\(\int (a+b x)^5 (c+d x)^7 \, dx\) [71]

Optimal result

Integrand size = 15, antiderivative size = 144 \[ \int (a+b x)^5 (c+d x)^7 \, dx=-\frac {(b c-a d)^5 (c+d x)^8}{8 d^6}+\frac {5 b (b c-a d)^4 (c+d x)^9}{9 d^6}-\frac {b^2 (b c-a d)^3 (c+d x)^{10}}{d^6}+\frac {10 b^3 (b c-a d)^2 (c+d x)^{11}}{11 d^6}-\frac {5 b^4 (b c-a d) (c+d x)^{12}}{12 d^6}+\frac {b^5 (c+d x)^{13}}{13 d^6} \] Output:

-1/8*(-a*d+b*c)^5*(d*x+c)^8/d^6+5/9*b*(-a*d+b*c)^4*(d*x+c)^9/d^6-b^2*(-a*d 
+b*c)^3*(d*x+c)^10/d^6+10/11*b^3*(-a*d+b*c)^2*(d*x+c)^11/d^6-5/12*b^4*(-a* 
d+b*c)*(d*x+c)^12/d^6+1/13*b^5*(d*x+c)^13/d^6
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(574\) vs. \(2(144)=288\).

Time = 0.03 (sec) , antiderivative size = 574, normalized size of antiderivative = 3.99 \[ \int (a+b x)^5 (c+d x)^7 \, dx=a^5 c^7 x+\frac {1}{2} a^4 c^6 (5 b c+7 a d) x^2+\frac {1}{3} a^3 c^5 \left (10 b^2 c^2+35 a b c d+21 a^2 d^2\right ) x^3+\frac {5}{4} a^2 c^4 \left (2 b^3 c^3+14 a b^2 c^2 d+21 a^2 b c d^2+7 a^3 d^3\right ) x^4+a c^3 \left (b^4 c^4+14 a b^3 c^3 d+42 a^2 b^2 c^2 d^2+35 a^3 b c d^3+7 a^4 d^4\right ) x^5+\frac {1}{6} c^2 \left (b^5 c^5+35 a b^4 c^4 d+210 a^2 b^3 c^3 d^2+350 a^3 b^2 c^2 d^3+175 a^4 b c d^4+21 a^5 d^5\right ) x^6+c d \left (b^5 c^5+15 a b^4 c^4 d+50 a^2 b^3 c^3 d^2+50 a^3 b^2 c^2 d^3+15 a^4 b c d^4+a^5 d^5\right ) x^7+\frac {1}{8} d^2 \left (21 b^5 c^5+175 a b^4 c^4 d+350 a^2 b^3 c^3 d^2+210 a^3 b^2 c^2 d^3+35 a^4 b c d^4+a^5 d^5\right ) x^8+\frac {5}{9} b d^3 \left (7 b^4 c^4+35 a b^3 c^3 d+42 a^2 b^2 c^2 d^2+14 a^3 b c d^3+a^4 d^4\right ) x^9+\frac {1}{2} b^2 d^4 \left (7 b^3 c^3+21 a b^2 c^2 d+14 a^2 b c d^2+2 a^3 d^3\right ) x^{10}+\frac {1}{11} b^3 d^5 \left (21 b^2 c^2+35 a b c d+10 a^2 d^2\right ) x^{11}+\frac {1}{12} b^4 d^6 (7 b c+5 a d) x^{12}+\frac {1}{13} b^5 d^7 x^{13} \] Input:

Integrate[(a + b*x)^5*(c + d*x)^7,x]
 

Output:

a^5*c^7*x + (a^4*c^6*(5*b*c + 7*a*d)*x^2)/2 + (a^3*c^5*(10*b^2*c^2 + 35*a* 
b*c*d + 21*a^2*d^2)*x^3)/3 + (5*a^2*c^4*(2*b^3*c^3 + 14*a*b^2*c^2*d + 21*a 
^2*b*c*d^2 + 7*a^3*d^3)*x^4)/4 + a*c^3*(b^4*c^4 + 14*a*b^3*c^3*d + 42*a^2* 
b^2*c^2*d^2 + 35*a^3*b*c*d^3 + 7*a^4*d^4)*x^5 + (c^2*(b^5*c^5 + 35*a*b^4*c 
^4*d + 210*a^2*b^3*c^3*d^2 + 350*a^3*b^2*c^2*d^3 + 175*a^4*b*c*d^4 + 21*a^ 
5*d^5)*x^6)/6 + c*d*(b^5*c^5 + 15*a*b^4*c^4*d + 50*a^2*b^3*c^3*d^2 + 50*a^ 
3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 + a^5*d^5)*x^7 + (d^2*(21*b^5*c^5 + 175*a*b 
^4*c^4*d + 350*a^2*b^3*c^3*d^2 + 210*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 + a^ 
5*d^5)*x^8)/8 + (5*b*d^3*(7*b^4*c^4 + 35*a*b^3*c^3*d + 42*a^2*b^2*c^2*d^2 
+ 14*a^3*b*c*d^3 + a^4*d^4)*x^9)/9 + (b^2*d^4*(7*b^3*c^3 + 21*a*b^2*c^2*d 
+ 14*a^2*b*c*d^2 + 2*a^3*d^3)*x^10)/2 + (b^3*d^5*(21*b^2*c^2 + 35*a*b*c*d 
+ 10*a^2*d^2)*x^11)/11 + (b^4*d^6*(7*b*c + 5*a*d)*x^12)/12 + (b^5*d^7*x^13 
)/13
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 144, 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.

\(\displaystyle \int (a+b x)^5 (c+d x)^7 \, dx\)

\(\Big \downarrow \) 49

\(\displaystyle \int \left (-\frac {5 b^4 (c+d x)^{11} (b c-a d)}{d^5}+\frac {10 b^3 (c+d x)^{10} (b c-a d)^2}{d^5}-\frac {10 b^2 (c+d x)^9 (b c-a d)^3}{d^5}+\frac {5 b (c+d x)^8 (b c-a d)^4}{d^5}+\frac {(c+d x)^7 (a d-b c)^5}{d^5}+\frac {b^5 (c+d x)^{12}}{d^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5 b^4 (c+d x)^{12} (b c-a d)}{12 d^6}+\frac {10 b^3 (c+d x)^{11} (b c-a d)^2}{11 d^6}-\frac {b^2 (c+d x)^{10} (b c-a d)^3}{d^6}+\frac {5 b (c+d x)^9 (b c-a d)^4}{9 d^6}-\frac {(c+d x)^8 (b c-a d)^5}{8 d^6}+\frac {b^5 (c+d x)^{13}}{13 d^6}\)

Input:

Int[(a + b*x)^5*(c + d*x)^7,x]
 

Output:

-1/8*((b*c - a*d)^5*(c + d*x)^8)/d^6 + (5*b*(b*c - a*d)^4*(c + d*x)^9)/(9* 
d^6) - (b^2*(b*c - a*d)^3*(c + d*x)^10)/d^6 + (10*b^3*(b*c - a*d)^2*(c + d 
*x)^11)/(11*d^6) - (5*b^4*(b*c - a*d)*(c + d*x)^12)/(12*d^6) + (b^5*(c + d 
*x)^13)/(13*d^6)
 

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. \(587\) vs. \(2(134)=268\).

Time = 0.11 (sec) , antiderivative size = 588, normalized size of antiderivative = 4.08

method	result	size
norman	\(\frac {b^{5} d^{7} x^{13}}{13}+\left (\frac {5}{12} a \,b^{4} d^{7}+\frac {7}{12} b^{5} c \,d^{6}\right ) x^{12}+\left (\frac {10}{11} a^{2} b^{3} d^{7}+\frac {35}{11} a \,b^{4} c \,d^{6}+\frac {21}{11} b^{5} c^{2} d^{5}\right ) x^{11}+\left (a^{3} b^{2} d^{7}+7 a^{2} b^{3} c \,d^{6}+\frac {21}{2} a \,b^{4} c^{2} d^{5}+\frac {7}{2} b^{5} c^{3} d^{4}\right ) x^{10}+\left (\frac {5}{9} a^{4} b \,d^{7}+\frac {70}{9} a^{3} b^{2} c \,d^{6}+\frac {70}{3} a^{2} b^{3} c^{2} d^{5}+\frac {175}{9} a \,b^{4} c^{3} d^{4}+\frac {35}{9} b^{5} c^{4} d^{3}\right ) x^{9}+\left (\frac {1}{8} a^{5} d^{7}+\frac {35}{8} a^{4} b c \,d^{6}+\frac {105}{4} a^{3} b^{2} c^{2} d^{5}+\frac {175}{4} a^{2} b^{3} c^{3} d^{4}+\frac {175}{8} a \,b^{4} c^{4} d^{3}+\frac {21}{8} b^{5} c^{5} d^{2}\right ) x^{8}+\left (a^{5} c \,d^{6}+15 a^{4} b \,c^{2} d^{5}+50 a^{3} b^{2} c^{3} d^{4}+50 a^{2} b^{3} c^{4} d^{3}+15 a \,b^{4} c^{5} d^{2}+b^{5} c^{6} d \right ) x^{7}+\left (\frac {7}{2} a^{5} c^{2} d^{5}+\frac {175}{6} a^{4} b \,c^{3} d^{4}+\frac {175}{3} a^{3} b^{2} c^{4} d^{3}+35 a^{2} b^{3} c^{5} d^{2}+\frac {35}{6} a \,b^{4} c^{6} d +\frac {1}{6} b^{5} c^{7}\right ) x^{6}+\left (7 a^{5} c^{3} d^{4}+35 a^{4} b \,c^{4} d^{3}+42 a^{3} b^{2} c^{5} d^{2}+14 a^{2} b^{3} c^{6} d +a \,b^{4} c^{7}\right ) x^{5}+\left (\frac {35}{4} a^{5} c^{4} d^{3}+\frac {105}{4} a^{4} b \,c^{5} d^{2}+\frac {35}{2} a^{3} b^{2} c^{6} d +\frac {5}{2} a^{2} b^{3} c^{7}\right ) x^{4}+\left (7 a^{5} c^{5} d^{2}+\frac {35}{3} a^{4} b \,c^{6} d +\frac {10}{3} a^{3} b^{2} c^{7}\right ) x^{3}+\left (\frac {7}{2} a^{5} c^{6} d +\frac {5}{2} a^{4} b \,c^{7}\right ) x^{2}+a^{5} c^{7} x\)	\(588\)
default	\(\frac {b^{5} d^{7} x^{13}}{13}+\frac {\left (5 a \,b^{4} d^{7}+7 b^{5} c \,d^{6}\right ) x^{12}}{12}+\frac {\left (10 a^{2} b^{3} d^{7}+35 a \,b^{4} c \,d^{6}+21 b^{5} c^{2} d^{5}\right ) x^{11}}{11}+\frac {\left (10 a^{3} b^{2} d^{7}+70 a^{2} b^{3} c \,d^{6}+105 a \,b^{4} c^{2} d^{5}+35 b^{5} c^{3} d^{4}\right ) x^{10}}{10}+\frac {\left (5 a^{4} b \,d^{7}+70 a^{3} b^{2} c \,d^{6}+210 a^{2} b^{3} c^{2} d^{5}+175 a \,b^{4} c^{3} d^{4}+35 b^{5} c^{4} d^{3}\right ) x^{9}}{9}+\frac {\left (a^{5} d^{7}+35 a^{4} b c \,d^{6}+210 a^{3} b^{2} c^{2} d^{5}+350 a^{2} b^{3} c^{3} d^{4}+175 a \,b^{4} c^{4} d^{3}+21 b^{5} c^{5} d^{2}\right ) x^{8}}{8}+\frac {\left (7 a^{5} c \,d^{6}+105 a^{4} b \,c^{2} d^{5}+350 a^{3} b^{2} c^{3} d^{4}+350 a^{2} b^{3} c^{4} d^{3}+105 a \,b^{4} c^{5} d^{2}+7 b^{5} c^{6} d \right ) x^{7}}{7}+\frac {\left (21 a^{5} c^{2} d^{5}+175 a^{4} b \,c^{3} d^{4}+350 a^{3} b^{2} c^{4} d^{3}+210 a^{2} b^{3} c^{5} d^{2}+35 a \,b^{4} c^{6} d +b^{5} c^{7}\right ) x^{6}}{6}+\frac {\left (35 a^{5} c^{3} d^{4}+175 a^{4} b \,c^{4} d^{3}+210 a^{3} b^{2} c^{5} d^{2}+70 a^{2} b^{3} c^{6} d +5 a \,b^{4} c^{7}\right ) x^{5}}{5}+\frac {\left (35 a^{5} c^{4} d^{3}+105 a^{4} b \,c^{5} d^{2}+70 a^{3} b^{2} c^{6} d +10 a^{2} b^{3} c^{7}\right ) x^{4}}{4}+\frac {\left (21 a^{5} c^{5} d^{2}+35 a^{4} b \,c^{6} d +10 a^{3} b^{2} c^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{5} c^{6} d +5 a^{4} b \,c^{7}\right ) x^{2}}{2}+a^{5} c^{7} x\)	\(601\)
gosper	\(\frac {5}{12} x^{12} a \,b^{4} d^{7}+\frac {7}{12} x^{12} b^{5} c \,d^{6}+\frac {10}{11} x^{11} a^{2} b^{3} d^{7}+\frac {21}{11} x^{11} b^{5} c^{2} d^{5}+x^{10} a^{3} b^{2} d^{7}+\frac {7}{2} x^{10} b^{5} c^{3} d^{4}+\frac {5}{9} x^{9} a^{4} b \,d^{7}+\frac {35}{9} x^{9} b^{5} c^{4} d^{3}+\frac {21}{8} x^{8} b^{5} c^{5} d^{2}+\frac {7}{2} x^{6} a^{5} c^{2} d^{5}+\frac {35}{4} x^{4} a^{5} c^{4} d^{3}+\frac {5}{2} x^{4} a^{2} b^{3} c^{7}+7 x^{3} a^{5} c^{5} d^{2}+\frac {10}{3} x^{3} a^{3} b^{2} c^{7}+\frac {7}{2} x^{2} a^{5} c^{6} d +\frac {5}{2} x^{2} a^{4} b \,c^{7}+a^{5} c \,d^{6} x^{7}+b^{5} c^{6} d \,x^{7}+7 a^{5} c^{3} d^{4} x^{5}+a \,b^{4} c^{7} x^{5}+42 a^{3} b^{2} c^{5} d^{2} x^{5}+14 a^{2} b^{3} c^{6} d \,x^{5}+35 a^{4} b \,c^{4} d^{3} x^{5}+15 a \,b^{4} c^{5} d^{2} x^{7}+50 a^{3} b^{2} c^{3} d^{4} x^{7}+50 a^{2} b^{3} c^{4} d^{3} x^{7}+\frac {35}{2} x^{4} a^{3} b^{2} c^{6} d +\frac {35}{3} x^{3} a^{4} b \,c^{6} d +15 a^{4} b \,c^{2} d^{5} x^{7}+\frac {1}{6} x^{6} b^{5} c^{7}+\frac {175}{9} x^{9} a \,b^{4} c^{3} d^{4}+\frac {35}{8} x^{8} a^{4} b c \,d^{6}+\frac {105}{4} x^{8} a^{3} b^{2} c^{2} d^{5}+\frac {175}{4} x^{8} a^{2} b^{3} c^{3} d^{4}+\frac {175}{8} x^{8} a \,b^{4} c^{4} d^{3}+\frac {175}{6} x^{6} a^{4} b \,c^{3} d^{4}+\frac {35}{11} x^{11} a \,b^{4} c \,d^{6}+7 x^{10} a^{2} b^{3} c \,d^{6}+\frac {21}{2} x^{10} a \,b^{4} c^{2} d^{5}+\frac {70}{9} x^{9} a^{3} b^{2} c \,d^{6}+\frac {70}{3} x^{9} a^{2} b^{3} c^{2} d^{5}+\frac {175}{3} x^{6} a^{3} b^{2} c^{4} d^{3}+35 x^{6} a^{2} b^{3} c^{5} d^{2}+\frac {35}{6} x^{6} a \,b^{4} c^{6} d +\frac {105}{4} x^{4} a^{4} b \,c^{5} d^{2}+\frac {1}{13} b^{5} d^{7} x^{13}+a^{5} c^{7} x +\frac {1}{8} x^{8} a^{5} d^{7}\)	\(671\)
risch	\(\frac {5}{12} x^{12} a \,b^{4} d^{7}+\frac {7}{12} x^{12} b^{5} c \,d^{6}+\frac {10}{11} x^{11} a^{2} b^{3} d^{7}+\frac {21}{11} x^{11} b^{5} c^{2} d^{5}+x^{10} a^{3} b^{2} d^{7}+\frac {7}{2} x^{10} b^{5} c^{3} d^{4}+\frac {5}{9} x^{9} a^{4} b \,d^{7}+\frac {35}{9} x^{9} b^{5} c^{4} d^{3}+\frac {21}{8} x^{8} b^{5} c^{5} d^{2}+\frac {7}{2} x^{6} a^{5} c^{2} d^{5}+\frac {35}{4} x^{4} a^{5} c^{4} d^{3}+\frac {5}{2} x^{4} a^{2} b^{3} c^{7}+7 x^{3} a^{5} c^{5} d^{2}+\frac {10}{3} x^{3} a^{3} b^{2} c^{7}+\frac {7}{2} x^{2} a^{5} c^{6} d +\frac {5}{2} x^{2} a^{4} b \,c^{7}+a^{5} c \,d^{6} x^{7}+b^{5} c^{6} d \,x^{7}+7 a^{5} c^{3} d^{4} x^{5}+a \,b^{4} c^{7} x^{5}+42 a^{3} b^{2} c^{5} d^{2} x^{5}+14 a^{2} b^{3} c^{6} d \,x^{5}+35 a^{4} b \,c^{4} d^{3} x^{5}+15 a \,b^{4} c^{5} d^{2} x^{7}+50 a^{3} b^{2} c^{3} d^{4} x^{7}+50 a^{2} b^{3} c^{4} d^{3} x^{7}+\frac {35}{2} x^{4} a^{3} b^{2} c^{6} d +\frac {35}{3} x^{3} a^{4} b \,c^{6} d +15 a^{4} b \,c^{2} d^{5} x^{7}+\frac {1}{6} x^{6} b^{5} c^{7}+\frac {175}{9} x^{9} a \,b^{4} c^{3} d^{4}+\frac {35}{8} x^{8} a^{4} b c \,d^{6}+\frac {105}{4} x^{8} a^{3} b^{2} c^{2} d^{5}+\frac {175}{4} x^{8} a^{2} b^{3} c^{3} d^{4}+\frac {175}{8} x^{8} a \,b^{4} c^{4} d^{3}+\frac {175}{6} x^{6} a^{4} b \,c^{3} d^{4}+\frac {35}{11} x^{11} a \,b^{4} c \,d^{6}+7 x^{10} a^{2} b^{3} c \,d^{6}+\frac {21}{2} x^{10} a \,b^{4} c^{2} d^{5}+\frac {70}{9} x^{9} a^{3} b^{2} c \,d^{6}+\frac {70}{3} x^{9} a^{2} b^{3} c^{2} d^{5}+\frac {175}{3} x^{6} a^{3} b^{2} c^{4} d^{3}+35 x^{6} a^{2} b^{3} c^{5} d^{2}+\frac {35}{6} x^{6} a \,b^{4} c^{6} d +\frac {105}{4} x^{4} a^{4} b \,c^{5} d^{2}+\frac {1}{13} b^{5} d^{7} x^{13}+a^{5} c^{7} x +\frac {1}{8} x^{8} a^{5} d^{7}\)	\(671\)
parallelrisch	\(\frac {5}{12} x^{12} a \,b^{4} d^{7}+\frac {7}{12} x^{12} b^{5} c \,d^{6}+\frac {10}{11} x^{11} a^{2} b^{3} d^{7}+\frac {21}{11} x^{11} b^{5} c^{2} d^{5}+x^{10} a^{3} b^{2} d^{7}+\frac {7}{2} x^{10} b^{5} c^{3} d^{4}+\frac {5}{9} x^{9} a^{4} b \,d^{7}+\frac {35}{9} x^{9} b^{5} c^{4} d^{3}+\frac {21}{8} x^{8} b^{5} c^{5} d^{2}+\frac {7}{2} x^{6} a^{5} c^{2} d^{5}+\frac {35}{4} x^{4} a^{5} c^{4} d^{3}+\frac {5}{2} x^{4} a^{2} b^{3} c^{7}+7 x^{3} a^{5} c^{5} d^{2}+\frac {10}{3} x^{3} a^{3} b^{2} c^{7}+\frac {7}{2} x^{2} a^{5} c^{6} d +\frac {5}{2} x^{2} a^{4} b \,c^{7}+a^{5} c \,d^{6} x^{7}+b^{5} c^{6} d \,x^{7}+7 a^{5} c^{3} d^{4} x^{5}+a \,b^{4} c^{7} x^{5}+42 a^{3} b^{2} c^{5} d^{2} x^{5}+14 a^{2} b^{3} c^{6} d \,x^{5}+35 a^{4} b \,c^{4} d^{3} x^{5}+15 a \,b^{4} c^{5} d^{2} x^{7}+50 a^{3} b^{2} c^{3} d^{4} x^{7}+50 a^{2} b^{3} c^{4} d^{3} x^{7}+\frac {35}{2} x^{4} a^{3} b^{2} c^{6} d +\frac {35}{3} x^{3} a^{4} b \,c^{6} d +15 a^{4} b \,c^{2} d^{5} x^{7}+\frac {1}{6} x^{6} b^{5} c^{7}+\frac {175}{9} x^{9} a \,b^{4} c^{3} d^{4}+\frac {35}{8} x^{8} a^{4} b c \,d^{6}+\frac {105}{4} x^{8} a^{3} b^{2} c^{2} d^{5}+\frac {175}{4} x^{8} a^{2} b^{3} c^{3} d^{4}+\frac {175}{8} x^{8} a \,b^{4} c^{4} d^{3}+\frac {175}{6} x^{6} a^{4} b \,c^{3} d^{4}+\frac {35}{11} x^{11} a \,b^{4} c \,d^{6}+7 x^{10} a^{2} b^{3} c \,d^{6}+\frac {21}{2} x^{10} a \,b^{4} c^{2} d^{5}+\frac {70}{9} x^{9} a^{3} b^{2} c \,d^{6}+\frac {70}{3} x^{9} a^{2} b^{3} c^{2} d^{5}+\frac {175}{3} x^{6} a^{3} b^{2} c^{4} d^{3}+35 x^{6} a^{2} b^{3} c^{5} d^{2}+\frac {35}{6} x^{6} a \,b^{4} c^{6} d +\frac {105}{4} x^{4} a^{4} b \,c^{5} d^{2}+\frac {1}{13} b^{5} d^{7} x^{13}+a^{5} c^{7} x +\frac {1}{8} x^{8} a^{5} d^{7}\)	\(671\)
orering	\(\frac {x \left (792 b^{5} d^{7} x^{12}+4290 a \,b^{4} d^{7} x^{11}+6006 b^{5} c \,d^{6} x^{11}+9360 a^{2} b^{3} d^{7} x^{10}+32760 a \,b^{4} c \,d^{6} x^{10}+19656 b^{5} c^{2} d^{5} x^{10}+10296 a^{3} b^{2} d^{7} x^{9}+72072 a^{2} b^{3} c \,d^{6} x^{9}+108108 a \,b^{4} c^{2} d^{5} x^{9}+36036 b^{5} c^{3} d^{4} x^{9}+5720 a^{4} b \,d^{7} x^{8}+80080 a^{3} b^{2} c \,d^{6} x^{8}+240240 a^{2} b^{3} c^{2} d^{5} x^{8}+200200 a \,b^{4} c^{3} d^{4} x^{8}+40040 b^{5} c^{4} d^{3} x^{8}+1287 a^{5} d^{7} x^{7}+45045 a^{4} b c \,d^{6} x^{7}+270270 a^{3} b^{2} c^{2} d^{5} x^{7}+450450 a^{2} b^{3} c^{3} d^{4} x^{7}+225225 a \,b^{4} c^{4} d^{3} x^{7}+27027 b^{5} c^{5} d^{2} x^{7}+10296 a^{5} c \,d^{6} x^{6}+154440 a^{4} b \,c^{2} d^{5} x^{6}+514800 a^{3} b^{2} c^{3} d^{4} x^{6}+514800 a^{2} b^{3} c^{4} d^{3} x^{6}+154440 a \,b^{4} c^{5} d^{2} x^{6}+10296 b^{5} c^{6} d \,x^{6}+36036 a^{5} c^{2} d^{5} x^{5}+300300 a^{4} b \,c^{3} d^{4} x^{5}+600600 a^{3} b^{2} c^{4} d^{3} x^{5}+360360 a^{2} b^{3} c^{5} d^{2} x^{5}+60060 a \,b^{4} c^{6} d \,x^{5}+1716 b^{5} c^{7} x^{5}+72072 a^{5} c^{3} d^{4} x^{4}+360360 a^{4} b \,c^{4} d^{3} x^{4}+432432 a^{3} b^{2} c^{5} d^{2} x^{4}+144144 a^{2} b^{3} c^{6} d \,x^{4}+10296 a \,b^{4} c^{7} x^{4}+90090 a^{5} c^{4} d^{3} x^{3}+270270 a^{4} b \,c^{5} d^{2} x^{3}+180180 a^{3} b^{2} c^{6} d \,x^{3}+25740 a^{2} b^{3} c^{7} x^{3}+72072 a^{5} c^{5} d^{2} x^{2}+120120 a^{4} b \,c^{6} d \,x^{2}+34320 a^{3} b^{2} c^{7} x^{2}+36036 a^{5} c^{6} d x +25740 a^{4} b \,c^{7} x +10296 a^{5} c^{7}\right )}{10296}\)	\(674\)

Input:

int((b*x+a)^5*(d*x+c)^7,x,method=_RETURNVERBOSE)
 

Output:

1/13*b^5*d^7*x^13+(5/12*a*b^4*d^7+7/12*b^5*c*d^6)*x^12+(10/11*a^2*b^3*d^7+ 
35/11*a*b^4*c*d^6+21/11*b^5*c^2*d^5)*x^11+(a^3*b^2*d^7+7*a^2*b^3*c*d^6+21/ 
2*a*b^4*c^2*d^5+7/2*b^5*c^3*d^4)*x^10+(5/9*a^4*b*d^7+70/9*a^3*b^2*c*d^6+70 
/3*a^2*b^3*c^2*d^5+175/9*a*b^4*c^3*d^4+35/9*b^5*c^4*d^3)*x^9+(1/8*a^5*d^7+ 
35/8*a^4*b*c*d^6+105/4*a^3*b^2*c^2*d^5+175/4*a^2*b^3*c^3*d^4+175/8*a*b^4*c 
^4*d^3+21/8*b^5*c^5*d^2)*x^8+(a^5*c*d^6+15*a^4*b*c^2*d^5+50*a^3*b^2*c^3*d^ 
4+50*a^2*b^3*c^4*d^3+15*a*b^4*c^5*d^2+b^5*c^6*d)*x^7+(7/2*a^5*c^2*d^5+175/ 
6*a^4*b*c^3*d^4+175/3*a^3*b^2*c^4*d^3+35*a^2*b^3*c^5*d^2+35/6*a*b^4*c^6*d+ 
1/6*b^5*c^7)*x^6+(7*a^5*c^3*d^4+35*a^4*b*c^4*d^3+42*a^3*b^2*c^5*d^2+14*a^2 
*b^3*c^6*d+a*b^4*c^7)*x^5+(35/4*a^5*c^4*d^3+105/4*a^4*b*c^5*d^2+35/2*a^3*b 
^2*c^6*d+5/2*a^2*b^3*c^7)*x^4+(7*a^5*c^5*d^2+35/3*a^4*b*c^6*d+10/3*a^3*b^2 
*c^7)*x^3+(7/2*a^5*c^6*d+5/2*a^4*b*c^7)*x^2+a^5*c^7*x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (134) = 268\).

Time = 0.08 (sec) , antiderivative size = 594, normalized size of antiderivative = 4.12 \[ \int (a+b x)^5 (c+d x)^7 \, dx=\frac {1}{13} \, b^{5} d^{7} x^{13} + a^{5} c^{7} x + \frac {1}{12} \, {\left (7 \, b^{5} c d^{6} + 5 \, a b^{4} d^{7}\right )} x^{12} + \frac {1}{11} \, {\left (21 \, b^{5} c^{2} d^{5} + 35 \, a b^{4} c d^{6} + 10 \, a^{2} b^{3} d^{7}\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{5} c^{3} d^{4} + 21 \, a b^{4} c^{2} d^{5} + 14 \, a^{2} b^{3} c d^{6} + 2 \, a^{3} b^{2} d^{7}\right )} x^{10} + \frac {5}{9} \, {\left (7 \, b^{5} c^{4} d^{3} + 35 \, a b^{4} c^{3} d^{4} + 42 \, a^{2} b^{3} c^{2} d^{5} + 14 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (21 \, b^{5} c^{5} d^{2} + 175 \, a b^{4} c^{4} d^{3} + 350 \, a^{2} b^{3} c^{3} d^{4} + 210 \, a^{3} b^{2} c^{2} d^{5} + 35 \, a^{4} b c d^{6} + a^{5} d^{7}\right )} x^{8} + {\left (b^{5} c^{6} d + 15 \, a b^{4} c^{5} d^{2} + 50 \, a^{2} b^{3} c^{4} d^{3} + 50 \, a^{3} b^{2} c^{3} d^{4} + 15 \, a^{4} b c^{2} d^{5} + a^{5} c d^{6}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{7} + 35 \, a b^{4} c^{6} d + 210 \, a^{2} b^{3} c^{5} d^{2} + 350 \, a^{3} b^{2} c^{4} d^{3} + 175 \, a^{4} b c^{3} d^{4} + 21 \, a^{5} c^{2} d^{5}\right )} x^{6} + {\left (a b^{4} c^{7} + 14 \, a^{2} b^{3} c^{6} d + 42 \, a^{3} b^{2} c^{5} d^{2} + 35 \, a^{4} b c^{4} d^{3} + 7 \, a^{5} c^{3} d^{4}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} c^{7} + 14 \, a^{3} b^{2} c^{6} d + 21 \, a^{4} b c^{5} d^{2} + 7 \, a^{5} c^{4} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} c^{7} + 35 \, a^{4} b c^{6} d + 21 \, a^{5} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b c^{7} + 7 \, a^{5} c^{6} d\right )} x^{2} \] Input:

integrate((b*x+a)^5*(d*x+c)^7,x, algorithm="fricas")
 

Output:

1/13*b^5*d^7*x^13 + a^5*c^7*x + 1/12*(7*b^5*c*d^6 + 5*a*b^4*d^7)*x^12 + 1/ 
11*(21*b^5*c^2*d^5 + 35*a*b^4*c*d^6 + 10*a^2*b^3*d^7)*x^11 + 1/2*(7*b^5*c^ 
3*d^4 + 21*a*b^4*c^2*d^5 + 14*a^2*b^3*c*d^6 + 2*a^3*b^2*d^7)*x^10 + 5/9*(7 
*b^5*c^4*d^3 + 35*a*b^4*c^3*d^4 + 42*a^2*b^3*c^2*d^5 + 14*a^3*b^2*c*d^6 + 
a^4*b*d^7)*x^9 + 1/8*(21*b^5*c^5*d^2 + 175*a*b^4*c^4*d^3 + 350*a^2*b^3*c^3 
*d^4 + 210*a^3*b^2*c^2*d^5 + 35*a^4*b*c*d^6 + a^5*d^7)*x^8 + (b^5*c^6*d + 
15*a*b^4*c^5*d^2 + 50*a^2*b^3*c^4*d^3 + 50*a^3*b^2*c^3*d^4 + 15*a^4*b*c^2* 
d^5 + a^5*c*d^6)*x^7 + 1/6*(b^5*c^7 + 35*a*b^4*c^6*d + 210*a^2*b^3*c^5*d^2 
 + 350*a^3*b^2*c^4*d^3 + 175*a^4*b*c^3*d^4 + 21*a^5*c^2*d^5)*x^6 + (a*b^4* 
c^7 + 14*a^2*b^3*c^6*d + 42*a^3*b^2*c^5*d^2 + 35*a^4*b*c^4*d^3 + 7*a^5*c^3 
*d^4)*x^5 + 5/4*(2*a^2*b^3*c^7 + 14*a^3*b^2*c^6*d + 21*a^4*b*c^5*d^2 + 7*a 
^5*c^4*d^3)*x^4 + 1/3*(10*a^3*b^2*c^7 + 35*a^4*b*c^6*d + 21*a^5*c^5*d^2)*x 
^3 + 1/2*(5*a^4*b*c^7 + 7*a^5*c^6*d)*x^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (129) = 258\).

Time = 0.05 (sec) , antiderivative size = 673, normalized size of antiderivative = 4.67 \[ \int (a+b x)^5 (c+d x)^7 \, dx=a^{5} c^{7} x + \frac {b^{5} d^{7} x^{13}}{13} + x^{12} \cdot \left (\frac {5 a b^{4} d^{7}}{12} + \frac {7 b^{5} c d^{6}}{12}\right ) + x^{11} \cdot \left (\frac {10 a^{2} b^{3} d^{7}}{11} + \frac {35 a b^{4} c d^{6}}{11} + \frac {21 b^{5} c^{2} d^{5}}{11}\right ) + x^{10} \left (a^{3} b^{2} d^{7} + 7 a^{2} b^{3} c d^{6} + \frac {21 a b^{4} c^{2} d^{5}}{2} + \frac {7 b^{5} c^{3} d^{4}}{2}\right ) + x^{9} \cdot \left (\frac {5 a^{4} b d^{7}}{9} + \frac {70 a^{3} b^{2} c d^{6}}{9} + \frac {70 a^{2} b^{3} c^{2} d^{5}}{3} + \frac {175 a b^{4} c^{3} d^{4}}{9} + \frac {35 b^{5} c^{4} d^{3}}{9}\right ) + x^{8} \left (\frac {a^{5} d^{7}}{8} + \frac {35 a^{4} b c d^{6}}{8} + \frac {105 a^{3} b^{2} c^{2} d^{5}}{4} + \frac {175 a^{2} b^{3} c^{3} d^{4}}{4} + \frac {175 a b^{4} c^{4} d^{3}}{8} + \frac {21 b^{5} c^{5} d^{2}}{8}\right ) + x^{7} \left (a^{5} c d^{6} + 15 a^{4} b c^{2} d^{5} + 50 a^{3} b^{2} c^{3} d^{4} + 50 a^{2} b^{3} c^{4} d^{3} + 15 a b^{4} c^{5} d^{2} + b^{5} c^{6} d\right ) + x^{6} \cdot \left (\frac {7 a^{5} c^{2} d^{5}}{2} + \frac {175 a^{4} b c^{3} d^{4}}{6} + \frac {175 a^{3} b^{2} c^{4} d^{3}}{3} + 35 a^{2} b^{3} c^{5} d^{2} + \frac {35 a b^{4} c^{6} d}{6} + \frac {b^{5} c^{7}}{6}\right ) + x^{5} \cdot \left (7 a^{5} c^{3} d^{4} + 35 a^{4} b c^{4} d^{3} + 42 a^{3} b^{2} c^{5} d^{2} + 14 a^{2} b^{3} c^{6} d + a b^{4} c^{7}\right ) + x^{4} \cdot \left (\frac {35 a^{5} c^{4} d^{3}}{4} + \frac {105 a^{4} b c^{5} d^{2}}{4} + \frac {35 a^{3} b^{2} c^{6} d}{2} + \frac {5 a^{2} b^{3} c^{7}}{2}\right ) + x^{3} \cdot \left (7 a^{5} c^{5} d^{2} + \frac {35 a^{4} b c^{6} d}{3} + \frac {10 a^{3} b^{2} c^{7}}{3}\right ) + x^{2} \cdot \left (\frac {7 a^{5} c^{6} d}{2} + \frac {5 a^{4} b c^{7}}{2}\right ) \] Input:

integrate((b*x+a)**5*(d*x+c)**7,x)
 

Output:

a**5*c**7*x + b**5*d**7*x**13/13 + x**12*(5*a*b**4*d**7/12 + 7*b**5*c*d**6 
/12) + x**11*(10*a**2*b**3*d**7/11 + 35*a*b**4*c*d**6/11 + 21*b**5*c**2*d* 
*5/11) + x**10*(a**3*b**2*d**7 + 7*a**2*b**3*c*d**6 + 21*a*b**4*c**2*d**5/ 
2 + 7*b**5*c**3*d**4/2) + x**9*(5*a**4*b*d**7/9 + 70*a**3*b**2*c*d**6/9 + 
70*a**2*b**3*c**2*d**5/3 + 175*a*b**4*c**3*d**4/9 + 35*b**5*c**4*d**3/9) + 
 x**8*(a**5*d**7/8 + 35*a**4*b*c*d**6/8 + 105*a**3*b**2*c**2*d**5/4 + 175* 
a**2*b**3*c**3*d**4/4 + 175*a*b**4*c**4*d**3/8 + 21*b**5*c**5*d**2/8) + x* 
*7*(a**5*c*d**6 + 15*a**4*b*c**2*d**5 + 50*a**3*b**2*c**3*d**4 + 50*a**2*b 
**3*c**4*d**3 + 15*a*b**4*c**5*d**2 + b**5*c**6*d) + x**6*(7*a**5*c**2*d** 
5/2 + 175*a**4*b*c**3*d**4/6 + 175*a**3*b**2*c**4*d**3/3 + 35*a**2*b**3*c* 
*5*d**2 + 35*a*b**4*c**6*d/6 + b**5*c**7/6) + x**5*(7*a**5*c**3*d**4 + 35* 
a**4*b*c**4*d**3 + 42*a**3*b**2*c**5*d**2 + 14*a**2*b**3*c**6*d + a*b**4*c 
**7) + x**4*(35*a**5*c**4*d**3/4 + 105*a**4*b*c**5*d**2/4 + 35*a**3*b**2*c 
**6*d/2 + 5*a**2*b**3*c**7/2) + x**3*(7*a**5*c**5*d**2 + 35*a**4*b*c**6*d/ 
3 + 10*a**3*b**2*c**7/3) + x**2*(7*a**5*c**6*d/2 + 5*a**4*b*c**7/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (134) = 268\).

Time = 0.04 (sec) , antiderivative size = 594, normalized size of antiderivative = 4.12 \[ \int (a+b x)^5 (c+d x)^7 \, dx=\frac {1}{13} \, b^{5} d^{7} x^{13} + a^{5} c^{7} x + \frac {1}{12} \, {\left (7 \, b^{5} c d^{6} + 5 \, a b^{4} d^{7}\right )} x^{12} + \frac {1}{11} \, {\left (21 \, b^{5} c^{2} d^{5} + 35 \, a b^{4} c d^{6} + 10 \, a^{2} b^{3} d^{7}\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{5} c^{3} d^{4} + 21 \, a b^{4} c^{2} d^{5} + 14 \, a^{2} b^{3} c d^{6} + 2 \, a^{3} b^{2} d^{7}\right )} x^{10} + \frac {5}{9} \, {\left (7 \, b^{5} c^{4} d^{3} + 35 \, a b^{4} c^{3} d^{4} + 42 \, a^{2} b^{3} c^{2} d^{5} + 14 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (21 \, b^{5} c^{5} d^{2} + 175 \, a b^{4} c^{4} d^{3} + 350 \, a^{2} b^{3} c^{3} d^{4} + 210 \, a^{3} b^{2} c^{2} d^{5} + 35 \, a^{4} b c d^{6} + a^{5} d^{7}\right )} x^{8} + {\left (b^{5} c^{6} d + 15 \, a b^{4} c^{5} d^{2} + 50 \, a^{2} b^{3} c^{4} d^{3} + 50 \, a^{3} b^{2} c^{3} d^{4} + 15 \, a^{4} b c^{2} d^{5} + a^{5} c d^{6}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{7} + 35 \, a b^{4} c^{6} d + 210 \, a^{2} b^{3} c^{5} d^{2} + 350 \, a^{3} b^{2} c^{4} d^{3} + 175 \, a^{4} b c^{3} d^{4} + 21 \, a^{5} c^{2} d^{5}\right )} x^{6} + {\left (a b^{4} c^{7} + 14 \, a^{2} b^{3} c^{6} d + 42 \, a^{3} b^{2} c^{5} d^{2} + 35 \, a^{4} b c^{4} d^{3} + 7 \, a^{5} c^{3} d^{4}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} c^{7} + 14 \, a^{3} b^{2} c^{6} d + 21 \, a^{4} b c^{5} d^{2} + 7 \, a^{5} c^{4} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} c^{7} + 35 \, a^{4} b c^{6} d + 21 \, a^{5} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b c^{7} + 7 \, a^{5} c^{6} d\right )} x^{2} \] Input:

integrate((b*x+a)^5*(d*x+c)^7,x, algorithm="maxima")
 

Output:

1/13*b^5*d^7*x^13 + a^5*c^7*x + 1/12*(7*b^5*c*d^6 + 5*a*b^4*d^7)*x^12 + 1/ 
11*(21*b^5*c^2*d^5 + 35*a*b^4*c*d^6 + 10*a^2*b^3*d^7)*x^11 + 1/2*(7*b^5*c^ 
3*d^4 + 21*a*b^4*c^2*d^5 + 14*a^2*b^3*c*d^6 + 2*a^3*b^2*d^7)*x^10 + 5/9*(7 
*b^5*c^4*d^3 + 35*a*b^4*c^3*d^4 + 42*a^2*b^3*c^2*d^5 + 14*a^3*b^2*c*d^6 + 
a^4*b*d^7)*x^9 + 1/8*(21*b^5*c^5*d^2 + 175*a*b^4*c^4*d^3 + 350*a^2*b^3*c^3 
*d^4 + 210*a^3*b^2*c^2*d^5 + 35*a^4*b*c*d^6 + a^5*d^7)*x^8 + (b^5*c^6*d + 
15*a*b^4*c^5*d^2 + 50*a^2*b^3*c^4*d^3 + 50*a^3*b^2*c^3*d^4 + 15*a^4*b*c^2* 
d^5 + a^5*c*d^6)*x^7 + 1/6*(b^5*c^7 + 35*a*b^4*c^6*d + 210*a^2*b^3*c^5*d^2 
 + 350*a^3*b^2*c^4*d^3 + 175*a^4*b*c^3*d^4 + 21*a^5*c^2*d^5)*x^6 + (a*b^4* 
c^7 + 14*a^2*b^3*c^6*d + 42*a^3*b^2*c^5*d^2 + 35*a^4*b*c^4*d^3 + 7*a^5*c^3 
*d^4)*x^5 + 5/4*(2*a^2*b^3*c^7 + 14*a^3*b^2*c^6*d + 21*a^4*b*c^5*d^2 + 7*a 
^5*c^4*d^3)*x^4 + 1/3*(10*a^3*b^2*c^7 + 35*a^4*b*c^6*d + 21*a^5*c^5*d^2)*x 
^3 + 1/2*(5*a^4*b*c^7 + 7*a^5*c^6*d)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 670 vs. \(2 (134) = 268\).

Time = 0.11 (sec) , antiderivative size = 670, normalized size of antiderivative = 4.65 \[ \int (a+b x)^5 (c+d x)^7 \, dx=\frac {1}{13} \, b^{5} d^{7} x^{13} + \frac {7}{12} \, b^{5} c d^{6} x^{12} + \frac {5}{12} \, a b^{4} d^{7} x^{12} + \frac {21}{11} \, b^{5} c^{2} d^{5} x^{11} + \frac {35}{11} \, a b^{4} c d^{6} x^{11} + \frac {10}{11} \, a^{2} b^{3} d^{7} x^{11} + \frac {7}{2} \, b^{5} c^{3} d^{4} x^{10} + \frac {21}{2} \, a b^{4} c^{2} d^{5} x^{10} + 7 \, a^{2} b^{3} c d^{6} x^{10} + a^{3} b^{2} d^{7} x^{10} + \frac {35}{9} \, b^{5} c^{4} d^{3} x^{9} + \frac {175}{9} \, a b^{4} c^{3} d^{4} x^{9} + \frac {70}{3} \, a^{2} b^{3} c^{2} d^{5} x^{9} + \frac {70}{9} \, a^{3} b^{2} c d^{6} x^{9} + \frac {5}{9} \, a^{4} b d^{7} x^{9} + \frac {21}{8} \, b^{5} c^{5} d^{2} x^{8} + \frac {175}{8} \, a b^{4} c^{4} d^{3} x^{8} + \frac {175}{4} \, a^{2} b^{3} c^{3} d^{4} x^{8} + \frac {105}{4} \, a^{3} b^{2} c^{2} d^{5} x^{8} + \frac {35}{8} \, a^{4} b c d^{6} x^{8} + \frac {1}{8} \, a^{5} d^{7} x^{8} + b^{5} c^{6} d x^{7} + 15 \, a b^{4} c^{5} d^{2} x^{7} + 50 \, a^{2} b^{3} c^{4} d^{3} x^{7} + 50 \, a^{3} b^{2} c^{3} d^{4} x^{7} + 15 \, a^{4} b c^{2} d^{5} x^{7} + a^{5} c d^{6} x^{7} + \frac {1}{6} \, b^{5} c^{7} x^{6} + \frac {35}{6} \, a b^{4} c^{6} d x^{6} + 35 \, a^{2} b^{3} c^{5} d^{2} x^{6} + \frac {175}{3} \, a^{3} b^{2} c^{4} d^{3} x^{6} + \frac {175}{6} \, a^{4} b c^{3} d^{4} x^{6} + \frac {7}{2} \, a^{5} c^{2} d^{5} x^{6} + a b^{4} c^{7} x^{5} + 14 \, a^{2} b^{3} c^{6} d x^{5} + 42 \, a^{3} b^{2} c^{5} d^{2} x^{5} + 35 \, a^{4} b c^{4} d^{3} x^{5} + 7 \, a^{5} c^{3} d^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} c^{7} x^{4} + \frac {35}{2} \, a^{3} b^{2} c^{6} d x^{4} + \frac {105}{4} \, a^{4} b c^{5} d^{2} x^{4} + \frac {35}{4} \, a^{5} c^{4} d^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} c^{7} x^{3} + \frac {35}{3} \, a^{4} b c^{6} d x^{3} + 7 \, a^{5} c^{5} d^{2} x^{3} + \frac {5}{2} \, a^{4} b c^{7} x^{2} + \frac {7}{2} \, a^{5} c^{6} d x^{2} + a^{5} c^{7} x \] Input:

integrate((b*x+a)^5*(d*x+c)^7,x, algorithm="giac")
 

Output:

1/13*b^5*d^7*x^13 + 7/12*b^5*c*d^6*x^12 + 5/12*a*b^4*d^7*x^12 + 21/11*b^5* 
c^2*d^5*x^11 + 35/11*a*b^4*c*d^6*x^11 + 10/11*a^2*b^3*d^7*x^11 + 7/2*b^5*c 
^3*d^4*x^10 + 21/2*a*b^4*c^2*d^5*x^10 + 7*a^2*b^3*c*d^6*x^10 + a^3*b^2*d^7 
*x^10 + 35/9*b^5*c^4*d^3*x^9 + 175/9*a*b^4*c^3*d^4*x^9 + 70/3*a^2*b^3*c^2* 
d^5*x^9 + 70/9*a^3*b^2*c*d^6*x^9 + 5/9*a^4*b*d^7*x^9 + 21/8*b^5*c^5*d^2*x^ 
8 + 175/8*a*b^4*c^4*d^3*x^8 + 175/4*a^2*b^3*c^3*d^4*x^8 + 105/4*a^3*b^2*c^ 
2*d^5*x^8 + 35/8*a^4*b*c*d^6*x^8 + 1/8*a^5*d^7*x^8 + b^5*c^6*d*x^7 + 15*a* 
b^4*c^5*d^2*x^7 + 50*a^2*b^3*c^4*d^3*x^7 + 50*a^3*b^2*c^3*d^4*x^7 + 15*a^4 
*b*c^2*d^5*x^7 + a^5*c*d^6*x^7 + 1/6*b^5*c^7*x^6 + 35/6*a*b^4*c^6*d*x^6 + 
35*a^2*b^3*c^5*d^2*x^6 + 175/3*a^3*b^2*c^4*d^3*x^6 + 175/6*a^4*b*c^3*d^4*x 
^6 + 7/2*a^5*c^2*d^5*x^6 + a*b^4*c^7*x^5 + 14*a^2*b^3*c^6*d*x^5 + 42*a^3*b 
^2*c^5*d^2*x^5 + 35*a^4*b*c^4*d^3*x^5 + 7*a^5*c^3*d^4*x^5 + 5/2*a^2*b^3*c^ 
7*x^4 + 35/2*a^3*b^2*c^6*d*x^4 + 105/4*a^4*b*c^5*d^2*x^4 + 35/4*a^5*c^4*d^ 
3*x^4 + 10/3*a^3*b^2*c^7*x^3 + 35/3*a^4*b*c^6*d*x^3 + 7*a^5*c^5*d^2*x^3 + 
5/2*a^4*b*c^7*x^2 + 7/2*a^5*c^6*d*x^2 + a^5*c^7*x
 

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 570, normalized size of antiderivative = 3.96 \[ \int (a+b x)^5 (c+d x)^7 \, dx=x^7\,\left (a^5\,c\,d^6+15\,a^4\,b\,c^2\,d^5+50\,a^3\,b^2\,c^3\,d^4+50\,a^2\,b^3\,c^4\,d^3+15\,a\,b^4\,c^5\,d^2+b^5\,c^6\,d\right )+x^6\,\left (\frac {7\,a^5\,c^2\,d^5}{2}+\frac {175\,a^4\,b\,c^3\,d^4}{6}+\frac {175\,a^3\,b^2\,c^4\,d^3}{3}+35\,a^2\,b^3\,c^5\,d^2+\frac {35\,a\,b^4\,c^6\,d}{6}+\frac {b^5\,c^7}{6}\right )+x^8\,\left (\frac {a^5\,d^7}{8}+\frac {35\,a^4\,b\,c\,d^6}{8}+\frac {105\,a^3\,b^2\,c^2\,d^5}{4}+\frac {175\,a^2\,b^3\,c^3\,d^4}{4}+\frac {175\,a\,b^4\,c^4\,d^3}{8}+\frac {21\,b^5\,c^5\,d^2}{8}\right )+x^5\,\left (7\,a^5\,c^3\,d^4+35\,a^4\,b\,c^4\,d^3+42\,a^3\,b^2\,c^5\,d^2+14\,a^2\,b^3\,c^6\,d+a\,b^4\,c^7\right )+x^9\,\left (\frac {5\,a^4\,b\,d^7}{9}+\frac {70\,a^3\,b^2\,c\,d^6}{9}+\frac {70\,a^2\,b^3\,c^2\,d^5}{3}+\frac {175\,a\,b^4\,c^3\,d^4}{9}+\frac {35\,b^5\,c^4\,d^3}{9}\right )+a^5\,c^7\,x+\frac {b^5\,d^7\,x^{13}}{13}+\frac {5\,a^2\,c^4\,x^4\,\left (7\,a^3\,d^3+21\,a^2\,b\,c\,d^2+14\,a\,b^2\,c^2\,d+2\,b^3\,c^3\right )}{4}+\frac {b^2\,d^4\,x^{10}\,\left (2\,a^3\,d^3+14\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d+7\,b^3\,c^3\right )}{2}+\frac {a^4\,c^6\,x^2\,\left (7\,a\,d+5\,b\,c\right )}{2}+\frac {b^4\,d^6\,x^{12}\,\left (5\,a\,d+7\,b\,c\right )}{12}+\frac {a^3\,c^5\,x^3\,\left (21\,a^2\,d^2+35\,a\,b\,c\,d+10\,b^2\,c^2\right )}{3}+\frac {b^3\,d^5\,x^{11}\,\left (10\,a^2\,d^2+35\,a\,b\,c\,d+21\,b^2\,c^2\right )}{11} \] Input:

int((a + b*x)^5*(c + d*x)^7,x)
 

Output:

x^7*(a^5*c*d^6 + b^5*c^6*d + 15*a*b^4*c^5*d^2 + 15*a^4*b*c^2*d^5 + 50*a^2* 
b^3*c^4*d^3 + 50*a^3*b^2*c^3*d^4) + x^6*((b^5*c^7)/6 + (7*a^5*c^2*d^5)/2 + 
 (175*a^4*b*c^3*d^4)/6 + 35*a^2*b^3*c^5*d^2 + (175*a^3*b^2*c^4*d^3)/3 + (3 
5*a*b^4*c^6*d)/6) + x^8*((a^5*d^7)/8 + (21*b^5*c^5*d^2)/8 + (175*a*b^4*c^4 
*d^3)/8 + (175*a^2*b^3*c^3*d^4)/4 + (105*a^3*b^2*c^2*d^5)/4 + (35*a^4*b*c* 
d^6)/8) + x^5*(a*b^4*c^7 + 7*a^5*c^3*d^4 + 14*a^2*b^3*c^6*d + 35*a^4*b*c^4 
*d^3 + 42*a^3*b^2*c^5*d^2) + x^9*((5*a^4*b*d^7)/9 + (35*b^5*c^4*d^3)/9 + ( 
175*a*b^4*c^3*d^4)/9 + (70*a^3*b^2*c*d^6)/9 + (70*a^2*b^3*c^2*d^5)/3) + a^ 
5*c^7*x + (b^5*d^7*x^13)/13 + (5*a^2*c^4*x^4*(7*a^3*d^3 + 2*b^3*c^3 + 14*a 
*b^2*c^2*d + 21*a^2*b*c*d^2))/4 + (b^2*d^4*x^10*(2*a^3*d^3 + 7*b^3*c^3 + 2 
1*a*b^2*c^2*d + 14*a^2*b*c*d^2))/2 + (a^4*c^6*x^2*(7*a*d + 5*b*c))/2 + (b^ 
4*d^6*x^12*(5*a*d + 7*b*c))/12 + (a^3*c^5*x^3*(21*a^2*d^2 + 10*b^2*c^2 + 3 
5*a*b*c*d))/3 + (b^3*d^5*x^11*(10*a^2*d^2 + 21*b^2*c^2 + 35*a*b*c*d))/11
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 673, normalized size of antiderivative = 4.67 \[ \int (a+b x)^5 (c+d x)^7 \, dx=\frac {x \left (792 b^{5} d^{7} x^{12}+4290 a \,b^{4} d^{7} x^{11}+6006 b^{5} c \,d^{6} x^{11}+9360 a^{2} b^{3} d^{7} x^{10}+32760 a \,b^{4} c \,d^{6} x^{10}+19656 b^{5} c^{2} d^{5} x^{10}+10296 a^{3} b^{2} d^{7} x^{9}+72072 a^{2} b^{3} c \,d^{6} x^{9}+108108 a \,b^{4} c^{2} d^{5} x^{9}+36036 b^{5} c^{3} d^{4} x^{9}+5720 a^{4} b \,d^{7} x^{8}+80080 a^{3} b^{2} c \,d^{6} x^{8}+240240 a^{2} b^{3} c^{2} d^{5} x^{8}+200200 a \,b^{4} c^{3} d^{4} x^{8}+40040 b^{5} c^{4} d^{3} x^{8}+1287 a^{5} d^{7} x^{7}+45045 a^{4} b c \,d^{6} x^{7}+270270 a^{3} b^{2} c^{2} d^{5} x^{7}+450450 a^{2} b^{3} c^{3} d^{4} x^{7}+225225 a \,b^{4} c^{4} d^{3} x^{7}+27027 b^{5} c^{5} d^{2} x^{7}+10296 a^{5} c \,d^{6} x^{6}+154440 a^{4} b \,c^{2} d^{5} x^{6}+514800 a^{3} b^{2} c^{3} d^{4} x^{6}+514800 a^{2} b^{3} c^{4} d^{3} x^{6}+154440 a \,b^{4} c^{5} d^{2} x^{6}+10296 b^{5} c^{6} d \,x^{6}+36036 a^{5} c^{2} d^{5} x^{5}+300300 a^{4} b \,c^{3} d^{4} x^{5}+600600 a^{3} b^{2} c^{4} d^{3} x^{5}+360360 a^{2} b^{3} c^{5} d^{2} x^{5}+60060 a \,b^{4} c^{6} d \,x^{5}+1716 b^{5} c^{7} x^{5}+72072 a^{5} c^{3} d^{4} x^{4}+360360 a^{4} b \,c^{4} d^{3} x^{4}+432432 a^{3} b^{2} c^{5} d^{2} x^{4}+144144 a^{2} b^{3} c^{6} d \,x^{4}+10296 a \,b^{4} c^{7} x^{4}+90090 a^{5} c^{4} d^{3} x^{3}+270270 a^{4} b \,c^{5} d^{2} x^{3}+180180 a^{3} b^{2} c^{6} d \,x^{3}+25740 a^{2} b^{3} c^{7} x^{3}+72072 a^{5} c^{5} d^{2} x^{2}+120120 a^{4} b \,c^{6} d \,x^{2}+34320 a^{3} b^{2} c^{7} x^{2}+36036 a^{5} c^{6} d x +25740 a^{4} b \,c^{7} x +10296 a^{5} c^{7}\right )}{10296} \] Input:

int((b*x+a)^5*(d*x+c)^7,x)
 

Output:

(x*(10296*a**5*c**7 + 36036*a**5*c**6*d*x + 72072*a**5*c**5*d**2*x**2 + 90 
090*a**5*c**4*d**3*x**3 + 72072*a**5*c**3*d**4*x**4 + 36036*a**5*c**2*d**5 
*x**5 + 10296*a**5*c*d**6*x**6 + 1287*a**5*d**7*x**7 + 25740*a**4*b*c**7*x 
 + 120120*a**4*b*c**6*d*x**2 + 270270*a**4*b*c**5*d**2*x**3 + 360360*a**4* 
b*c**4*d**3*x**4 + 300300*a**4*b*c**3*d**4*x**5 + 154440*a**4*b*c**2*d**5* 
x**6 + 45045*a**4*b*c*d**6*x**7 + 5720*a**4*b*d**7*x**8 + 34320*a**3*b**2* 
c**7*x**2 + 180180*a**3*b**2*c**6*d*x**3 + 432432*a**3*b**2*c**5*d**2*x**4 
 + 600600*a**3*b**2*c**4*d**3*x**5 + 514800*a**3*b**2*c**3*d**4*x**6 + 270 
270*a**3*b**2*c**2*d**5*x**7 + 80080*a**3*b**2*c*d**6*x**8 + 10296*a**3*b* 
*2*d**7*x**9 + 25740*a**2*b**3*c**7*x**3 + 144144*a**2*b**3*c**6*d*x**4 + 
360360*a**2*b**3*c**5*d**2*x**5 + 514800*a**2*b**3*c**4*d**3*x**6 + 450450 
*a**2*b**3*c**3*d**4*x**7 + 240240*a**2*b**3*c**2*d**5*x**8 + 72072*a**2*b 
**3*c*d**6*x**9 + 9360*a**2*b**3*d**7*x**10 + 10296*a*b**4*c**7*x**4 + 600 
60*a*b**4*c**6*d*x**5 + 154440*a*b**4*c**5*d**2*x**6 + 225225*a*b**4*c**4* 
d**3*x**7 + 200200*a*b**4*c**3*d**4*x**8 + 108108*a*b**4*c**2*d**5*x**9 + 
32760*a*b**4*c*d**6*x**10 + 4290*a*b**4*d**7*x**11 + 1716*b**5*c**7*x**5 + 
 10296*b**5*c**6*d*x**6 + 27027*b**5*c**5*d**2*x**7 + 40040*b**5*c**4*d**3 
*x**8 + 36036*b**5*c**3*d**4*x**9 + 19656*b**5*c**2*d**5*x**10 + 6006*b**5 
*c*d**6*x**11 + 792*b**5*d**7*x**12))/10296