\(\int (a+b x)^n (c+d x)^3 \, dx\) [551]

Optimal result

Integrand size = 15, antiderivative size = 110 \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {(b c-a d)^3 (a+b x)^{1+n}}{b^4 (1+n)}+\frac {3 d (b c-a d)^2 (a+b x)^{2+n}}{b^4 (2+n)}+\frac {3 d^2 (b c-a d) (a+b x)^{3+n}}{b^4 (3+n)}+\frac {d^3 (a+b x)^{4+n}}{b^4 (4+n)} \] Output:

(-a*d+b*c)^3*(b*x+a)^(1+n)/b^4/(1+n)+3*d*(-a*d+b*c)^2*(b*x+a)^(2+n)/b^4/(2 
+n)+3*d^2*(-a*d+b*c)*(b*x+a)^(3+n)/b^4/(3+n)+d^3*(b*x+a)^(4+n)/b^4/(4+n)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {(a+b x)^{1+n} \left (\frac {(b c-a d)^3}{1+n}+\frac {3 d (b c-a d)^2 (a+b x)}{2+n}+\frac {3 d^2 (b c-a d) (a+b x)^2}{3+n}+\frac {d^3 (a+b x)^3}{4+n}\right )}{b^4} \] Input:

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

Output:

((a + b*x)^(1 + n)*((b*c - a*d)^3/(1 + n) + (3*d*(b*c - a*d)^2*(a + b*x))/ 
(2 + n) + (3*d^2*(b*c - a*d)*(a + b*x)^2)/(3 + n) + (d^3*(a + b*x)^3)/(4 + 
 n)))/b^4
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 110, 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 = {53, 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.

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

\(\Big \downarrow \) 53

\(\displaystyle \int \left (\frac {3 d^2 (b c-a d) (a+b x)^{n+2}}{b^3}+\frac {(b c-a d)^3 (a+b x)^n}{b^3}+\frac {3 d (b c-a d)^2 (a+b x)^{n+1}}{b^3}+\frac {d^3 (a+b x)^{n+3}}{b^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 d^2 (b c-a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac {(b c-a d)^3 (a+b x)^{n+1}}{b^4 (n+1)}+\frac {3 d (b c-a d)^2 (a+b x)^{n+2}}{b^4 (n+2)}+\frac {d^3 (a+b x)^{n+4}}{b^4 (n+4)}\)

Input:

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

Output:

((b*c - a*d)^3*(a + b*x)^(1 + n))/(b^4*(1 + n)) + (3*d*(b*c - a*d)^2*(a + 
b*x)^(2 + n))/(b^4*(2 + n)) + (3*d^2*(b*c - a*d)*(a + b*x)^(3 + n))/(b^4*( 
3 + n)) + (d^3*(a + b*x)^(4 + n))/(b^4*(4 + n))
 

Defintions of rubi rules used

rule 53

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, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[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. \(388\) vs. \(2(110)=220\).

Time = 0.28 (sec) , antiderivative size = 389, normalized size of antiderivative = 3.54

method	result	size
gosper	\(-\frac {\left (b x +a \right )^{1+n} \left (-b^{3} d^{3} n^{3} x^{3}-3 b^{3} c \,d^{2} n^{3} x^{2}-6 b^{3} d^{3} n^{2} x^{3}+3 a \,b^{2} d^{3} n^{2} x^{2}-3 b^{3} c^{2} d \,n^{3} x -21 b^{3} c \,d^{2} n^{2} x^{2}-11 b^{3} d^{3} n \,x^{3}+6 a \,b^{2} c \,d^{2} n^{2} x +9 a \,b^{2} d^{3} n \,x^{2}-b^{3} c^{3} n^{3}-24 b^{3} c^{2} d \,n^{2} x -42 b^{3} c \,d^{2} n \,x^{2}-6 d^{3} x^{3} b^{3}-6 a^{2} b \,d^{3} n x +3 a \,b^{2} c^{2} d \,n^{2}+30 a \,b^{2} c \,d^{2} n x +6 x^{2} a \,b^{2} d^{3}-9 b^{3} c^{3} n^{2}-57 b^{3} c^{2} d n x -24 x^{2} b^{3} c \,d^{2}-6 a^{2} b c \,d^{2} n -6 x \,a^{2} b \,d^{3}+21 a \,b^{2} c^{2} d n +24 x a \,b^{2} c \,d^{2}-26 b^{3} c^{3} n -36 x \,b^{3} c^{2} d +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right )}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\)	\(389\)
orering	\(-\frac {\left (b x +a \right )^{n} \left (-b^{3} d^{3} n^{3} x^{3}-3 b^{3} c \,d^{2} n^{3} x^{2}-6 b^{3} d^{3} n^{2} x^{3}+3 a \,b^{2} d^{3} n^{2} x^{2}-3 b^{3} c^{2} d \,n^{3} x -21 b^{3} c \,d^{2} n^{2} x^{2}-11 b^{3} d^{3} n \,x^{3}+6 a \,b^{2} c \,d^{2} n^{2} x +9 a \,b^{2} d^{3} n \,x^{2}-b^{3} c^{3} n^{3}-24 b^{3} c^{2} d \,n^{2} x -42 b^{3} c \,d^{2} n \,x^{2}-6 d^{3} x^{3} b^{3}-6 a^{2} b \,d^{3} n x +3 a \,b^{2} c^{2} d \,n^{2}+30 a \,b^{2} c \,d^{2} n x +6 x^{2} a \,b^{2} d^{3}-9 b^{3} c^{3} n^{2}-57 b^{3} c^{2} d n x -24 x^{2} b^{3} c \,d^{2}-6 a^{2} b c \,d^{2} n -6 x \,a^{2} b \,d^{3}+21 a \,b^{2} c^{2} d n +24 x a \,b^{2} c \,d^{2}-26 b^{3} c^{3} n -36 x \,b^{3} c^{2} d +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right ) \left (b x +a \right )}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\)	\(392\)
norman	\(\frac {d^{3} x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{4+n}+\frac {\left (3 a \,b^{2} c^{2} d \,n^{3}+b^{3} c^{3} n^{3}-6 a^{2} b c \,d^{2} n^{2}+21 a \,b^{2} c^{2} d \,n^{2}+9 b^{3} c^{3} n^{2}+6 a^{3} d^{3} n -24 a^{2} b c \,d^{2} n +36 a \,b^{2} c^{2} d n +26 b^{3} c^{3} n +24 b^{3} c^{3}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}+\frac {\left (a d n +3 b c n +12 b c \right ) d^{2} x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+7 n +12\right )}-\frac {a \left (-b^{3} c^{3} n^{3}+3 a \,b^{2} c^{2} d \,n^{2}-9 b^{3} c^{3} n^{2}-6 a^{2} b c \,d^{2} n +21 a \,b^{2} c^{2} d n -26 b^{3} c^{3} n +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}-\frac {3 \left (-a b c d \,n^{2}-b^{2} c^{2} n^{2}+a^{2} d^{2} n -4 a b c d n -7 b^{2} c^{2} n -12 b^{2} c^{2}\right ) d \,x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+9 n^{2}+26 n +24\right )}\)	\(433\)
risch	\(-\frac {\left (-b^{4} d^{3} n^{3} x^{4}-a \,b^{3} d^{3} n^{3} x^{3}-3 b^{4} c \,d^{2} n^{3} x^{3}-6 b^{4} d^{3} n^{2} x^{4}-3 a \,b^{3} c \,d^{2} n^{3} x^{2}-3 a \,b^{3} d^{3} n^{2} x^{3}-3 b^{4} c^{2} d \,n^{3} x^{2}-21 b^{4} c \,d^{2} n^{2} x^{3}-11 b^{4} d^{3} n \,x^{4}+3 a^{2} b^{2} d^{3} n^{2} x^{2}-3 a \,b^{3} c^{2} d \,n^{3} x -15 a \,b^{3} c \,d^{2} n^{2} x^{2}-2 a \,b^{3} d^{3} n \,x^{3}-b^{4} c^{3} n^{3} x -24 b^{4} c^{2} d \,n^{2} x^{2}-42 b^{4} c \,d^{2} n \,x^{3}-6 d^{3} x^{4} b^{4}+6 a^{2} b^{2} c \,d^{2} n^{2} x +3 a^{2} b^{2} d^{3} n \,x^{2}-a \,b^{3} c^{3} n^{3}-21 a \,b^{3} c^{2} d \,n^{2} x -12 a \,b^{3} c \,d^{2} n \,x^{2}-9 b^{4} c^{3} n^{2} x -57 b^{4} c^{2} d n \,x^{2}-24 b^{4} c \,d^{2} x^{3}-6 a^{3} b \,d^{3} n x +3 a^{2} b^{2} c^{2} d \,n^{2}+24 a^{2} b^{2} c \,d^{2} n x -9 a \,b^{3} c^{3} n^{2}-36 a \,b^{3} c^{2} d n x -26 b^{4} c^{3} n x -36 b^{4} c^{2} d \,x^{2}-6 a^{3} b c \,d^{2} n +21 a^{2} b^{2} c^{2} d n -26 a \,b^{3} c^{3} n -24 b^{4} c^{3} x +6 a^{4} d^{3}-24 a^{3} b c \,d^{2}+36 a^{2} b^{2} c^{2} d -24 a \,b^{3} c^{3}\right ) \left (b x +a \right )^{n}}{\left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) b^{4}}\)	\(552\)
parallelrisch	\(\frac {6 x^{4} \left (b x +a \right )^{n} a \,b^{4} d^{3}+24 x \left (b x +a \right )^{n} a \,b^{4} c^{3}+x^{4} \left (b x +a \right )^{n} a \,b^{4} d^{3} n^{3}+6 x^{4} \left (b x +a \right )^{n} a \,b^{4} d^{3} n^{2}+x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d^{3} n^{3}+11 x^{4} \left (b x +a \right )^{n} a \,b^{4} d^{3} n +3 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d^{3} n^{2}+24 x^{3} \left (b x +a \right )^{n} a \,b^{4} c \,d^{2}+36 x^{2} \left (b x +a \right )^{n} a \,b^{4} c^{2} d +24 \left (b x +a \right )^{n} a^{2} b^{3} c^{3}+3 x^{3} \left (b x +a \right )^{n} a \,b^{4} c \,d^{2} n^{3}+21 x^{3} \left (b x +a \right )^{n} a \,b^{4} c \,d^{2} n^{2}+3 x^{2} \left (b x +a \right )^{n} a^{2} b^{3} c \,d^{2} n^{3}+3 x^{2} \left (b x +a \right )^{n} a \,b^{4} c^{2} d \,n^{3}+2 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d^{3} n -3 x^{2} \left (b x +a \right )^{n} a^{3} b^{2} d^{3} n^{2}+x \left (b x +a \right )^{n} a \,b^{4} c^{3} n^{3}-6 \left (b x +a \right )^{n} a^{5} d^{3}-3 x^{2} \left (b x +a \right )^{n} a^{3} b^{2} d^{3} n +9 x \left (b x +a \right )^{n} a \,b^{4} c^{3} n^{2}+6 x \left (b x +a \right )^{n} a^{4} b \,d^{3} n +26 x \left (b x +a \right )^{n} a \,b^{4} c^{3} n -3 \left (b x +a \right )^{n} a^{3} b^{2} c^{2} d \,n^{2}+6 \left (b x +a \right )^{n} a^{4} b c \,d^{2} n -21 \left (b x +a \right )^{n} a^{3} b^{2} c^{2} d n +24 \left (b x +a \right )^{n} a^{4} b c \,d^{2}-36 \left (b x +a \right )^{n} a^{3} b^{2} c^{2} d +42 x^{3} \left (b x +a \right )^{n} a \,b^{4} c \,d^{2} n +15 x^{2} \left (b x +a \right )^{n} a^{2} b^{3} c \,d^{2} n^{2}+24 x^{2} \left (b x +a \right )^{n} a \,b^{4} c^{2} d \,n^{2}+3 x \left (b x +a \right )^{n} a^{2} b^{3} c^{2} d \,n^{3}+12 x^{2} \left (b x +a \right )^{n} a^{2} b^{3} c \,d^{2} n +57 x^{2} \left (b x +a \right )^{n} a \,b^{4} c^{2} d n -6 x \left (b x +a \right )^{n} a^{3} b^{2} c \,d^{2} n^{2}+21 x \left (b x +a \right )^{n} a^{2} b^{3} c^{2} d \,n^{2}-24 x \left (b x +a \right )^{n} a^{3} b^{2} c \,d^{2} n +36 x \left (b x +a \right )^{n} a^{2} b^{3} c^{2} d n +\left (b x +a \right )^{n} a^{2} b^{3} c^{3} n^{3}+9 \left (b x +a \right )^{n} a^{2} b^{3} c^{3} n^{2}+26 \left (b x +a \right )^{n} a^{2} b^{3} c^{3} n}{\left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) a \,b^{4}}\)	\(865\)

Input:

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

Output:

-1/b^4*(b*x+a)^(1+n)/(n^4+10*n^3+35*n^2+50*n+24)*(-b^3*d^3*n^3*x^3-3*b^3*c 
*d^2*n^3*x^2-6*b^3*d^3*n^2*x^3+3*a*b^2*d^3*n^2*x^2-3*b^3*c^2*d*n^3*x-21*b^ 
3*c*d^2*n^2*x^2-11*b^3*d^3*n*x^3+6*a*b^2*c*d^2*n^2*x+9*a*b^2*d^3*n*x^2-b^3 
*c^3*n^3-24*b^3*c^2*d*n^2*x-42*b^3*c*d^2*n*x^2-6*b^3*d^3*x^3-6*a^2*b*d^3*n 
*x+3*a*b^2*c^2*d*n^2+30*a*b^2*c*d^2*n*x+6*a*b^2*d^3*x^2-9*b^3*c^3*n^2-57*b 
^3*c^2*d*n*x-24*b^3*c*d^2*x^2-6*a^2*b*c*d^2*n-6*a^2*b*d^3*x+21*a*b^2*c^2*d 
*n+24*a*b^2*c*d^2*x-26*b^3*c^3*n-36*b^3*c^2*d*x+6*a^3*d^3-24*a^2*b*c*d^2+3 
6*a*b^2*c^2*d-24*b^3*c^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (110) = 220\).

Time = 0.12 (sec) , antiderivative size = 497, normalized size of antiderivative = 4.52 \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {{\left (a b^{3} c^{3} n^{3} + 24 \, a b^{3} c^{3} - 36 \, a^{2} b^{2} c^{2} d + 24 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} + {\left (b^{4} d^{3} n^{3} + 6 \, b^{4} d^{3} n^{2} + 11 \, b^{4} d^{3} n + 6 \, b^{4} d^{3}\right )} x^{4} + {\left (24 \, b^{4} c d^{2} + {\left (3 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} n^{3} + 3 \, {\left (7 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} n^{2} + 2 \, {\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} n\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )} n^{2} + 3 \, {\left (12 \, b^{4} c^{2} d + {\left (b^{4} c^{2} d + a b^{3} c d^{2}\right )} n^{3} + {\left (8 \, b^{4} c^{2} d + 5 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} n^{2} + {\left (19 \, b^{4} c^{2} d + 4 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} n\right )} x^{2} + {\left (26 \, a b^{3} c^{3} - 21 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2}\right )} n + {\left (24 \, b^{4} c^{3} + {\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d\right )} n^{3} + 3 \, {\left (3 \, b^{4} c^{3} + 7 \, a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2}\right )} n^{2} + 2 \, {\left (13 \, b^{4} c^{3} + 18 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 3 \, a^{3} b d^{3}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \] Input:

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

Output:

(a*b^3*c^3*n^3 + 24*a*b^3*c^3 - 36*a^2*b^2*c^2*d + 24*a^3*b*c*d^2 - 6*a^4* 
d^3 + (b^4*d^3*n^3 + 6*b^4*d^3*n^2 + 11*b^4*d^3*n + 6*b^4*d^3)*x^4 + (24*b 
^4*c*d^2 + (3*b^4*c*d^2 + a*b^3*d^3)*n^3 + 3*(7*b^4*c*d^2 + a*b^3*d^3)*n^2 
 + 2*(21*b^4*c*d^2 + a*b^3*d^3)*n)*x^3 + 3*(3*a*b^3*c^3 - a^2*b^2*c^2*d)*n 
^2 + 3*(12*b^4*c^2*d + (b^4*c^2*d + a*b^3*c*d^2)*n^3 + (8*b^4*c^2*d + 5*a* 
b^3*c*d^2 - a^2*b^2*d^3)*n^2 + (19*b^4*c^2*d + 4*a*b^3*c*d^2 - a^2*b^2*d^3 
)*n)*x^2 + (26*a*b^3*c^3 - 21*a^2*b^2*c^2*d + 6*a^3*b*c*d^2)*n + (24*b^4*c 
^3 + (b^4*c^3 + 3*a*b^3*c^2*d)*n^3 + 3*(3*b^4*c^3 + 7*a*b^3*c^2*d - 2*a^2* 
b^2*c*d^2)*n^2 + 2*(13*b^4*c^3 + 18*a*b^3*c^2*d - 12*a^2*b^2*c*d^2 + 3*a^3 
*b*d^3)*n)*x)*(b*x + a)^n/(b^4*n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b^4*n + 
24*b^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4058 vs. \(2 (95) = 190\).

Time = 1.12 (sec) , antiderivative size = 4058, normalized size of antiderivative = 36.89 \[ \int (a+b x)^n (c+d x)^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((a**n*(c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4), Eq 
(b, 0)), (6*a**3*d**3*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b* 
*6*x**2 + 6*b**7*x**3) + 11*a**3*d**3/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a 
*b**6*x**2 + 6*b**7*x**3) - 6*a**2*b*c*d**2/(6*a**3*b**4 + 18*a**2*b**5*x 
+ 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a**2*b*d**3*x*log(a/b + x)/(6*a**3*b* 
*4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 27*a**2*b*d**3*x/(6* 
a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 3*a*b**2*c**2 
*d/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 18*a*b* 
*2*c*d**2*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) 
+ 18*a*b**2*d**3*x**2*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b* 
*6*x**2 + 6*b**7*x**3) + 18*a*b**2*d**3*x**2/(6*a**3*b**4 + 18*a**2*b**5*x 
 + 18*a*b**6*x**2 + 6*b**7*x**3) - 2*b**3*c**3/(6*a**3*b**4 + 18*a**2*b**5 
*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 9*b**3*c**2*d*x/(6*a**3*b**4 + 18*a** 
2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 18*b**3*c*d**2*x**2/(6*a**3*b** 
4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 6*b**3*d**3*x**3*log( 
a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3), Eq 
(n, -4)), (-6*a**3*d**3*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x* 
*2) - 9*a**3*d**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 6*a**2*b*c*d* 
*2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 9*a**2*b*c*d**2 
/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*d**3*x*log(a/b + ...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (110) = 220\).

Time = 0.04 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.24 \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {3 \, {\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c^{2} d}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {{\left (b x + a\right )}^{n + 1} c^{3}}{b {\left (n + 1\right )}} + \frac {3 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} d^{3}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \] Input:

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

Output:

3*(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^2*d/((n^2 + 3*n + 2)*b^2 
) + (b*x + a)^(n + 1)*c^3/(b*(n + 1)) + 3*((n^2 + 3*n + 2)*b^3*x^3 + (n^2 
+ n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x + a)^n*c*d^2/((n^3 + 6*n^2 + 11 
*n + 6)*b^3) + ((n^3 + 6*n^2 + 11*n + 6)*b^4*x^4 + (n^3 + 3*n^2 + 2*n)*a*b 
^3*x^3 - 3*(n^2 + n)*a^2*b^2*x^2 + 6*a^3*b*n*x - 6*a^4)*(b*x + a)^n*d^3/(( 
n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 833 vs. \(2 (110) = 220\).

Time = 0.13 (sec) , antiderivative size = 833, normalized size of antiderivative = 7.57 \[ \int (a+b x)^n (c+d x)^3 \, dx =\text {Too large to display} \] Input:

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

Output:

((b*x + a)^n*b^4*d^3*n^3*x^4 + 3*(b*x + a)^n*b^4*c*d^2*n^3*x^3 + (b*x + a) 
^n*a*b^3*d^3*n^3*x^3 + 6*(b*x + a)^n*b^4*d^3*n^2*x^4 + 3*(b*x + a)^n*b^4*c 
^2*d*n^3*x^2 + 3*(b*x + a)^n*a*b^3*c*d^2*n^3*x^2 + 21*(b*x + a)^n*b^4*c*d^ 
2*n^2*x^3 + 3*(b*x + a)^n*a*b^3*d^3*n^2*x^3 + 11*(b*x + a)^n*b^4*d^3*n*x^4 
 + (b*x + a)^n*b^4*c^3*n^3*x + 3*(b*x + a)^n*a*b^3*c^2*d*n^3*x + 24*(b*x + 
 a)^n*b^4*c^2*d*n^2*x^2 + 15*(b*x + a)^n*a*b^3*c*d^2*n^2*x^2 - 3*(b*x + a) 
^n*a^2*b^2*d^3*n^2*x^2 + 42*(b*x + a)^n*b^4*c*d^2*n*x^3 + 2*(b*x + a)^n*a* 
b^3*d^3*n*x^3 + 6*(b*x + a)^n*b^4*d^3*x^4 + (b*x + a)^n*a*b^3*c^3*n^3 + 9* 
(b*x + a)^n*b^4*c^3*n^2*x + 21*(b*x + a)^n*a*b^3*c^2*d*n^2*x - 6*(b*x + a) 
^n*a^2*b^2*c*d^2*n^2*x + 57*(b*x + a)^n*b^4*c^2*d*n*x^2 + 12*(b*x + a)^n*a 
*b^3*c*d^2*n*x^2 - 3*(b*x + a)^n*a^2*b^2*d^3*n*x^2 + 24*(b*x + a)^n*b^4*c* 
d^2*x^3 + 9*(b*x + a)^n*a*b^3*c^3*n^2 - 3*(b*x + a)^n*a^2*b^2*c^2*d*n^2 + 
26*(b*x + a)^n*b^4*c^3*n*x + 36*(b*x + a)^n*a*b^3*c^2*d*n*x - 24*(b*x + a) 
^n*a^2*b^2*c*d^2*n*x + 6*(b*x + a)^n*a^3*b*d^3*n*x + 36*(b*x + a)^n*b^4*c^ 
2*d*x^2 + 26*(b*x + a)^n*a*b^3*c^3*n - 21*(b*x + a)^n*a^2*b^2*c^2*d*n + 6* 
(b*x + a)^n*a^3*b*c*d^2*n + 24*(b*x + a)^n*b^4*c^3*x + 24*(b*x + a)^n*a*b^ 
3*c^3 - 36*(b*x + a)^n*a^2*b^2*c^2*d + 24*(b*x + a)^n*a^3*b*c*d^2 - 6*(b*x 
 + a)^n*a^4*d^3)/(b^4*n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b^4*n + 24*b^4)
 

Mupad [B] (verification not implemented)

Time = 1.23 (sec) , antiderivative size = 478, normalized size of antiderivative = 4.35 \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {x\,{\left (a+b\,x\right )}^n\,\left (6\,a^3\,b\,d^3\,n-6\,a^2\,b^2\,c\,d^2\,n^2-24\,a^2\,b^2\,c\,d^2\,n+3\,a\,b^3\,c^2\,d\,n^3+21\,a\,b^3\,c^2\,d\,n^2+36\,a\,b^3\,c^2\,d\,n+b^4\,c^3\,n^3+9\,b^4\,c^3\,n^2+26\,b^4\,c^3\,n+24\,b^4\,c^3\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {d^3\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}+\frac {a\,{\left (a+b\,x\right )}^n\,\left (-6\,a^3\,d^3+6\,a^2\,b\,c\,d^2\,n+24\,a^2\,b\,c\,d^2-3\,a\,b^2\,c^2\,d\,n^2-21\,a\,b^2\,c^2\,d\,n-36\,a\,b^2\,c^2\,d+b^3\,c^3\,n^3+9\,b^3\,c^3\,n^2+26\,b^3\,c^3\,n+24\,b^3\,c^3\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {3\,d\,x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (-a^2\,d^2\,n+a\,b\,c\,d\,n^2+4\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+7\,b^2\,c^2\,n+12\,b^2\,c^2\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {d^2\,x^3\,{\left (a+b\,x\right )}^n\,\left (12\,b\,c+a\,d\,n+3\,b\,c\,n\right )\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )} \] Input:

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

Output:

(x*(a + b*x)^n*(24*b^4*c^3 + 26*b^4*c^3*n + 9*b^4*c^3*n^2 + b^4*c^3*n^3 + 
6*a^3*b*d^3*n + 36*a*b^3*c^2*d*n - 24*a^2*b^2*c*d^2*n + 21*a*b^3*c^2*d*n^2 
 + 3*a*b^3*c^2*d*n^3 - 6*a^2*b^2*c*d^2*n^2))/(b^4*(50*n + 35*n^2 + 10*n^3 
+ n^4 + 24)) + (d^3*x^4*(a + b*x)^n*(11*n + 6*n^2 + n^3 + 6))/(50*n + 35*n 
^2 + 10*n^3 + n^4 + 24) + (a*(a + b*x)^n*(24*b^3*c^3 - 6*a^3*d^3 + 26*b^3* 
c^3*n + 9*b^3*c^3*n^2 + b^3*c^3*n^3 - 36*a*b^2*c^2*d + 24*a^2*b*c*d^2 - 21 
*a*b^2*c^2*d*n + 6*a^2*b*c*d^2*n - 3*a*b^2*c^2*d*n^2))/(b^4*(50*n + 35*n^2 
 + 10*n^3 + n^4 + 24)) + (3*d*x^2*(n + 1)*(a + b*x)^n*(12*b^2*c^2 - a^2*d^ 
2*n + 7*b^2*c^2*n + b^2*c^2*n^2 + 4*a*b*c*d*n + a*b*c*d*n^2))/(b^2*(50*n + 
 35*n^2 + 10*n^3 + n^4 + 24)) + (d^2*x^3*(a + b*x)^n*(12*b*c + a*d*n + 3*b 
*c*n)*(3*n + n^2 + 2))/(b*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 546, normalized size of antiderivative = 4.96 \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {\left (b x +a \right )^{n} \left (b^{4} d^{3} n^{3} x^{4}+a \,b^{3} d^{3} n^{3} x^{3}+3 b^{4} c \,d^{2} n^{3} x^{3}+6 b^{4} d^{3} n^{2} x^{4}+3 a \,b^{3} c \,d^{2} n^{3} x^{2}+3 a \,b^{3} d^{3} n^{2} x^{3}+3 b^{4} c^{2} d \,n^{3} x^{2}+21 b^{4} c \,d^{2} n^{2} x^{3}+11 b^{4} d^{3} n \,x^{4}-3 a^{2} b^{2} d^{3} n^{2} x^{2}+3 a \,b^{3} c^{2} d \,n^{3} x +15 a \,b^{3} c \,d^{2} n^{2} x^{2}+2 a \,b^{3} d^{3} n \,x^{3}+b^{4} c^{3} n^{3} x +24 b^{4} c^{2} d \,n^{2} x^{2}+42 b^{4} c \,d^{2} n \,x^{3}+6 b^{4} d^{3} x^{4}-6 a^{2} b^{2} c \,d^{2} n^{2} x -3 a^{2} b^{2} d^{3} n \,x^{2}+a \,b^{3} c^{3} n^{3}+21 a \,b^{3} c^{2} d \,n^{2} x +12 a \,b^{3} c \,d^{2} n \,x^{2}+9 b^{4} c^{3} n^{2} x +57 b^{4} c^{2} d n \,x^{2}+24 b^{4} c \,d^{2} x^{3}+6 a^{3} b \,d^{3} n x -3 a^{2} b^{2} c^{2} d \,n^{2}-24 a^{2} b^{2} c \,d^{2} n x +9 a \,b^{3} c^{3} n^{2}+36 a \,b^{3} c^{2} d n x +26 b^{4} c^{3} n x +36 b^{4} c^{2} d \,x^{2}+6 a^{3} b c \,d^{2} n -21 a^{2} b^{2} c^{2} d n +26 a \,b^{3} c^{3} n +24 b^{4} c^{3} x -6 a^{4} d^{3}+24 a^{3} b c \,d^{2}-36 a^{2} b^{2} c^{2} d +24 a \,b^{3} c^{3}\right )}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )} \] Input:

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

Output:

((a + b*x)**n*( - 6*a**4*d**3 + 6*a**3*b*c*d**2*n + 24*a**3*b*c*d**2 + 6*a 
**3*b*d**3*n*x - 3*a**2*b**2*c**2*d*n**2 - 21*a**2*b**2*c**2*d*n - 36*a**2 
*b**2*c**2*d - 6*a**2*b**2*c*d**2*n**2*x - 24*a**2*b**2*c*d**2*n*x - 3*a** 
2*b**2*d**3*n**2*x**2 - 3*a**2*b**2*d**3*n*x**2 + a*b**3*c**3*n**3 + 9*a*b 
**3*c**3*n**2 + 26*a*b**3*c**3*n + 24*a*b**3*c**3 + 3*a*b**3*c**2*d*n**3*x 
 + 21*a*b**3*c**2*d*n**2*x + 36*a*b**3*c**2*d*n*x + 3*a*b**3*c*d**2*n**3*x 
**2 + 15*a*b**3*c*d**2*n**2*x**2 + 12*a*b**3*c*d**2*n*x**2 + a*b**3*d**3*n 
**3*x**3 + 3*a*b**3*d**3*n**2*x**3 + 2*a*b**3*d**3*n*x**3 + b**4*c**3*n**3 
*x + 9*b**4*c**3*n**2*x + 26*b**4*c**3*n*x + 24*b**4*c**3*x + 3*b**4*c**2* 
d*n**3*x**2 + 24*b**4*c**2*d*n**2*x**2 + 57*b**4*c**2*d*n*x**2 + 36*b**4*c 
**2*d*x**2 + 3*b**4*c*d**2*n**3*x**3 + 21*b**4*c*d**2*n**2*x**3 + 42*b**4* 
c*d**2*n*x**3 + 24*b**4*c*d**2*x**3 + b**4*d**3*n**3*x**4 + 6*b**4*d**3*n* 
*2*x**4 + 11*b**4*d**3*n*x**4 + 6*b**4*d**3*x**4))/(b**4*(n**4 + 10*n**3 + 
 35*n**2 + 50*n + 24))