\(\int (a+b x)^3 (c+d x)^7 \, dx\) [73]

Optimal result

Integrand size = 15, antiderivative size = 92 \[ \int (a+b x)^3 (c+d x)^7 \, dx=-\frac {(b c-a d)^3 (c+d x)^8}{8 d^4}+\frac {b (b c-a d)^2 (c+d x)^9}{3 d^4}-\frac {3 b^2 (b c-a d) (c+d x)^{10}}{10 d^4}+\frac {b^3 (c+d x)^{11}}{11 d^4} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(360\) vs. \(2(92)=184\).

Time = 0.02 (sec) , antiderivative size = 360, normalized size of antiderivative = 3.91 \[ \int (a+b x)^3 (c+d x)^7 \, dx=a^3 c^7 x+\frac {1}{2} a^2 c^6 (3 b c+7 a d) x^2+a c^5 \left (b^2 c^2+7 a b c d+7 a^2 d^2\right ) x^3+\frac {1}{4} c^4 \left (b^3 c^3+21 a b^2 c^2 d+63 a^2 b c d^2+35 a^3 d^3\right ) x^4+\frac {7}{5} c^3 d \left (b^3 c^3+9 a b^2 c^2 d+15 a^2 b c d^2+5 a^3 d^3\right ) x^5+\frac {7}{2} c^2 d^2 \left (b^3 c^3+5 a b^2 c^2 d+5 a^2 b c d^2+a^3 d^3\right ) x^6+c d^3 \left (5 b^3 c^3+15 a b^2 c^2 d+9 a^2 b c d^2+a^3 d^3\right ) x^7+\frac {1}{8} d^4 \left (35 b^3 c^3+63 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) x^8+\frac {1}{3} b d^5 \left (7 b^2 c^2+7 a b c d+a^2 d^2\right ) x^9+\frac {1}{10} b^2 d^6 (7 b c+3 a d) x^{10}+\frac {1}{11} b^3 d^7 x^{11} \] Input:

Integrate[(a + b*x)^3*(c + d*x)^7,x]
 

Output:

a^3*c^7*x + (a^2*c^6*(3*b*c + 7*a*d)*x^2)/2 + a*c^5*(b^2*c^2 + 7*a*b*c*d + 
 7*a^2*d^2)*x^3 + (c^4*(b^3*c^3 + 21*a*b^2*c^2*d + 63*a^2*b*c*d^2 + 35*a^3 
*d^3)*x^4)/4 + (7*c^3*d*(b^3*c^3 + 9*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 5*a^3* 
d^3)*x^5)/5 + (7*c^2*d^2*(b^3*c^3 + 5*a*b^2*c^2*d + 5*a^2*b*c*d^2 + a^3*d^ 
3)*x^6)/2 + c*d^3*(5*b^3*c^3 + 15*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*x 
^7 + (d^4*(35*b^3*c^3 + 63*a*b^2*c^2*d + 21*a^2*b*c*d^2 + a^3*d^3)*x^8)/8 
+ (b*d^5*(7*b^2*c^2 + 7*a*b*c*d + a^2*d^2)*x^9)/3 + (b^2*d^6*(7*b*c + 3*a* 
d)*x^10)/10 + (b^3*d^7*x^11)/11
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {49, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (a+b x)^3 (c+d x)^7 \, dx\)

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*x)^3*(c + d*x)^7,x]
 

Output:

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

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(375\) vs. \(2(84)=168\).

Time = 0.08 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.09

method	result	size
norman	\(\frac {b^{3} d^{7} x^{11}}{11}+\left (\frac {3}{10} a \,b^{2} d^{7}+\frac {7}{10} b^{3} c \,d^{6}\right ) x^{10}+\left (\frac {1}{3} a^{2} b \,d^{7}+\frac {7}{3} a \,b^{2} c \,d^{6}+\frac {7}{3} b^{3} c^{2} d^{5}\right ) x^{9}+\left (\frac {1}{8} a^{3} d^{7}+\frac {21}{8} a^{2} b c \,d^{6}+\frac {63}{8} a \,b^{2} c^{2} d^{5}+\frac {35}{8} b^{3} c^{3} d^{4}\right ) x^{8}+\left (a^{3} c \,d^{6}+9 a^{2} b \,c^{2} d^{5}+15 a \,b^{2} c^{3} d^{4}+5 b^{3} c^{4} d^{3}\right ) x^{7}+\left (\frac {7}{2} a^{3} c^{2} d^{5}+\frac {35}{2} a^{2} b \,c^{3} d^{4}+\frac {35}{2} a \,b^{2} c^{4} d^{3}+\frac {7}{2} b^{3} c^{5} d^{2}\right ) x^{6}+\left (7 a^{3} c^{3} d^{4}+21 a^{2} b \,c^{4} d^{3}+\frac {63}{5} a \,b^{2} c^{5} d^{2}+\frac {7}{5} b^{3} c^{6} d \right ) x^{5}+\left (\frac {35}{4} a^{3} c^{4} d^{3}+\frac {63}{4} a^{2} b \,c^{5} d^{2}+\frac {21}{4} a \,b^{2} c^{6} d +\frac {1}{4} b^{3} c^{7}\right ) x^{4}+\left (7 a^{3} c^{5} d^{2}+7 a^{2} b \,c^{6} d +a \,b^{2} c^{7}\right ) x^{3}+\left (\frac {7}{2} a^{3} c^{6} d +\frac {3}{2} a^{2} b \,c^{7}\right ) x^{2}+a^{3} c^{7} x\)	\(376\)
default	\(\frac {b^{3} d^{7} x^{11}}{11}+\frac {\left (3 a \,b^{2} d^{7}+7 b^{3} c \,d^{6}\right ) x^{10}}{10}+\frac {\left (3 a^{2} b \,d^{7}+21 a \,b^{2} c \,d^{6}+21 b^{3} c^{2} d^{5}\right ) x^{9}}{9}+\frac {\left (a^{3} d^{7}+21 a^{2} b c \,d^{6}+63 a \,b^{2} c^{2} d^{5}+35 b^{3} c^{3} d^{4}\right ) x^{8}}{8}+\frac {\left (7 a^{3} c \,d^{6}+63 a^{2} b \,c^{2} d^{5}+105 a \,b^{2} c^{3} d^{4}+35 b^{3} c^{4} d^{3}\right ) x^{7}}{7}+\frac {\left (21 a^{3} c^{2} d^{5}+105 a^{2} b \,c^{3} d^{4}+105 a \,b^{2} c^{4} d^{3}+21 b^{3} c^{5} d^{2}\right ) x^{6}}{6}+\frac {\left (35 a^{3} c^{3} d^{4}+105 a^{2} b \,c^{4} d^{3}+63 a \,b^{2} c^{5} d^{2}+7 b^{3} c^{6} d \right ) x^{5}}{5}+\frac {\left (35 a^{3} c^{4} d^{3}+63 a^{2} b \,c^{5} d^{2}+21 a \,b^{2} c^{6} d +b^{3} c^{7}\right ) x^{4}}{4}+\frac {\left (21 a^{3} c^{5} d^{2}+21 a^{2} b \,c^{6} d +3 a \,b^{2} c^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{3} c^{6} d +3 a^{2} b \,c^{7}\right ) x^{2}}{2}+a^{3} c^{7} x\)	\(385\)
gosper	\(\frac {1}{8} x^{8} a^{3} d^{7}+\frac {1}{4} x^{4} b^{3} c^{7}+\frac {1}{11} b^{3} d^{7} x^{11}+a^{3} c^{7} x +\frac {7}{10} x^{10} b^{3} c \,d^{6}+\frac {3}{10} x^{10} a \,b^{2} d^{7}+\frac {1}{3} x^{9} a^{2} b \,d^{7}+\frac {7}{3} x^{9} b^{3} c^{2} d^{5}+\frac {35}{8} x^{8} b^{3} c^{3} d^{4}+\frac {7}{2} x^{6} a^{3} c^{2} d^{5}+\frac {7}{2} x^{6} b^{3} c^{5} d^{2}+7 x^{5} a^{3} c^{3} d^{4}+\frac {7}{5} x^{5} b^{3} c^{6} d +\frac {35}{4} x^{4} a^{3} c^{4} d^{3}+\frac {7}{2} x^{2} a^{3} c^{6} d +\frac {3}{2} x^{2} a^{2} b \,c^{7}+a^{3} c \,d^{6} x^{7}+5 b^{3} c^{4} d^{3} x^{7}+7 a^{3} c^{5} d^{2} x^{3}+a \,b^{2} c^{7} x^{3}+15 a \,b^{2} c^{3} d^{4} x^{7}+7 a^{2} b \,c^{6} d \,x^{3}+\frac {7}{3} x^{9} a \,b^{2} c \,d^{6}+\frac {21}{8} x^{8} a^{2} b c \,d^{6}+\frac {63}{8} x^{8} a \,b^{2} c^{2} d^{5}+\frac {35}{2} x^{6} a^{2} b \,c^{3} d^{4}+\frac {35}{2} x^{6} a \,b^{2} c^{4} d^{3}+21 x^{5} a^{2} b \,c^{4} d^{3}+\frac {63}{5} x^{5} a \,b^{2} c^{5} d^{2}+\frac {63}{4} x^{4} a^{2} b \,c^{5} d^{2}+\frac {21}{4} x^{4} a \,b^{2} c^{6} d +9 a^{2} b \,c^{2} d^{5} x^{7}\)	\(421\)
risch	\(\frac {1}{8} x^{8} a^{3} d^{7}+\frac {1}{4} x^{4} b^{3} c^{7}+\frac {1}{11} b^{3} d^{7} x^{11}+a^{3} c^{7} x +\frac {7}{10} x^{10} b^{3} c \,d^{6}+\frac {3}{10} x^{10} a \,b^{2} d^{7}+\frac {1}{3} x^{9} a^{2} b \,d^{7}+\frac {7}{3} x^{9} b^{3} c^{2} d^{5}+\frac {35}{8} x^{8} b^{3} c^{3} d^{4}+\frac {7}{2} x^{6} a^{3} c^{2} d^{5}+\frac {7}{2} x^{6} b^{3} c^{5} d^{2}+7 x^{5} a^{3} c^{3} d^{4}+\frac {7}{5} x^{5} b^{3} c^{6} d +\frac {35}{4} x^{4} a^{3} c^{4} d^{3}+\frac {7}{2} x^{2} a^{3} c^{6} d +\frac {3}{2} x^{2} a^{2} b \,c^{7}+a^{3} c \,d^{6} x^{7}+5 b^{3} c^{4} d^{3} x^{7}+7 a^{3} c^{5} d^{2} x^{3}+a \,b^{2} c^{7} x^{3}+15 a \,b^{2} c^{3} d^{4} x^{7}+7 a^{2} b \,c^{6} d \,x^{3}+\frac {7}{3} x^{9} a \,b^{2} c \,d^{6}+\frac {21}{8} x^{8} a^{2} b c \,d^{6}+\frac {63}{8} x^{8} a \,b^{2} c^{2} d^{5}+\frac {35}{2} x^{6} a^{2} b \,c^{3} d^{4}+\frac {35}{2} x^{6} a \,b^{2} c^{4} d^{3}+21 x^{5} a^{2} b \,c^{4} d^{3}+\frac {63}{5} x^{5} a \,b^{2} c^{5} d^{2}+\frac {63}{4} x^{4} a^{2} b \,c^{5} d^{2}+\frac {21}{4} x^{4} a \,b^{2} c^{6} d +9 a^{2} b \,c^{2} d^{5} x^{7}\)	\(421\)
parallelrisch	\(\frac {1}{8} x^{8} a^{3} d^{7}+\frac {1}{4} x^{4} b^{3} c^{7}+\frac {1}{11} b^{3} d^{7} x^{11}+a^{3} c^{7} x +\frac {7}{10} x^{10} b^{3} c \,d^{6}+\frac {3}{10} x^{10} a \,b^{2} d^{7}+\frac {1}{3} x^{9} a^{2} b \,d^{7}+\frac {7}{3} x^{9} b^{3} c^{2} d^{5}+\frac {35}{8} x^{8} b^{3} c^{3} d^{4}+\frac {7}{2} x^{6} a^{3} c^{2} d^{5}+\frac {7}{2} x^{6} b^{3} c^{5} d^{2}+7 x^{5} a^{3} c^{3} d^{4}+\frac {7}{5} x^{5} b^{3} c^{6} d +\frac {35}{4} x^{4} a^{3} c^{4} d^{3}+\frac {7}{2} x^{2} a^{3} c^{6} d +\frac {3}{2} x^{2} a^{2} b \,c^{7}+a^{3} c \,d^{6} x^{7}+5 b^{3} c^{4} d^{3} x^{7}+7 a^{3} c^{5} d^{2} x^{3}+a \,b^{2} c^{7} x^{3}+15 a \,b^{2} c^{3} d^{4} x^{7}+7 a^{2} b \,c^{6} d \,x^{3}+\frac {7}{3} x^{9} a \,b^{2} c \,d^{6}+\frac {21}{8} x^{8} a^{2} b c \,d^{6}+\frac {63}{8} x^{8} a \,b^{2} c^{2} d^{5}+\frac {35}{2} x^{6} a^{2} b \,c^{3} d^{4}+\frac {35}{2} x^{6} a \,b^{2} c^{4} d^{3}+21 x^{5} a^{2} b \,c^{4} d^{3}+\frac {63}{5} x^{5} a \,b^{2} c^{5} d^{2}+\frac {63}{4} x^{4} a^{2} b \,c^{5} d^{2}+\frac {21}{4} x^{4} a \,b^{2} c^{6} d +9 a^{2} b \,c^{2} d^{5} x^{7}\)	\(421\)
orering	\(\frac {x \left (120 b^{3} d^{7} x^{10}+396 a \,b^{2} d^{7} x^{9}+924 b^{3} c \,d^{6} x^{9}+440 a^{2} b \,d^{7} x^{8}+3080 a \,b^{2} c \,d^{6} x^{8}+3080 b^{3} c^{2} d^{5} x^{8}+165 a^{3} d^{7} x^{7}+3465 a^{2} b c \,d^{6} x^{7}+10395 a \,b^{2} c^{2} d^{5} x^{7}+5775 b^{3} c^{3} d^{4} x^{7}+1320 a^{3} c \,d^{6} x^{6}+11880 a^{2} b \,c^{2} d^{5} x^{6}+19800 a \,b^{2} c^{3} d^{4} x^{6}+6600 b^{3} c^{4} d^{3} x^{6}+4620 a^{3} c^{2} d^{5} x^{5}+23100 a^{2} b \,c^{3} d^{4} x^{5}+23100 a \,b^{2} c^{4} d^{3} x^{5}+4620 b^{3} c^{5} d^{2} x^{5}+9240 a^{3} c^{3} d^{4} x^{4}+27720 a^{2} b \,c^{4} d^{3} x^{4}+16632 a \,b^{2} c^{5} d^{2} x^{4}+1848 b^{3} c^{6} d \,x^{4}+11550 a^{3} c^{4} d^{3} x^{3}+20790 a^{2} b \,c^{5} d^{2} x^{3}+6930 a \,b^{2} c^{6} d \,x^{3}+330 b^{3} c^{7} x^{3}+9240 a^{3} c^{5} d^{2} x^{2}+9240 a^{2} b \,c^{6} d \,x^{2}+1320 a \,b^{2} c^{7} x^{2}+4620 a^{3} c^{6} d x +1980 a^{2} b \,c^{7} x +1320 a^{3} c^{7}\right )}{1320}\)	\(422\)

Input:

int((b*x+a)^3*(d*x+c)^7,x,method=_RETURNVERBOSE)
 

Output:

1/11*b^3*d^7*x^11+(3/10*a*b^2*d^7+7/10*b^3*c*d^6)*x^10+(1/3*a^2*b*d^7+7/3* 
a*b^2*c*d^6+7/3*b^3*c^2*d^5)*x^9+(1/8*a^3*d^7+21/8*a^2*b*c*d^6+63/8*a*b^2* 
c^2*d^5+35/8*b^3*c^3*d^4)*x^8+(a^3*c*d^6+9*a^2*b*c^2*d^5+15*a*b^2*c^3*d^4+ 
5*b^3*c^4*d^3)*x^7+(7/2*a^3*c^2*d^5+35/2*a^2*b*c^3*d^4+35/2*a*b^2*c^4*d^3+ 
7/2*b^3*c^5*d^2)*x^6+(7*a^3*c^3*d^4+21*a^2*b*c^4*d^3+63/5*a*b^2*c^5*d^2+7/ 
5*b^3*c^6*d)*x^5+(35/4*a^3*c^4*d^3+63/4*a^2*b*c^5*d^2+21/4*a*b^2*c^6*d+1/4 
*b^3*c^7)*x^4+(7*a^3*c^5*d^2+7*a^2*b*c^6*d+a*b^2*c^7)*x^3+(7/2*a^3*c^6*d+3 
/2*a^2*b*c^7)*x^2+a^3*c^7*x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (84) = 168\).

Time = 0.09 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.09 \[ \int (a+b x)^3 (c+d x)^7 \, dx=\frac {1}{11} \, b^{3} d^{7} x^{11} + a^{3} c^{7} x + \frac {1}{10} \, {\left (7 \, b^{3} c d^{6} + 3 \, a b^{2} d^{7}\right )} x^{10} + \frac {1}{3} \, {\left (7 \, b^{3} c^{2} d^{5} + 7 \, a b^{2} c d^{6} + a^{2} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (35 \, b^{3} c^{3} d^{4} + 63 \, a b^{2} c^{2} d^{5} + 21 \, a^{2} b c d^{6} + a^{3} d^{7}\right )} x^{8} + {\left (5 \, b^{3} c^{4} d^{3} + 15 \, a b^{2} c^{3} d^{4} + 9 \, a^{2} b c^{2} d^{5} + a^{3} c d^{6}\right )} x^{7} + \frac {7}{2} \, {\left (b^{3} c^{5} d^{2} + 5 \, a b^{2} c^{4} d^{3} + 5 \, a^{2} b c^{3} d^{4} + a^{3} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (b^{3} c^{6} d + 9 \, a b^{2} c^{5} d^{2} + 15 \, a^{2} b c^{4} d^{3} + 5 \, a^{3} c^{3} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{7} + 21 \, a b^{2} c^{6} d + 63 \, a^{2} b c^{5} d^{2} + 35 \, a^{3} c^{4} d^{3}\right )} x^{4} + {\left (a b^{2} c^{7} + 7 \, a^{2} b c^{6} d + 7 \, a^{3} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{7} + 7 \, a^{3} c^{6} d\right )} x^{2} \] Input:

integrate((b*x+a)^3*(d*x+c)^7,x, algorithm="fricas")
 

Output:

1/11*b^3*d^7*x^11 + a^3*c^7*x + 1/10*(7*b^3*c*d^6 + 3*a*b^2*d^7)*x^10 + 1/ 
3*(7*b^3*c^2*d^5 + 7*a*b^2*c*d^6 + a^2*b*d^7)*x^9 + 1/8*(35*b^3*c^3*d^4 + 
63*a*b^2*c^2*d^5 + 21*a^2*b*c*d^6 + a^3*d^7)*x^8 + (5*b^3*c^4*d^3 + 15*a*b 
^2*c^3*d^4 + 9*a^2*b*c^2*d^5 + a^3*c*d^6)*x^7 + 7/2*(b^3*c^5*d^2 + 5*a*b^2 
*c^4*d^3 + 5*a^2*b*c^3*d^4 + a^3*c^2*d^5)*x^6 + 7/5*(b^3*c^6*d + 9*a*b^2*c 
^5*d^2 + 15*a^2*b*c^4*d^3 + 5*a^3*c^3*d^4)*x^5 + 1/4*(b^3*c^7 + 21*a*b^2*c 
^6*d + 63*a^2*b*c^5*d^2 + 35*a^3*c^4*d^3)*x^4 + (a*b^2*c^7 + 7*a^2*b*c^6*d 
 + 7*a^3*c^5*d^2)*x^3 + 1/2*(3*a^2*b*c^7 + 7*a^3*c^6*d)*x^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (80) = 160\).

Time = 0.04 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.64 \[ \int (a+b x)^3 (c+d x)^7 \, dx=a^{3} c^{7} x + \frac {b^{3} d^{7} x^{11}}{11} + x^{10} \cdot \left (\frac {3 a b^{2} d^{7}}{10} + \frac {7 b^{3} c d^{6}}{10}\right ) + x^{9} \left (\frac {a^{2} b d^{7}}{3} + \frac {7 a b^{2} c d^{6}}{3} + \frac {7 b^{3} c^{2} d^{5}}{3}\right ) + x^{8} \left (\frac {a^{3} d^{7}}{8} + \frac {21 a^{2} b c d^{6}}{8} + \frac {63 a b^{2} c^{2} d^{5}}{8} + \frac {35 b^{3} c^{3} d^{4}}{8}\right ) + x^{7} \left (a^{3} c d^{6} + 9 a^{2} b c^{2} d^{5} + 15 a b^{2} c^{3} d^{4} + 5 b^{3} c^{4} d^{3}\right ) + x^{6} \cdot \left (\frac {7 a^{3} c^{2} d^{5}}{2} + \frac {35 a^{2} b c^{3} d^{4}}{2} + \frac {35 a b^{2} c^{4} d^{3}}{2} + \frac {7 b^{3} c^{5} d^{2}}{2}\right ) + x^{5} \cdot \left (7 a^{3} c^{3} d^{4} + 21 a^{2} b c^{4} d^{3} + \frac {63 a b^{2} c^{5} d^{2}}{5} + \frac {7 b^{3} c^{6} d}{5}\right ) + x^{4} \cdot \left (\frac {35 a^{3} c^{4} d^{3}}{4} + \frac {63 a^{2} b c^{5} d^{2}}{4} + \frac {21 a b^{2} c^{6} d}{4} + \frac {b^{3} c^{7}}{4}\right ) + x^{3} \cdot \left (7 a^{3} c^{5} d^{2} + 7 a^{2} b c^{6} d + a b^{2} c^{7}\right ) + x^{2} \cdot \left (\frac {7 a^{3} c^{6} d}{2} + \frac {3 a^{2} b c^{7}}{2}\right ) \] Input:

integrate((b*x+a)**3*(d*x+c)**7,x)
 

Output:

a**3*c**7*x + b**3*d**7*x**11/11 + x**10*(3*a*b**2*d**7/10 + 7*b**3*c*d**6 
/10) + x**9*(a**2*b*d**7/3 + 7*a*b**2*c*d**6/3 + 7*b**3*c**2*d**5/3) + x** 
8*(a**3*d**7/8 + 21*a**2*b*c*d**6/8 + 63*a*b**2*c**2*d**5/8 + 35*b**3*c**3 
*d**4/8) + x**7*(a**3*c*d**6 + 9*a**2*b*c**2*d**5 + 15*a*b**2*c**3*d**4 + 
5*b**3*c**4*d**3) + x**6*(7*a**3*c**2*d**5/2 + 35*a**2*b*c**3*d**4/2 + 35* 
a*b**2*c**4*d**3/2 + 7*b**3*c**5*d**2/2) + x**5*(7*a**3*c**3*d**4 + 21*a** 
2*b*c**4*d**3 + 63*a*b**2*c**5*d**2/5 + 7*b**3*c**6*d/5) + x**4*(35*a**3*c 
**4*d**3/4 + 63*a**2*b*c**5*d**2/4 + 21*a*b**2*c**6*d/4 + b**3*c**7/4) + x 
**3*(7*a**3*c**5*d**2 + 7*a**2*b*c**6*d + a*b**2*c**7) + x**2*(7*a**3*c**6 
*d/2 + 3*a**2*b*c**7/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (84) = 168\).

Time = 0.03 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.09 \[ \int (a+b x)^3 (c+d x)^7 \, dx=\frac {1}{11} \, b^{3} d^{7} x^{11} + a^{3} c^{7} x + \frac {1}{10} \, {\left (7 \, b^{3} c d^{6} + 3 \, a b^{2} d^{7}\right )} x^{10} + \frac {1}{3} \, {\left (7 \, b^{3} c^{2} d^{5} + 7 \, a b^{2} c d^{6} + a^{2} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (35 \, b^{3} c^{3} d^{4} + 63 \, a b^{2} c^{2} d^{5} + 21 \, a^{2} b c d^{6} + a^{3} d^{7}\right )} x^{8} + {\left (5 \, b^{3} c^{4} d^{3} + 15 \, a b^{2} c^{3} d^{4} + 9 \, a^{2} b c^{2} d^{5} + a^{3} c d^{6}\right )} x^{7} + \frac {7}{2} \, {\left (b^{3} c^{5} d^{2} + 5 \, a b^{2} c^{4} d^{3} + 5 \, a^{2} b c^{3} d^{4} + a^{3} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (b^{3} c^{6} d + 9 \, a b^{2} c^{5} d^{2} + 15 \, a^{2} b c^{4} d^{3} + 5 \, a^{3} c^{3} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{7} + 21 \, a b^{2} c^{6} d + 63 \, a^{2} b c^{5} d^{2} + 35 \, a^{3} c^{4} d^{3}\right )} x^{4} + {\left (a b^{2} c^{7} + 7 \, a^{2} b c^{6} d + 7 \, a^{3} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{7} + 7 \, a^{3} c^{6} d\right )} x^{2} \] Input:

integrate((b*x+a)^3*(d*x+c)^7,x, algorithm="maxima")
 

Output:

1/11*b^3*d^7*x^11 + a^3*c^7*x + 1/10*(7*b^3*c*d^6 + 3*a*b^2*d^7)*x^10 + 1/ 
3*(7*b^3*c^2*d^5 + 7*a*b^2*c*d^6 + a^2*b*d^7)*x^9 + 1/8*(35*b^3*c^3*d^4 + 
63*a*b^2*c^2*d^5 + 21*a^2*b*c*d^6 + a^3*d^7)*x^8 + (5*b^3*c^4*d^3 + 15*a*b 
^2*c^3*d^4 + 9*a^2*b*c^2*d^5 + a^3*c*d^6)*x^7 + 7/2*(b^3*c^5*d^2 + 5*a*b^2 
*c^4*d^3 + 5*a^2*b*c^3*d^4 + a^3*c^2*d^5)*x^6 + 7/5*(b^3*c^6*d + 9*a*b^2*c 
^5*d^2 + 15*a^2*b*c^4*d^3 + 5*a^3*c^3*d^4)*x^5 + 1/4*(b^3*c^7 + 21*a*b^2*c 
^6*d + 63*a^2*b*c^5*d^2 + 35*a^3*c^4*d^3)*x^4 + (a*b^2*c^7 + 7*a^2*b*c^6*d 
 + 7*a^3*c^5*d^2)*x^3 + 1/2*(3*a^2*b*c^7 + 7*a^3*c^6*d)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (84) = 168\).

Time = 0.13 (sec) , antiderivative size = 420, normalized size of antiderivative = 4.57 \[ \int (a+b x)^3 (c+d x)^7 \, dx=\frac {1}{11} \, b^{3} d^{7} x^{11} + \frac {7}{10} \, b^{3} c d^{6} x^{10} + \frac {3}{10} \, a b^{2} d^{7} x^{10} + \frac {7}{3} \, b^{3} c^{2} d^{5} x^{9} + \frac {7}{3} \, a b^{2} c d^{6} x^{9} + \frac {1}{3} \, a^{2} b d^{7} x^{9} + \frac {35}{8} \, b^{3} c^{3} d^{4} x^{8} + \frac {63}{8} \, a b^{2} c^{2} d^{5} x^{8} + \frac {21}{8} \, a^{2} b c d^{6} x^{8} + \frac {1}{8} \, a^{3} d^{7} x^{8} + 5 \, b^{3} c^{4} d^{3} x^{7} + 15 \, a b^{2} c^{3} d^{4} x^{7} + 9 \, a^{2} b c^{2} d^{5} x^{7} + a^{3} c d^{6} x^{7} + \frac {7}{2} \, b^{3} c^{5} d^{2} x^{6} + \frac {35}{2} \, a b^{2} c^{4} d^{3} x^{6} + \frac {35}{2} \, a^{2} b c^{3} d^{4} x^{6} + \frac {7}{2} \, a^{3} c^{2} d^{5} x^{6} + \frac {7}{5} \, b^{3} c^{6} d x^{5} + \frac {63}{5} \, a b^{2} c^{5} d^{2} x^{5} + 21 \, a^{2} b c^{4} d^{3} x^{5} + 7 \, a^{3} c^{3} d^{4} x^{5} + \frac {1}{4} \, b^{3} c^{7} x^{4} + \frac {21}{4} \, a b^{2} c^{6} d x^{4} + \frac {63}{4} \, a^{2} b c^{5} d^{2} x^{4} + \frac {35}{4} \, a^{3} c^{4} d^{3} x^{4} + a b^{2} c^{7} x^{3} + 7 \, a^{2} b c^{6} d x^{3} + 7 \, a^{3} c^{5} d^{2} x^{3} + \frac {3}{2} \, a^{2} b c^{7} x^{2} + \frac {7}{2} \, a^{3} c^{6} d x^{2} + a^{3} c^{7} x \] Input:

integrate((b*x+a)^3*(d*x+c)^7,x, algorithm="giac")
 

Output:

1/11*b^3*d^7*x^11 + 7/10*b^3*c*d^6*x^10 + 3/10*a*b^2*d^7*x^10 + 7/3*b^3*c^ 
2*d^5*x^9 + 7/3*a*b^2*c*d^6*x^9 + 1/3*a^2*b*d^7*x^9 + 35/8*b^3*c^3*d^4*x^8 
 + 63/8*a*b^2*c^2*d^5*x^8 + 21/8*a^2*b*c*d^6*x^8 + 1/8*a^3*d^7*x^8 + 5*b^3 
*c^4*d^3*x^7 + 15*a*b^2*c^3*d^4*x^7 + 9*a^2*b*c^2*d^5*x^7 + a^3*c*d^6*x^7 
+ 7/2*b^3*c^5*d^2*x^6 + 35/2*a*b^2*c^4*d^3*x^6 + 35/2*a^2*b*c^3*d^4*x^6 + 
7/2*a^3*c^2*d^5*x^6 + 7/5*b^3*c^6*d*x^5 + 63/5*a*b^2*c^5*d^2*x^5 + 21*a^2* 
b*c^4*d^3*x^5 + 7*a^3*c^3*d^4*x^5 + 1/4*b^3*c^7*x^4 + 21/4*a*b^2*c^6*d*x^4 
 + 63/4*a^2*b*c^5*d^2*x^4 + 35/4*a^3*c^4*d^3*x^4 + a*b^2*c^7*x^3 + 7*a^2*b 
*c^6*d*x^3 + 7*a^3*c^5*d^2*x^3 + 3/2*a^2*b*c^7*x^2 + 7/2*a^3*c^6*d*x^2 + a 
^3*c^7*x
 

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 356, normalized size of antiderivative = 3.87 \[ \int (a+b x)^3 (c+d x)^7 \, dx=x^7\,\left (a^3\,c\,d^6+9\,a^2\,b\,c^2\,d^5+15\,a\,b^2\,c^3\,d^4+5\,b^3\,c^4\,d^3\right )+x^5\,\left (7\,a^3\,c^3\,d^4+21\,a^2\,b\,c^4\,d^3+\frac {63\,a\,b^2\,c^5\,d^2}{5}+\frac {7\,b^3\,c^6\,d}{5}\right )+x^4\,\left (\frac {35\,a^3\,c^4\,d^3}{4}+\frac {63\,a^2\,b\,c^5\,d^2}{4}+\frac {21\,a\,b^2\,c^6\,d}{4}+\frac {b^3\,c^7}{4}\right )+x^8\,\left (\frac {a^3\,d^7}{8}+\frac {21\,a^2\,b\,c\,d^6}{8}+\frac {63\,a\,b^2\,c^2\,d^5}{8}+\frac {35\,b^3\,c^3\,d^4}{8}\right )+a^3\,c^7\,x+\frac {b^3\,d^7\,x^{11}}{11}+\frac {7\,c^2\,d^2\,x^6\,\left (a^3\,d^3+5\,a^2\,b\,c\,d^2+5\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{2}+\frac {a^2\,c^6\,x^2\,\left (7\,a\,d+3\,b\,c\right )}{2}+\frac {b^2\,d^6\,x^{10}\,\left (3\,a\,d+7\,b\,c\right )}{10}+a\,c^5\,x^3\,\left (7\,a^2\,d^2+7\,a\,b\,c\,d+b^2\,c^2\right )+\frac {b\,d^5\,x^9\,\left (a^2\,d^2+7\,a\,b\,c\,d+7\,b^2\,c^2\right )}{3} \] Input:

int((a + b*x)^3*(c + d*x)^7,x)
 

Output:

x^7*(a^3*c*d^6 + 5*b^3*c^4*d^3 + 15*a*b^2*c^3*d^4 + 9*a^2*b*c^2*d^5) + x^5 
*((7*b^3*c^6*d)/5 + 7*a^3*c^3*d^4 + (63*a*b^2*c^5*d^2)/5 + 21*a^2*b*c^4*d^ 
3) + x^4*((b^3*c^7)/4 + (35*a^3*c^4*d^3)/4 + (63*a^2*b*c^5*d^2)/4 + (21*a* 
b^2*c^6*d)/4) + x^8*((a^3*d^7)/8 + (35*b^3*c^3*d^4)/8 + (63*a*b^2*c^2*d^5) 
/8 + (21*a^2*b*c*d^6)/8) + a^3*c^7*x + (b^3*d^7*x^11)/11 + (7*c^2*d^2*x^6* 
(a^3*d^3 + b^3*c^3 + 5*a*b^2*c^2*d + 5*a^2*b*c*d^2))/2 + (a^2*c^6*x^2*(7*a 
*d + 3*b*c))/2 + (b^2*d^6*x^10*(3*a*d + 7*b*c))/10 + a*c^5*x^3*(7*a^2*d^2 
+ b^2*c^2 + 7*a*b*c*d) + (b*d^5*x^9*(a^2*d^2 + 7*b^2*c^2 + 7*a*b*c*d))/3
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 421, normalized size of antiderivative = 4.58 \[ \int (a+b x)^3 (c+d x)^7 \, dx=\frac {x \left (120 b^{3} d^{7} x^{10}+396 a \,b^{2} d^{7} x^{9}+924 b^{3} c \,d^{6} x^{9}+440 a^{2} b \,d^{7} x^{8}+3080 a \,b^{2} c \,d^{6} x^{8}+3080 b^{3} c^{2} d^{5} x^{8}+165 a^{3} d^{7} x^{7}+3465 a^{2} b c \,d^{6} x^{7}+10395 a \,b^{2} c^{2} d^{5} x^{7}+5775 b^{3} c^{3} d^{4} x^{7}+1320 a^{3} c \,d^{6} x^{6}+11880 a^{2} b \,c^{2} d^{5} x^{6}+19800 a \,b^{2} c^{3} d^{4} x^{6}+6600 b^{3} c^{4} d^{3} x^{6}+4620 a^{3} c^{2} d^{5} x^{5}+23100 a^{2} b \,c^{3} d^{4} x^{5}+23100 a \,b^{2} c^{4} d^{3} x^{5}+4620 b^{3} c^{5} d^{2} x^{5}+9240 a^{3} c^{3} d^{4} x^{4}+27720 a^{2} b \,c^{4} d^{3} x^{4}+16632 a \,b^{2} c^{5} d^{2} x^{4}+1848 b^{3} c^{6} d \,x^{4}+11550 a^{3} c^{4} d^{3} x^{3}+20790 a^{2} b \,c^{5} d^{2} x^{3}+6930 a \,b^{2} c^{6} d \,x^{3}+330 b^{3} c^{7} x^{3}+9240 a^{3} c^{5} d^{2} x^{2}+9240 a^{2} b \,c^{6} d \,x^{2}+1320 a \,b^{2} c^{7} x^{2}+4620 a^{3} c^{6} d x +1980 a^{2} b \,c^{7} x +1320 a^{3} c^{7}\right )}{1320} \] Input:

int((b*x+a)^3*(d*x+c)^7,x)
 

Output:

(x*(1320*a**3*c**7 + 4620*a**3*c**6*d*x + 9240*a**3*c**5*d**2*x**2 + 11550 
*a**3*c**4*d**3*x**3 + 9240*a**3*c**3*d**4*x**4 + 4620*a**3*c**2*d**5*x**5 
 + 1320*a**3*c*d**6*x**6 + 165*a**3*d**7*x**7 + 1980*a**2*b*c**7*x + 9240* 
a**2*b*c**6*d*x**2 + 20790*a**2*b*c**5*d**2*x**3 + 27720*a**2*b*c**4*d**3* 
x**4 + 23100*a**2*b*c**3*d**4*x**5 + 11880*a**2*b*c**2*d**5*x**6 + 3465*a* 
*2*b*c*d**6*x**7 + 440*a**2*b*d**7*x**8 + 1320*a*b**2*c**7*x**2 + 6930*a*b 
**2*c**6*d*x**3 + 16632*a*b**2*c**5*d**2*x**4 + 23100*a*b**2*c**4*d**3*x** 
5 + 19800*a*b**2*c**3*d**4*x**6 + 10395*a*b**2*c**2*d**5*x**7 + 3080*a*b** 
2*c*d**6*x**8 + 396*a*b**2*d**7*x**9 + 330*b**3*c**7*x**3 + 1848*b**3*c**6 
*d*x**4 + 4620*b**3*c**5*d**2*x**5 + 6600*b**3*c**4*d**3*x**6 + 5775*b**3* 
c**3*d**4*x**7 + 3080*b**3*c**2*d**5*x**8 + 924*b**3*c*d**6*x**9 + 120*b** 
3*d**7*x**10))/1320