\(\int (a+b x)^6 (c+d x)^{10} \, dx\) [99]

Optimal result

Integrand size = 15, antiderivative size = 170 \[ \int (a+b x)^6 (c+d x)^{10} \, dx=\frac {(b c-a d)^6 (c+d x)^{11}}{11 d^7}-\frac {b (b c-a d)^5 (c+d x)^{12}}{2 d^7}+\frac {15 b^2 (b c-a d)^4 (c+d x)^{13}}{13 d^7}-\frac {10 b^3 (b c-a d)^3 (c+d x)^{14}}{7 d^7}+\frac {b^4 (b c-a d)^2 (c+d x)^{15}}{d^7}-\frac {3 b^5 (b c-a d) (c+d x)^{16}}{8 d^7}+\frac {b^6 (c+d x)^{17}}{17 d^7} \] Output:

1/11*(-a*d+b*c)^6*(d*x+c)^11/d^7-1/2*b*(-a*d+b*c)^5*(d*x+c)^12/d^7+15/13*b 
^2*(-a*d+b*c)^4*(d*x+c)^13/d^7-10/7*b^3*(-a*d+b*c)^3*(d*x+c)^14/d^7+b^4*(- 
a*d+b*c)^2*(d*x+c)^15/d^7-3/8*b^5*(-a*d+b*c)*(d*x+c)^16/d^7+1/17*b^6*(d*x+ 
c)^17/d^7
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(939\) vs. \(2(170)=340\).

Time = 0.07 (sec) , antiderivative size = 939, normalized size of antiderivative = 5.52 \[ \int (a+b x)^6 (c+d x)^{10} \, dx=a^6 c^{10} x+a^5 c^9 (3 b c+5 a d) x^2+5 a^4 c^8 \left (b^2 c^2+4 a b c d+3 a^2 d^2\right ) x^3+\frac {5}{2} a^3 c^7 \left (2 b^3 c^3+15 a b^2 c^2 d+27 a^2 b c d^2+12 a^3 d^3\right ) x^4+a^2 c^6 \left (3 b^4 c^4+40 a b^3 c^3 d+135 a^2 b^2 c^2 d^2+144 a^3 b c d^3+42 a^4 d^4\right ) x^5+a c^5 \left (b^5 c^5+25 a b^4 c^4 d+150 a^2 b^3 c^3 d^2+300 a^3 b^2 c^2 d^3+210 a^4 b c d^4+42 a^5 d^5\right ) x^6+\frac {1}{7} c^4 \left (b^6 c^6+60 a b^5 c^5 d+675 a^2 b^4 c^4 d^2+2400 a^3 b^3 c^3 d^3+3150 a^4 b^2 c^2 d^4+1512 a^5 b c d^5+210 a^6 d^6\right ) x^7+\frac {5}{4} c^3 d \left (b^6 c^6+27 a b^5 c^5 d+180 a^2 b^4 c^4 d^2+420 a^3 b^3 c^3 d^3+378 a^4 b^2 c^2 d^4+126 a^5 b c d^5+12 a^6 d^6\right ) x^8+5 c^2 d^2 \left (b^6 c^6+16 a b^5 c^5 d+70 a^2 b^4 c^4 d^2+112 a^3 b^3 c^3 d^3+70 a^4 b^2 c^2 d^4+16 a^5 b c d^5+a^6 d^6\right ) x^9+c d^3 \left (12 b^6 c^6+126 a b^5 c^5 d+378 a^2 b^4 c^4 d^2+420 a^3 b^3 c^3 d^3+180 a^4 b^2 c^2 d^4+27 a^5 b c d^5+a^6 d^6\right ) x^{10}+\frac {1}{11} d^4 \left (210 b^6 c^6+1512 a b^5 c^5 d+3150 a^2 b^4 c^4 d^2+2400 a^3 b^3 c^3 d^3+675 a^4 b^2 c^2 d^4+60 a^5 b c d^5+a^6 d^6\right ) x^{11}+\frac {1}{2} b d^5 \left (42 b^5 c^5+210 a b^4 c^4 d+300 a^2 b^3 c^3 d^2+150 a^3 b^2 c^2 d^3+25 a^4 b c d^4+a^5 d^5\right ) x^{12}+\frac {5}{13} b^2 d^6 \left (42 b^4 c^4+144 a b^3 c^3 d+135 a^2 b^2 c^2 d^2+40 a^3 b c d^3+3 a^4 d^4\right ) x^{13}+\frac {5}{7} b^3 d^7 \left (12 b^3 c^3+27 a b^2 c^2 d+15 a^2 b c d^2+2 a^3 d^3\right ) x^{14}+b^4 d^8 \left (3 b^2 c^2+4 a b c d+a^2 d^2\right ) x^{15}+\frac {1}{8} b^5 d^9 (5 b c+3 a d) x^{16}+\frac {1}{17} b^6 d^{10} x^{17} \] Input:

Integrate[(a + b*x)^6*(c + d*x)^10,x]
 

Output:

a^6*c^10*x + a^5*c^9*(3*b*c + 5*a*d)*x^2 + 5*a^4*c^8*(b^2*c^2 + 4*a*b*c*d 
+ 3*a^2*d^2)*x^3 + (5*a^3*c^7*(2*b^3*c^3 + 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 
 + 12*a^3*d^3)*x^4)/2 + a^2*c^6*(3*b^4*c^4 + 40*a*b^3*c^3*d + 135*a^2*b^2* 
c^2*d^2 + 144*a^3*b*c*d^3 + 42*a^4*d^4)*x^5 + a*c^5*(b^5*c^5 + 25*a*b^4*c^ 
4*d + 150*a^2*b^3*c^3*d^2 + 300*a^3*b^2*c^2*d^3 + 210*a^4*b*c*d^4 + 42*a^5 
*d^5)*x^6 + (c^4*(b^6*c^6 + 60*a*b^5*c^5*d + 675*a^2*b^4*c^4*d^2 + 2400*a^ 
3*b^3*c^3*d^3 + 3150*a^4*b^2*c^2*d^4 + 1512*a^5*b*c*d^5 + 210*a^6*d^6)*x^7 
)/7 + (5*c^3*d*(b^6*c^6 + 27*a*b^5*c^5*d + 180*a^2*b^4*c^4*d^2 + 420*a^3*b 
^3*c^3*d^3 + 378*a^4*b^2*c^2*d^4 + 126*a^5*b*c*d^5 + 12*a^6*d^6)*x^8)/4 + 
5*c^2*d^2*(b^6*c^6 + 16*a*b^5*c^5*d + 70*a^2*b^4*c^4*d^2 + 112*a^3*b^3*c^3 
*d^3 + 70*a^4*b^2*c^2*d^4 + 16*a^5*b*c*d^5 + a^6*d^6)*x^9 + c*d^3*(12*b^6* 
c^6 + 126*a*b^5*c^5*d + 378*a^2*b^4*c^4*d^2 + 420*a^3*b^3*c^3*d^3 + 180*a^ 
4*b^2*c^2*d^4 + 27*a^5*b*c*d^5 + a^6*d^6)*x^10 + (d^4*(210*b^6*c^6 + 1512* 
a*b^5*c^5*d + 3150*a^2*b^4*c^4*d^2 + 2400*a^3*b^3*c^3*d^3 + 675*a^4*b^2*c^ 
2*d^4 + 60*a^5*b*c*d^5 + a^6*d^6)*x^11)/11 + (b*d^5*(42*b^5*c^5 + 210*a*b^ 
4*c^4*d + 300*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 + 25*a^4*b*c*d^4 + a^5 
*d^5)*x^12)/2 + (5*b^2*d^6*(42*b^4*c^4 + 144*a*b^3*c^3*d + 135*a^2*b^2*c^2 
*d^2 + 40*a^3*b*c*d^3 + 3*a^4*d^4)*x^13)/13 + (5*b^3*d^7*(12*b^3*c^3 + 27* 
a*b^2*c^2*d + 15*a^2*b*c*d^2 + 2*a^3*d^3)*x^14)/7 + b^4*d^8*(3*b^2*c^2 + 4 
*a*b*c*d + a^2*d^2)*x^15 + (b^5*d^9*(5*b*c + 3*a*d)*x^16)/8 + (b^6*d^10...
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 170, 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.

\(\displaystyle \int (a+b x)^6 (c+d x)^{10} \, dx\)

\(\Big \downarrow \) 49

\(\displaystyle \int \left (-\frac {6 b^5 (c+d x)^{15} (b c-a d)}{d^6}+\frac {15 b^4 (c+d x)^{14} (b c-a d)^2}{d^6}-\frac {20 b^3 (c+d x)^{13} (b c-a d)^3}{d^6}+\frac {15 b^2 (c+d x)^{12} (b c-a d)^4}{d^6}-\frac {6 b (c+d x)^{11} (b c-a d)^5}{d^6}+\frac {(c+d x)^{10} (a d-b c)^6}{d^6}+\frac {b^6 (c+d x)^{16}}{d^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 b^5 (c+d x)^{16} (b c-a d)}{8 d^7}+\frac {b^4 (c+d x)^{15} (b c-a d)^2}{d^7}-\frac {10 b^3 (c+d x)^{14} (b c-a d)^3}{7 d^7}+\frac {15 b^2 (c+d x)^{13} (b c-a d)^4}{13 d^7}-\frac {b (c+d x)^{12} (b c-a d)^5}{2 d^7}+\frac {(c+d x)^{11} (b c-a d)^6}{11 d^7}+\frac {b^6 (c+d x)^{17}}{17 d^7}\)

Input:

Int[(a + b*x)^6*(c + d*x)^10,x]
 

Output:

((b*c - a*d)^6*(c + d*x)^11)/(11*d^7) - (b*(b*c - a*d)^5*(c + d*x)^12)/(2* 
d^7) + (15*b^2*(b*c - a*d)^4*(c + d*x)^13)/(13*d^7) - (10*b^3*(b*c - a*d)^ 
3*(c + d*x)^14)/(7*d^7) + (b^4*(b*c - a*d)^2*(c + d*x)^15)/d^7 - (3*b^5*(b 
*c - a*d)*(c + d*x)^16)/(8*d^7) + (b^6*(c + d*x)^17)/(17*d^7)
 

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. \(974\) vs. \(2(158)=316\).

Time = 0.12 (sec) , antiderivative size = 975, normalized size of antiderivative = 5.74

method	result	size
norman	\(a^{6} c^{10} x +\left (5 a^{6} c^{9} d +3 a^{5} b \,c^{10}\right ) x^{2}+\left (15 a^{6} c^{8} d^{2}+20 a^{5} b \,c^{9} d +5 a^{4} b^{2} c^{10}\right ) x^{3}+\left (30 a^{6} c^{7} d^{3}+\frac {135}{2} a^{5} b \,c^{8} d^{2}+\frac {75}{2} a^{4} b^{2} c^{9} d +5 a^{3} b^{3} c^{10}\right ) x^{4}+\left (42 a^{6} c^{6} d^{4}+144 a^{5} b \,c^{7} d^{3}+135 a^{4} b^{2} c^{8} d^{2}+40 a^{3} b^{3} c^{9} d +3 a^{2} b^{4} c^{10}\right ) x^{5}+\left (42 a^{6} c^{5} d^{5}+210 a^{5} b \,c^{6} d^{4}+300 a^{4} b^{2} c^{7} d^{3}+150 a^{3} b^{3} c^{8} d^{2}+25 a^{2} b^{4} c^{9} d +a \,b^{5} c^{10}\right ) x^{6}+\left (30 a^{6} c^{4} d^{6}+216 a^{5} b \,c^{5} d^{5}+450 a^{4} b^{2} c^{6} d^{4}+\frac {2400}{7} a^{3} b^{3} c^{7} d^{3}+\frac {675}{7} a^{2} b^{4} c^{8} d^{2}+\frac {60}{7} a \,b^{5} c^{9} d +\frac {1}{7} b^{6} c^{10}\right ) x^{7}+\left (15 a^{6} c^{3} d^{7}+\frac {315}{2} a^{5} b \,c^{4} d^{6}+\frac {945}{2} a^{4} b^{2} c^{5} d^{5}+525 a^{3} b^{3} c^{6} d^{4}+225 a^{2} b^{4} c^{7} d^{3}+\frac {135}{4} a \,b^{5} c^{8} d^{2}+\frac {5}{4} b^{6} c^{9} d \right ) x^{8}+\left (5 a^{6} c^{2} d^{8}+80 a^{5} b \,c^{3} d^{7}+350 a^{4} b^{2} c^{4} d^{6}+560 a^{3} b^{3} c^{5} d^{5}+350 a^{2} b^{4} c^{6} d^{4}+80 a \,b^{5} c^{7} d^{3}+5 b^{6} c^{8} d^{2}\right ) x^{9}+\left (a^{6} c \,d^{9}+27 a^{5} b \,c^{2} d^{8}+180 a^{4} b^{2} c^{3} d^{7}+420 a^{3} b^{3} c^{4} d^{6}+378 a^{2} b^{4} c^{5} d^{5}+126 a \,b^{5} c^{6} d^{4}+12 b^{6} c^{7} d^{3}\right ) x^{10}+\left (\frac {1}{11} a^{6} d^{10}+\frac {60}{11} a^{5} b c \,d^{9}+\frac {675}{11} a^{4} b^{2} c^{2} d^{8}+\frac {2400}{11} a^{3} b^{3} c^{3} d^{7}+\frac {3150}{11} a^{2} b^{4} c^{4} d^{6}+\frac {1512}{11} a \,b^{5} c^{5} d^{5}+\frac {210}{11} b^{6} c^{6} d^{4}\right ) x^{11}+\left (\frac {1}{2} a^{5} b \,d^{10}+\frac {25}{2} a^{4} b^{2} c \,d^{9}+75 a^{3} b^{3} c^{2} d^{8}+150 a^{2} b^{4} c^{3} d^{7}+105 a \,b^{5} c^{4} d^{6}+21 b^{6} c^{5} d^{5}\right ) x^{12}+\left (\frac {15}{13} a^{4} b^{2} d^{10}+\frac {200}{13} a^{3} b^{3} c \,d^{9}+\frac {675}{13} a^{2} b^{4} c^{2} d^{8}+\frac {720}{13} a \,b^{5} c^{3} d^{7}+\frac {210}{13} b^{6} c^{4} d^{6}\right ) x^{13}+\left (\frac {10}{7} a^{3} b^{3} d^{10}+\frac {75}{7} a^{2} b^{4} c \,d^{9}+\frac {135}{7} a \,b^{5} c^{2} d^{8}+\frac {60}{7} b^{6} c^{3} d^{7}\right ) x^{14}+\left (a^{2} b^{4} d^{10}+4 a \,b^{5} c \,d^{9}+3 b^{6} c^{2} d^{8}\right ) x^{15}+\left (\frac {3}{8} a \,b^{5} d^{10}+\frac {5}{8} b^{6} c \,d^{9}\right ) x^{16}+\frac {b^{6} d^{10} x^{17}}{17}\)	\(975\)
default	\(\frac {b^{6} d^{10} x^{17}}{17}+\frac {\left (6 a \,b^{5} d^{10}+10 b^{6} c \,d^{9}\right ) x^{16}}{16}+\frac {\left (15 a^{2} b^{4} d^{10}+60 a \,b^{5} c \,d^{9}+45 b^{6} c^{2} d^{8}\right ) x^{15}}{15}+\frac {\left (20 a^{3} b^{3} d^{10}+150 a^{2} b^{4} c \,d^{9}+270 a \,b^{5} c^{2} d^{8}+120 b^{6} c^{3} d^{7}\right ) x^{14}}{14}+\frac {\left (15 a^{4} b^{2} d^{10}+200 a^{3} b^{3} c \,d^{9}+675 a^{2} b^{4} c^{2} d^{8}+720 a \,b^{5} c^{3} d^{7}+210 b^{6} c^{4} d^{6}\right ) x^{13}}{13}+\frac {\left (6 a^{5} b \,d^{10}+150 a^{4} b^{2} c \,d^{9}+900 a^{3} b^{3} c^{2} d^{8}+1800 a^{2} b^{4} c^{3} d^{7}+1260 a \,b^{5} c^{4} d^{6}+252 b^{6} c^{5} d^{5}\right ) x^{12}}{12}+\frac {\left (a^{6} d^{10}+60 a^{5} b c \,d^{9}+675 a^{4} b^{2} c^{2} d^{8}+2400 a^{3} b^{3} c^{3} d^{7}+3150 a^{2} b^{4} c^{4} d^{6}+1512 a \,b^{5} c^{5} d^{5}+210 b^{6} c^{6} d^{4}\right ) x^{11}}{11}+\frac {\left (10 a^{6} c \,d^{9}+270 a^{5} b \,c^{2} d^{8}+1800 a^{4} b^{2} c^{3} d^{7}+4200 a^{3} b^{3} c^{4} d^{6}+3780 a^{2} b^{4} c^{5} d^{5}+1260 a \,b^{5} c^{6} d^{4}+120 b^{6} c^{7} d^{3}\right ) x^{10}}{10}+\frac {\left (45 a^{6} c^{2} d^{8}+720 a^{5} b \,c^{3} d^{7}+3150 a^{4} b^{2} c^{4} d^{6}+5040 a^{3} b^{3} c^{5} d^{5}+3150 a^{2} b^{4} c^{6} d^{4}+720 a \,b^{5} c^{7} d^{3}+45 b^{6} c^{8} d^{2}\right ) x^{9}}{9}+\frac {\left (120 a^{6} c^{3} d^{7}+1260 a^{5} b \,c^{4} d^{6}+3780 a^{4} b^{2} c^{5} d^{5}+4200 a^{3} b^{3} c^{6} d^{4}+1800 a^{2} b^{4} c^{7} d^{3}+270 a \,b^{5} c^{8} d^{2}+10 b^{6} c^{9} d \right ) x^{8}}{8}+\frac {\left (210 a^{6} c^{4} d^{6}+1512 a^{5} b \,c^{5} d^{5}+3150 a^{4} b^{2} c^{6} d^{4}+2400 a^{3} b^{3} c^{7} d^{3}+675 a^{2} b^{4} c^{8} d^{2}+60 a \,b^{5} c^{9} d +b^{6} c^{10}\right ) x^{7}}{7}+\frac {\left (252 a^{6} c^{5} d^{5}+1260 a^{5} b \,c^{6} d^{4}+1800 a^{4} b^{2} c^{7} d^{3}+900 a^{3} b^{3} c^{8} d^{2}+150 a^{2} b^{4} c^{9} d +6 a \,b^{5} c^{10}\right ) x^{6}}{6}+\frac {\left (210 a^{6} c^{6} d^{4}+720 a^{5} b \,c^{7} d^{3}+675 a^{4} b^{2} c^{8} d^{2}+200 a^{3} b^{3} c^{9} d +15 a^{2} b^{4} c^{10}\right ) x^{5}}{5}+\frac {\left (120 a^{6} c^{7} d^{3}+270 a^{5} b \,c^{8} d^{2}+150 a^{4} b^{2} c^{9} d +20 a^{3} b^{3} c^{10}\right ) x^{4}}{4}+\frac {\left (45 a^{6} c^{8} d^{2}+60 a^{5} b \,c^{9} d +15 a^{4} b^{2} c^{10}\right ) x^{3}}{3}+\frac {\left (10 a^{6} c^{9} d +6 a^{5} b \,c^{10}\right ) x^{2}}{2}+a^{6} c^{10} x\)	\(991\)
gosper	\(\text {Expression too large to display}\)	\(1125\)
risch	\(\text {Expression too large to display}\)	\(1125\)
parallelrisch	\(\text {Expression too large to display}\)	\(1125\)
orering	\(\text {Expression too large to display}\)	\(1127\)

Input:

int((b*x+a)^6*(d*x+c)^10,x,method=_RETURNVERBOSE)
 

Output:

a^6*c^10*x+(5*a^6*c^9*d+3*a^5*b*c^10)*x^2+(15*a^6*c^8*d^2+20*a^5*b*c^9*d+5 
*a^4*b^2*c^10)*x^3+(30*a^6*c^7*d^3+135/2*a^5*b*c^8*d^2+75/2*a^4*b^2*c^9*d+ 
5*a^3*b^3*c^10)*x^4+(42*a^6*c^6*d^4+144*a^5*b*c^7*d^3+135*a^4*b^2*c^8*d^2+ 
40*a^3*b^3*c^9*d+3*a^2*b^4*c^10)*x^5+(42*a^6*c^5*d^5+210*a^5*b*c^6*d^4+300 
*a^4*b^2*c^7*d^3+150*a^3*b^3*c^8*d^2+25*a^2*b^4*c^9*d+a*b^5*c^10)*x^6+(30* 
a^6*c^4*d^6+216*a^5*b*c^5*d^5+450*a^4*b^2*c^6*d^4+2400/7*a^3*b^3*c^7*d^3+6 
75/7*a^2*b^4*c^8*d^2+60/7*a*b^5*c^9*d+1/7*b^6*c^10)*x^7+(15*a^6*c^3*d^7+31 
5/2*a^5*b*c^4*d^6+945/2*a^4*b^2*c^5*d^5+525*a^3*b^3*c^6*d^4+225*a^2*b^4*c^ 
7*d^3+135/4*a*b^5*c^8*d^2+5/4*b^6*c^9*d)*x^8+(5*a^6*c^2*d^8+80*a^5*b*c^3*d 
^7+350*a^4*b^2*c^4*d^6+560*a^3*b^3*c^5*d^5+350*a^2*b^4*c^6*d^4+80*a*b^5*c^ 
7*d^3+5*b^6*c^8*d^2)*x^9+(a^6*c*d^9+27*a^5*b*c^2*d^8+180*a^4*b^2*c^3*d^7+4 
20*a^3*b^3*c^4*d^6+378*a^2*b^4*c^5*d^5+126*a*b^5*c^6*d^4+12*b^6*c^7*d^3)*x 
^10+(1/11*a^6*d^10+60/11*a^5*b*c*d^9+675/11*a^4*b^2*c^2*d^8+2400/11*a^3*b^ 
3*c^3*d^7+3150/11*a^2*b^4*c^4*d^6+1512/11*a*b^5*c^5*d^5+210/11*b^6*c^6*d^4 
)*x^11+(1/2*a^5*b*d^10+25/2*a^4*b^2*c*d^9+75*a^3*b^3*c^2*d^8+150*a^2*b^4*c 
^3*d^7+105*a*b^5*c^4*d^6+21*b^6*c^5*d^5)*x^12+(15/13*a^4*b^2*d^10+200/13*a 
^3*b^3*c*d^9+675/13*a^2*b^4*c^2*d^8+720/13*a*b^5*c^3*d^7+210/13*b^6*c^4*d^ 
6)*x^13+(10/7*a^3*b^3*d^10+75/7*a^2*b^4*c*d^9+135/7*a*b^5*c^2*d^8+60/7*b^6 
*c^3*d^7)*x^14+(a^2*b^4*d^10+4*a*b^5*c*d^9+3*b^6*c^2*d^8)*x^15+(3/8*a*b^5* 
d^10+5/8*b^6*c*d^9)*x^16+1/17*b^6*d^10*x^17
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 977 vs. \(2 (158) = 316\).

Time = 0.08 (sec) , antiderivative size = 977, normalized size of antiderivative = 5.75 \[ \int (a+b x)^6 (c+d x)^{10} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(d*x+c)^10,x, algorithm="fricas")
 

Output:

1/17*b^6*d^10*x^17 + a^6*c^10*x + 1/8*(5*b^6*c*d^9 + 3*a*b^5*d^10)*x^16 + 
(3*b^6*c^2*d^8 + 4*a*b^5*c*d^9 + a^2*b^4*d^10)*x^15 + 5/7*(12*b^6*c^3*d^7 
+ 27*a*b^5*c^2*d^8 + 15*a^2*b^4*c*d^9 + 2*a^3*b^3*d^10)*x^14 + 5/13*(42*b^ 
6*c^4*d^6 + 144*a*b^5*c^3*d^7 + 135*a^2*b^4*c^2*d^8 + 40*a^3*b^3*c*d^9 + 3 
*a^4*b^2*d^10)*x^13 + 1/2*(42*b^6*c^5*d^5 + 210*a*b^5*c^4*d^6 + 300*a^2*b^ 
4*c^3*d^7 + 150*a^3*b^3*c^2*d^8 + 25*a^4*b^2*c*d^9 + a^5*b*d^10)*x^12 + 1/ 
11*(210*b^6*c^6*d^4 + 1512*a*b^5*c^5*d^5 + 3150*a^2*b^4*c^4*d^6 + 2400*a^3 
*b^3*c^3*d^7 + 675*a^4*b^2*c^2*d^8 + 60*a^5*b*c*d^9 + a^6*d^10)*x^11 + (12 
*b^6*c^7*d^3 + 126*a*b^5*c^6*d^4 + 378*a^2*b^4*c^5*d^5 + 420*a^3*b^3*c^4*d 
^6 + 180*a^4*b^2*c^3*d^7 + 27*a^5*b*c^2*d^8 + a^6*c*d^9)*x^10 + 5*(b^6*c^8 
*d^2 + 16*a*b^5*c^7*d^3 + 70*a^2*b^4*c^6*d^4 + 112*a^3*b^3*c^5*d^5 + 70*a^ 
4*b^2*c^4*d^6 + 16*a^5*b*c^3*d^7 + a^6*c^2*d^8)*x^9 + 5/4*(b^6*c^9*d + 27* 
a*b^5*c^8*d^2 + 180*a^2*b^4*c^7*d^3 + 420*a^3*b^3*c^6*d^4 + 378*a^4*b^2*c^ 
5*d^5 + 126*a^5*b*c^4*d^6 + 12*a^6*c^3*d^7)*x^8 + 1/7*(b^6*c^10 + 60*a*b^5 
*c^9*d + 675*a^2*b^4*c^8*d^2 + 2400*a^3*b^3*c^7*d^3 + 3150*a^4*b^2*c^6*d^4 
 + 1512*a^5*b*c^5*d^5 + 210*a^6*c^4*d^6)*x^7 + (a*b^5*c^10 + 25*a^2*b^4*c^ 
9*d + 150*a^3*b^3*c^8*d^2 + 300*a^4*b^2*c^7*d^3 + 210*a^5*b*c^6*d^4 + 42*a 
^6*c^5*d^5)*x^6 + (3*a^2*b^4*c^10 + 40*a^3*b^3*c^9*d + 135*a^4*b^2*c^8*d^2 
 + 144*a^5*b*c^7*d^3 + 42*a^6*c^6*d^4)*x^5 + 5/2*(2*a^3*b^3*c^10 + 15*a^4* 
b^2*c^9*d + 27*a^5*b*c^8*d^2 + 12*a^6*c^7*d^3)*x^4 + 5*(a^4*b^2*c^10 + ...
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1088 vs. \(2 (153) = 306\).

Time = 0.08 (sec) , antiderivative size = 1088, normalized size of antiderivative = 6.40 \[ \int (a+b x)^6 (c+d x)^{10} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)**6*(d*x+c)**10,x)
 

Output:

a**6*c**10*x + b**6*d**10*x**17/17 + x**16*(3*a*b**5*d**10/8 + 5*b**6*c*d* 
*9/8) + x**15*(a**2*b**4*d**10 + 4*a*b**5*c*d**9 + 3*b**6*c**2*d**8) + x** 
14*(10*a**3*b**3*d**10/7 + 75*a**2*b**4*c*d**9/7 + 135*a*b**5*c**2*d**8/7 
+ 60*b**6*c**3*d**7/7) + x**13*(15*a**4*b**2*d**10/13 + 200*a**3*b**3*c*d* 
*9/13 + 675*a**2*b**4*c**2*d**8/13 + 720*a*b**5*c**3*d**7/13 + 210*b**6*c* 
*4*d**6/13) + x**12*(a**5*b*d**10/2 + 25*a**4*b**2*c*d**9/2 + 75*a**3*b**3 
*c**2*d**8 + 150*a**2*b**4*c**3*d**7 + 105*a*b**5*c**4*d**6 + 21*b**6*c**5 
*d**5) + x**11*(a**6*d**10/11 + 60*a**5*b*c*d**9/11 + 675*a**4*b**2*c**2*d 
**8/11 + 2400*a**3*b**3*c**3*d**7/11 + 3150*a**2*b**4*c**4*d**6/11 + 1512* 
a*b**5*c**5*d**5/11 + 210*b**6*c**6*d**4/11) + x**10*(a**6*c*d**9 + 27*a** 
5*b*c**2*d**8 + 180*a**4*b**2*c**3*d**7 + 420*a**3*b**3*c**4*d**6 + 378*a* 
*2*b**4*c**5*d**5 + 126*a*b**5*c**6*d**4 + 12*b**6*c**7*d**3) + x**9*(5*a* 
*6*c**2*d**8 + 80*a**5*b*c**3*d**7 + 350*a**4*b**2*c**4*d**6 + 560*a**3*b* 
*3*c**5*d**5 + 350*a**2*b**4*c**6*d**4 + 80*a*b**5*c**7*d**3 + 5*b**6*c**8 
*d**2) + x**8*(15*a**6*c**3*d**7 + 315*a**5*b*c**4*d**6/2 + 945*a**4*b**2* 
c**5*d**5/2 + 525*a**3*b**3*c**6*d**4 + 225*a**2*b**4*c**7*d**3 + 135*a*b* 
*5*c**8*d**2/4 + 5*b**6*c**9*d/4) + x**7*(30*a**6*c**4*d**6 + 216*a**5*b*c 
**5*d**5 + 450*a**4*b**2*c**6*d**4 + 2400*a**3*b**3*c**7*d**3/7 + 675*a**2 
*b**4*c**8*d**2/7 + 60*a*b**5*c**9*d/7 + b**6*c**10/7) + x**6*(42*a**6*c** 
5*d**5 + 210*a**5*b*c**6*d**4 + 300*a**4*b**2*c**7*d**3 + 150*a**3*b**3...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 977 vs. \(2 (158) = 316\).

Time = 0.04 (sec) , antiderivative size = 977, normalized size of antiderivative = 5.75 \[ \int (a+b x)^6 (c+d x)^{10} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(d*x+c)^10,x, algorithm="maxima")
 

Output:

1/17*b^6*d^10*x^17 + a^6*c^10*x + 1/8*(5*b^6*c*d^9 + 3*a*b^5*d^10)*x^16 + 
(3*b^6*c^2*d^8 + 4*a*b^5*c*d^9 + a^2*b^4*d^10)*x^15 + 5/7*(12*b^6*c^3*d^7 
+ 27*a*b^5*c^2*d^8 + 15*a^2*b^4*c*d^9 + 2*a^3*b^3*d^10)*x^14 + 5/13*(42*b^ 
6*c^4*d^6 + 144*a*b^5*c^3*d^7 + 135*a^2*b^4*c^2*d^8 + 40*a^3*b^3*c*d^9 + 3 
*a^4*b^2*d^10)*x^13 + 1/2*(42*b^6*c^5*d^5 + 210*a*b^5*c^4*d^6 + 300*a^2*b^ 
4*c^3*d^7 + 150*a^3*b^3*c^2*d^8 + 25*a^4*b^2*c*d^9 + a^5*b*d^10)*x^12 + 1/ 
11*(210*b^6*c^6*d^4 + 1512*a*b^5*c^5*d^5 + 3150*a^2*b^4*c^4*d^6 + 2400*a^3 
*b^3*c^3*d^7 + 675*a^4*b^2*c^2*d^8 + 60*a^5*b*c*d^9 + a^6*d^10)*x^11 + (12 
*b^6*c^7*d^3 + 126*a*b^5*c^6*d^4 + 378*a^2*b^4*c^5*d^5 + 420*a^3*b^3*c^4*d 
^6 + 180*a^4*b^2*c^3*d^7 + 27*a^5*b*c^2*d^8 + a^6*c*d^9)*x^10 + 5*(b^6*c^8 
*d^2 + 16*a*b^5*c^7*d^3 + 70*a^2*b^4*c^6*d^4 + 112*a^3*b^3*c^5*d^5 + 70*a^ 
4*b^2*c^4*d^6 + 16*a^5*b*c^3*d^7 + a^6*c^2*d^8)*x^9 + 5/4*(b^6*c^9*d + 27* 
a*b^5*c^8*d^2 + 180*a^2*b^4*c^7*d^3 + 420*a^3*b^3*c^6*d^4 + 378*a^4*b^2*c^ 
5*d^5 + 126*a^5*b*c^4*d^6 + 12*a^6*c^3*d^7)*x^8 + 1/7*(b^6*c^10 + 60*a*b^5 
*c^9*d + 675*a^2*b^4*c^8*d^2 + 2400*a^3*b^3*c^7*d^3 + 3150*a^4*b^2*c^6*d^4 
 + 1512*a^5*b*c^5*d^5 + 210*a^6*c^4*d^6)*x^7 + (a*b^5*c^10 + 25*a^2*b^4*c^ 
9*d + 150*a^3*b^3*c^8*d^2 + 300*a^4*b^2*c^7*d^3 + 210*a^5*b*c^6*d^4 + 42*a 
^6*c^5*d^5)*x^6 + (3*a^2*b^4*c^10 + 40*a^3*b^3*c^9*d + 135*a^4*b^2*c^8*d^2 
 + 144*a^5*b*c^7*d^3 + 42*a^6*c^6*d^4)*x^5 + 5/2*(2*a^3*b^3*c^10 + 15*a^4* 
b^2*c^9*d + 27*a^5*b*c^8*d^2 + 12*a^6*c^7*d^3)*x^4 + 5*(a^4*b^2*c^10 + ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1124 vs. \(2 (158) = 316\).

Time = 0.13 (sec) , antiderivative size = 1124, normalized size of antiderivative = 6.61 \[ \int (a+b x)^6 (c+d x)^{10} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(d*x+c)^10,x, algorithm="giac")
 

Output:

1/17*b^6*d^10*x^17 + 5/8*b^6*c*d^9*x^16 + 3/8*a*b^5*d^10*x^16 + 3*b^6*c^2* 
d^8*x^15 + 4*a*b^5*c*d^9*x^15 + a^2*b^4*d^10*x^15 + 60/7*b^6*c^3*d^7*x^14 
+ 135/7*a*b^5*c^2*d^8*x^14 + 75/7*a^2*b^4*c*d^9*x^14 + 10/7*a^3*b^3*d^10*x 
^14 + 210/13*b^6*c^4*d^6*x^13 + 720/13*a*b^5*c^3*d^7*x^13 + 675/13*a^2*b^4 
*c^2*d^8*x^13 + 200/13*a^3*b^3*c*d^9*x^13 + 15/13*a^4*b^2*d^10*x^13 + 21*b 
^6*c^5*d^5*x^12 + 105*a*b^5*c^4*d^6*x^12 + 150*a^2*b^4*c^3*d^7*x^12 + 75*a 
^3*b^3*c^2*d^8*x^12 + 25/2*a^4*b^2*c*d^9*x^12 + 1/2*a^5*b*d^10*x^12 + 210/ 
11*b^6*c^6*d^4*x^11 + 1512/11*a*b^5*c^5*d^5*x^11 + 3150/11*a^2*b^4*c^4*d^6 
*x^11 + 2400/11*a^3*b^3*c^3*d^7*x^11 + 675/11*a^4*b^2*c^2*d^8*x^11 + 60/11 
*a^5*b*c*d^9*x^11 + 1/11*a^6*d^10*x^11 + 12*b^6*c^7*d^3*x^10 + 126*a*b^5*c 
^6*d^4*x^10 + 378*a^2*b^4*c^5*d^5*x^10 + 420*a^3*b^3*c^4*d^6*x^10 + 180*a^ 
4*b^2*c^3*d^7*x^10 + 27*a^5*b*c^2*d^8*x^10 + a^6*c*d^9*x^10 + 5*b^6*c^8*d^ 
2*x^9 + 80*a*b^5*c^7*d^3*x^9 + 350*a^2*b^4*c^6*d^4*x^9 + 560*a^3*b^3*c^5*d 
^5*x^9 + 350*a^4*b^2*c^4*d^6*x^9 + 80*a^5*b*c^3*d^7*x^9 + 5*a^6*c^2*d^8*x^ 
9 + 5/4*b^6*c^9*d*x^8 + 135/4*a*b^5*c^8*d^2*x^8 + 225*a^2*b^4*c^7*d^3*x^8 
+ 525*a^3*b^3*c^6*d^4*x^8 + 945/2*a^4*b^2*c^5*d^5*x^8 + 315/2*a^5*b*c^4*d^ 
6*x^8 + 15*a^6*c^3*d^7*x^8 + 1/7*b^6*c^10*x^7 + 60/7*a*b^5*c^9*d*x^7 + 675 
/7*a^2*b^4*c^8*d^2*x^7 + 2400/7*a^3*b^3*c^7*d^3*x^7 + 450*a^4*b^2*c^6*d^4* 
x^7 + 216*a^5*b*c^5*d^5*x^7 + 30*a^6*c^4*d^6*x^7 + a*b^5*c^10*x^6 + 25*a^2 
*b^4*c^9*d*x^6 + 150*a^3*b^3*c^8*d^2*x^6 + 300*a^4*b^2*c^7*d^3*x^6 + 21...
 

Mupad [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 953, normalized size of antiderivative = 5.61 \[ \int (a+b x)^6 (c+d x)^{10} \, dx =\text {Too large to display} \] Input:

int((a + b*x)^6*(c + d*x)^10,x)
 

Output:

x^7*((b^6*c^10)/7 + 30*a^6*c^4*d^6 + 216*a^5*b*c^5*d^5 + (675*a^2*b^4*c^8* 
d^2)/7 + (2400*a^3*b^3*c^7*d^3)/7 + 450*a^4*b^2*c^6*d^4 + (60*a*b^5*c^9*d) 
/7) + x^11*((a^6*d^10)/11 + (210*b^6*c^6*d^4)/11 + (1512*a*b^5*c^5*d^5)/11 
 + (3150*a^2*b^4*c^4*d^6)/11 + (2400*a^3*b^3*c^3*d^7)/11 + (675*a^4*b^2*c^ 
2*d^8)/11 + (60*a^5*b*c*d^9)/11) + x^9*(5*a^6*c^2*d^8 + 5*b^6*c^8*d^2 + 80 
*a*b^5*c^7*d^3 + 80*a^5*b*c^3*d^7 + 350*a^2*b^4*c^6*d^4 + 560*a^3*b^3*c^5* 
d^5 + 350*a^4*b^2*c^4*d^6) + x^5*(3*a^2*b^4*c^10 + 42*a^6*c^6*d^4 + 40*a^3 
*b^3*c^9*d + 144*a^5*b*c^7*d^3 + 135*a^4*b^2*c^8*d^2) + x^13*((15*a^4*b^2* 
d^10)/13 + (210*b^6*c^4*d^6)/13 + (720*a*b^5*c^3*d^7)/13 + (200*a^3*b^3*c* 
d^9)/13 + (675*a^2*b^4*c^2*d^8)/13) + x^6*(a*b^5*c^10 + 42*a^6*c^5*d^5 + 2 
5*a^2*b^4*c^9*d + 210*a^5*b*c^6*d^4 + 150*a^3*b^3*c^8*d^2 + 300*a^4*b^2*c^ 
7*d^3) + x^12*((a^5*b*d^10)/2 + 21*b^6*c^5*d^5 + 105*a*b^5*c^4*d^6 + (25*a 
^4*b^2*c*d^9)/2 + 150*a^2*b^4*c^3*d^7 + 75*a^3*b^3*c^2*d^8) + x^10*(a^6*c* 
d^9 + 12*b^6*c^7*d^3 + 126*a*b^5*c^6*d^4 + 27*a^5*b*c^2*d^8 + 378*a^2*b^4* 
c^5*d^5 + 420*a^3*b^3*c^4*d^6 + 180*a^4*b^2*c^3*d^7) + x^8*((5*b^6*c^9*d)/ 
4 + 15*a^6*c^3*d^7 + (135*a*b^5*c^8*d^2)/4 + (315*a^5*b*c^4*d^6)/2 + 225*a 
^2*b^4*c^7*d^3 + 525*a^3*b^3*c^6*d^4 + (945*a^4*b^2*c^5*d^5)/2) + a^6*c^10 
*x + (b^6*d^10*x^17)/17 + (5*a^3*c^7*x^4*(12*a^3*d^3 + 2*b^3*c^3 + 15*a*b^ 
2*c^2*d + 27*a^2*b*c*d^2))/2 + (5*b^3*d^7*x^14*(2*a^3*d^3 + 12*b^3*c^3 + 2 
7*a*b^2*c^2*d + 15*a^2*b*c*d^2))/7 + a^5*c^9*x^2*(5*a*d + 3*b*c) + (b^5...
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1126, normalized size of antiderivative = 6.62 \[ \int (a+b x)^6 (c+d x)^{10} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^6*(d*x+c)^10,x)
 

Output:

(x*(136136*a**6*c**10 + 680680*a**6*c**9*d*x + 2042040*a**6*c**8*d**2*x**2 
 + 4084080*a**6*c**7*d**3*x**3 + 5717712*a**6*c**6*d**4*x**4 + 5717712*a** 
6*c**5*d**5*x**5 + 4084080*a**6*c**4*d**6*x**6 + 2042040*a**6*c**3*d**7*x* 
*7 + 680680*a**6*c**2*d**8*x**8 + 136136*a**6*c*d**9*x**9 + 12376*a**6*d** 
10*x**10 + 408408*a**5*b*c**10*x + 2722720*a**5*b*c**9*d*x**2 + 9189180*a* 
*5*b*c**8*d**2*x**3 + 19603584*a**5*b*c**7*d**3*x**4 + 28588560*a**5*b*c** 
6*d**4*x**5 + 29405376*a**5*b*c**5*d**5*x**6 + 21441420*a**5*b*c**4*d**6*x 
**7 + 10890880*a**5*b*c**3*d**7*x**8 + 3675672*a**5*b*c**2*d**8*x**9 + 742 
560*a**5*b*c*d**9*x**10 + 68068*a**5*b*d**10*x**11 + 680680*a**4*b**2*c**1 
0*x**2 + 5105100*a**4*b**2*c**9*d*x**3 + 18378360*a**4*b**2*c**8*d**2*x**4 
 + 40840800*a**4*b**2*c**7*d**3*x**5 + 61261200*a**4*b**2*c**6*d**4*x**6 + 
 64324260*a**4*b**2*c**5*d**5*x**7 + 47647600*a**4*b**2*c**4*d**6*x**8 + 2 
4504480*a**4*b**2*c**3*d**7*x**9 + 8353800*a**4*b**2*c**2*d**8*x**10 + 170 
1700*a**4*b**2*c*d**9*x**11 + 157080*a**4*b**2*d**10*x**12 + 680680*a**3*b 
**3*c**10*x**3 + 5445440*a**3*b**3*c**9*d*x**4 + 20420400*a**3*b**3*c**8*d 
**2*x**5 + 46675200*a**3*b**3*c**7*d**3*x**6 + 71471400*a**3*b**3*c**6*d** 
4*x**7 + 76236160*a**3*b**3*c**5*d**5*x**8 + 57177120*a**3*b**3*c**4*d**6* 
x**9 + 29702400*a**3*b**3*c**3*d**7*x**10 + 10210200*a**3*b**3*c**2*d**8*x 
**11 + 2094400*a**3*b**3*c*d**9*x**12 + 194480*a**3*b**3*d**10*x**13 + 408 
408*a**2*b**4*c**10*x**4 + 3403400*a**2*b**4*c**9*d*x**5 + 13127400*a**...