x^{11}+75075 a^{2} b^{2} c \,d^{9} x^{11}+225225 a \,b^{3} c^{2} d^{8} x^{11}+150150 b^{4} c^{3} d^{7} x^{11}+1365 a^{4} d^{10} x^{10}+54600 a^{3} b c \,d^{9} x^{10}+368550 a^{2} b^{2} c^{2} d^{8} x^{10}+655200 a \,b^{3} c^{3} d^{7} x^{10}+286650 b^{4} c^{4} d^{6} x^{10}+15015 a^{4} c \,d^{9} x^{9}+270270 a^{3} b \,c^{2} d^{8} x^{9}+1081080 a^{2} b^{2} c^{3} d^{7} x^{9}+1261260 a \,b^{3} c^{4} d^{6} x^{9}+378378 b^{4} c^{5} d^{5} x^{9}+75075 a^{4} c^{2} d^{8} x^{8}+800800 a^{3} b \,c^{3} d^{7} x^{8}+2102100 a^{2} b^{2} c^{4} d^{6} x^{8}+1681680 a \,b^{3} c^{5} d^{5} x^{8}+350350 b^{4} c^{6} d^{4} x^{8}+225225 a^{4} c^{3} d^{7} x^{7}+1576575 a^{3} b \,c^{4} d^{6} x^{7}+2837835 a^{2} b^{2} c^{5} d^{5} x^{7}+1576575 a \,b^{3} c^{6} d^{4} x^{7}+225225 b^{4} c^{7} d^{3} x^{7}+450450 a^{4} c^{4} d^{6} x^{6}+2162160 a^{3} b \,c^{5} d^{5} x^{6}+2702700 a^{2} b^{2} c^{6} d^{4} x^{6}+1029600 a \,b^{3} c^{7} d^{3} x^{6}+96525 b^{4} c^{8} d^{2} x^{6}+630630 a^{4} c^{5} d^{5} x^{5}+2102100 a^{3} b \,c^{6} d^{4} x^{5}+1801800 a^{2} b^{2} c^{7} d^{3} x^{5}+450450 a \,b^{3} c^{8} d^{2} x^{5}+25025 b^{4} c^{9} d \,x^{5}+630630 a^{4} c^{6} d^{4} x^{4}+1441440 a^{3} b \,c^{7} d^{3} x^{4}+810810 a^{2} b^{2} c^{8} d^{2} x^{4}+120120 a \,b^{3} c^{9} d \,x^{4}+3003 b^{4} c^{10} x^{4}+450450 a^{4} c^{7} d^{3} x^{3}+675675 a^{3} b \,c^{8} d^{2} x^{3}+225225 a^{2} b^{2} c^{9} d \,x^{3}+15015 a \,b^{3} c^{10} x^{3}+225225 a^{4} c^{8} d^{2} x^{2}+200200 a^{3} b \,c^{9} d \,x^{2}+30030 a^{2} b^{2} c^{10} x^{2}+75075 a^{4} c^{9} d x +30030 a^{3} b \,c^{10} x +15015 a^{4} c^{10}\right )}{15015} \] Input:

int((b*x+a)^4*(d*x+c)^10,x)
 

Output:

(x*(15015*a**4*c**10 + 75075*a**4*c**9*d*x + 225225*a**4*c**8*d**2*x**2 + 
450450*a**4*c**7*d**3*x**3 + 630630*a**4*c**6*d**4*x**4 + 630630*a**4*c**5 
*d**5*x**5 + 450450*a**4*c**4*d**6*x**6 + 225225*a**4*c**3*d**7*x**7 + 750 
75*a**4*c**2*d**8*x**8 + 15015*a**4*c*d**9*x**9 + 1365*a**4*d**10*x**10 + 
30030*a**3*b*c**10*x + 200200*a**3*b*c**9*d*x**2 + 675675*a**3*b*c**8*d**2 
*x**3 + 1441440*a**3*b*c**7*d**3*x**4 + 2102100*a**3*b*c**6*d**4*x**5 + 21 
62160*a**3*b*c**5*d**5*x**6 + 1576575*a**3*b*c**4*d**6*x**7 + 800800*a**3* 
b*c**3*d**7*x**8 + 270270*a**3*b*c**2*d**8*x**9 + 54600*a**3*b*c*d**9*x**1 
0 + 5005*a**3*b*d**10*x**11 + 30030*a**2*b**2*c**10*x**2 + 225225*a**2*b** 
2*c**9*d*x**3 + 810810*a**2*b**2*c**8*d**2*x**4 + 1801800*a**2*b**2*c**7*d 
**3*x**5 + 2702700*a**2*b**2*c**6*d**4*x**6 + 2837835*a**2*b**2*c**5*d**5* 
x**7 + 2102100*a**2*b**2*c**4*d**6*x**8 + 1081080*a**2*b**2*c**3*d**7*x**9 
 + 368550*a**2*b**2*c**2*d**8*x**10 + 75075*a**2*b**2*c*d**9*x**11 + 6930* 
a**2*b**2*d**10*x**12 + 15015*a*b**3*c**10*x**3 + 120120*a*b**3*c**9*d*x** 
4 + 450450*a*b**3*c**8*d**2*x**5 + 1029600*a*b**3*c**7*d**3*x**6 + 1576575 
*a*b**3*c**6*d**4*x**7 + 1681680*a*b**3*c**5*d**5*x**8 + 1261260*a*b**3*c* 
*4*d**6*x**9 + 655200*a*b**3*c**3*d**7*x**10 + 225225*a*b**3*c**2*d**8*x** 
11 + 46200*a*b**3*c*d**9*x**12 + 4290*a*b**3*d**10*x**13 + 3003*b**4*c**10 
*x**4 + 25025*b**4*c**9*d*x**5 + 96525*b**4*c**8*d**2*x**6 + 225225*b**4*c 
**7*d**3*x**7 + 350350*b**4*c**6*d**4*x**8 + 378378*b**4*c**5*d**5*x**9...