Integrand size = 15, antiderivative size = 110 \[ \int (a+b x)^m (c+d x)^3 \, dx=\frac {(b c-a d)^3 (a+b x)^{1+m}}{b^4 (1+m)}+\frac {3 d (b c-a d)^2 (a+b x)^{2+m}}{b^4 (2+m)}+\frac {3 d^2 (b c-a d) (a+b x)^{3+m}}{b^4 (3+m)}+\frac {d^3 (a+b x)^{4+m}}{b^4 (4+m)} \]
(-a*d+b*c)^3*(b*x+a)^(1+m)/b^4/(1+m)+3*d*(-a*d+b*c)^2*(b*x+a)^(2+m)/b^4/(2 +m)+3*d^2*(-a*d+b*c)*(b*x+a)^(3+m)/b^4/(3+m)+d^3*(b*x+a)^(4+m)/b^4/(4+m)
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int (a+b x)^m (c+d x)^3 \, dx=\frac {(a+b x)^{1+m} \left (\frac {(b c-a d)^3}{1+m}+\frac {3 d (b c-a d)^2 (a+b x)}{2+m}+\frac {3 d^2 (b c-a d) (a+b x)^2}{3+m}+\frac {d^3 (a+b x)^3}{4+m}\right )}{b^4} \]
((a + b*x)^(1 + m)*((b*c - a*d)^3/(1 + m) + (3*d*(b*c - a*d)^2*(a + b*x))/ (2 + m) + (3*d^2*(b*c - a*d)*(a + b*x)^2)/(3 + m) + (d^3*(a + b*x)^3)/(4 + m)))/b^4
Time = 0.26 (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.
| \(\displaystyle \int (c+d x)^3 (a+b x)^m \, dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (\frac {3 d^2 (b c-a d) (a+b x)^{m+2}}{b^3}+\frac {(b c-a d)^3 (a+b x)^m}{b^3}+\frac {3 d (b c-a d)^2 (a+b x)^{m+1}}{b^3}+\frac {d^3 (a+b x)^{m+3}}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 d^2 (b c-a d) (a+b x)^{m+3}}{b^4 (m+3)}+\frac {(b c-a d)^3 (a+b x)^{m+1}}{b^4 (m+1)}+\frac {3 d (b c-a d)^2 (a+b x)^{m+2}}{b^4 (m+2)}+\frac {d^3 (a+b x)^{m+4}}{b^4 (m+4)}\) |
((b*c - a*d)^3*(a + b*x)^(1 + m))/(b^4*(1 + m)) + (3*d*(b*c - a*d)^2*(a + b*x)^(2 + m))/(b^4*(2 + m)) + (3*d^2*(b*c - a*d)*(a + b*x)^(3 + m))/(b^4*( 3 + m)) + (d^3*(a + b*x)^(4 + m))/(b^4*(4 + m))
3.19.46.3.1 Defintions of rubi rules used
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])
Leaf count of result is larger than twice the leaf count of optimal. \(388\) vs. \(2(110)=220\).
Time = 0.41 (sec) , antiderivative size = 389, normalized size of antiderivative = 3.54
| method | result | size |
| gosper | \(-\frac {\left (b x +a \right )^{1+m} \left (-b^{3} d^{3} m^{3} x^{3}-3 b^{3} c \,d^{2} m^{3} x^{2}-6 b^{3} d^{3} m^{2} x^{3}+3 a \,b^{2} d^{3} m^{2} x^{2}-3 b^{3} c^{2} d \,m^{3} x -21 b^{3} c \,d^{2} m^{2} x^{2}-11 b^{3} d^{3} m \,x^{3}+6 a \,b^{2} c \,d^{2} m^{2} x +9 a \,b^{2} d^{3} m \,x^{2}-b^{3} c^{3} m^{3}-24 b^{3} c^{2} d \,m^{2} x -42 b^{3} c \,d^{2} m \,x^{2}-6 d^{3} x^{3} b^{3}-6 a^{2} b \,d^{3} m x +3 a \,b^{2} c^{2} d \,m^{2}+30 a \,b^{2} c \,d^{2} m x +6 x^{2} a \,b^{2} d^{3}-9 b^{3} c^{3} m^{2}-57 b^{3} c^{2} d m x -24 x^{2} b^{3} c \,d^{2}-6 a^{2} b c \,d^{2} m -6 x \,a^{2} b \,d^{3}+21 a \,b^{2} c^{2} d m +24 x a \,b^{2} c \,d^{2}-26 b^{3} c^{3} m -36 x \,b^{3} c^{2} d +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right )}{b^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(389\) |
| norman | \(\frac {d^{3} x^{4} {\mathrm e}^{m \ln \left (b x +a \right )}}{4+m}+\frac {\left (3 a \,b^{2} c^{2} d \,m^{3}+b^{3} c^{3} m^{3}-6 a^{2} b c \,d^{2} m^{2}+21 a \,b^{2} c^{2} d \,m^{2}+9 b^{3} c^{3} m^{2}+6 a^{3} d^{3} m -24 a^{2} b c \,d^{2} m +36 a \,b^{2} c^{2} d m +26 b^{3} c^{3} m +24 b^{3} c^{3}\right ) x \,{\mathrm e}^{m \ln \left (b x +a \right )}}{b^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (a d m +3 b c m +12 b c \right ) d^{2} x^{3} {\mathrm e}^{m \ln \left (b x +a \right )}}{b \left (m^{2}+7 m +12\right )}-\frac {a \left (-b^{3} c^{3} m^{3}+3 a \,b^{2} c^{2} d \,m^{2}-9 b^{3} c^{3} m^{2}-6 a^{2} b c \,d^{2} m +21 a \,b^{2} c^{2} d m -26 b^{3} c^{3} m +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right ) {\mathrm e}^{m \ln \left (b x +a \right )}}{b^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}-\frac {3 \left (-a b c d \,m^{2}-b^{2} c^{2} m^{2}+a^{2} d^{2} m -4 a b c d m -7 b^{2} c^{2} m -12 b^{2} c^{2}\right ) d \,x^{2} {\mathrm e}^{m \ln \left (b x +a \right )}}{b^{2} \left (m^{3}+9 m^{2}+26 m +24\right )}\) | \(433\) |
| risch | \(-\frac {\left (-b^{4} d^{3} m^{3} x^{4}-a \,b^{3} d^{3} m^{3} x^{3}-3 b^{4} c \,d^{2} m^{3} x^{3}-6 b^{4} d^{3} m^{2} x^{4}-3 a \,b^{3} c \,d^{2} m^{3} x^{2}-3 a \,b^{3} d^{3} m^{2} x^{3}-3 b^{4} c^{2} d \,m^{3} x^{2}-21 b^{4} c \,d^{2} m^{2} x^{3}-11 b^{4} d^{3} m \,x^{4}+3 a^{2} b^{2} d^{3} m^{2} x^{2}-3 a \,b^{3} c^{2} d \,m^{3} x -15 a \,b^{3} c \,d^{2} m^{2} x^{2}-2 a \,b^{3} d^{3} m \,x^{3}-b^{4} c^{3} m^{3} x -24 b^{4} c^{2} d \,m^{2} x^{2}-42 b^{4} c \,d^{2} m \,x^{3}-6 d^{3} x^{4} b^{4}+6 a^{2} b^{2} c \,d^{2} m^{2} x +3 a^{2} b^{2} d^{3} m \,x^{2}-a \,b^{3} c^{3} m^{3}-21 a \,b^{3} c^{2} d \,m^{2} x -12 a \,b^{3} c \,d^{2} m \,x^{2}-9 b^{4} c^{3} m^{2} x -57 b^{4} c^{2} d m \,x^{2}-24 b^{4} c \,d^{2} x^{3}-6 a^{3} b \,d^{3} m x +3 a^{2} b^{2} c^{2} d \,m^{2}+24 a^{2} b^{2} c \,d^{2} m x -9 a \,b^{3} c^{3} m^{2}-36 a \,b^{3} c^{2} d m x -26 b^{4} c^{3} m x -36 b^{4} c^{2} d \,x^{2}-6 a^{3} b c \,d^{2} m +21 a^{2} b^{2} c^{2} d m -26 a \,b^{3} c^{3} m -24 b^{4} c^{3} x +6 a^{4} d^{3}-24 a^{3} b c \,d^{2}+36 a^{2} b^{2} c^{2} d -24 a \,b^{3} c^{3}\right ) \left (b x +a \right )^{m}}{\left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) b^{4}}\) | \(552\) |
| parallelrisch | \(\frac {24 \left (b x +a \right )^{m} a^{2} b^{3} c^{3}+x^{4} \left (b x +a \right )^{m} a \,b^{4} d^{3} m^{3}+6 x^{4} \left (b x +a \right )^{m} a \,b^{4} d^{3} m^{2}+x^{3} \left (b x +a \right )^{m} a^{2} b^{3} d^{3} m^{3}+11 x^{4} \left (b x +a \right )^{m} a \,b^{4} d^{3} m +3 x^{3} \left (b x +a \right )^{m} a^{2} b^{3} d^{3} m^{2}+2 x^{3} \left (b x +a \right )^{m} a^{2} b^{3} d^{3} m -3 x^{2} \left (b x +a \right )^{m} a^{3} b^{2} d^{3} m^{2}+x \left (b x +a \right )^{m} a \,b^{4} c^{3} m^{3}+24 x^{3} \left (b x +a \right )^{m} a \,b^{4} c \,d^{2}-3 x^{2} \left (b x +a \right )^{m} a^{3} b^{2} d^{3} m +9 x \left (b x +a \right )^{m} a \,b^{4} c^{3} m^{2}+36 x^{2} \left (b x +a \right )^{m} a \,b^{4} c^{2} d +6 x \left (b x +a \right )^{m} a^{4} b \,d^{3} m +26 x \left (b x +a \right )^{m} a \,b^{4} c^{3} m -3 \left (b x +a \right )^{m} a^{3} b^{2} c^{2} d \,m^{2}+6 \left (b x +a \right )^{m} a^{4} b c \,d^{2} m -21 \left (b x +a \right )^{m} a^{3} b^{2} c^{2} d m +6 x^{4} \left (b x +a \right )^{m} a \,b^{4} d^{3}+\left (b x +a \right )^{m} a^{2} b^{3} c^{3} m^{3}+9 \left (b x +a \right )^{m} a^{2} b^{3} c^{3} m^{2}+24 x \left (b x +a \right )^{m} a \,b^{4} c^{3}+26 \left (b x +a \right )^{m} a^{2} b^{3} c^{3} m +24 \left (b x +a \right )^{m} a^{4} b c \,d^{2}+21 x \left (b x +a \right )^{m} a^{2} b^{3} c^{2} d \,m^{2}-24 x \left (b x +a \right )^{m} a^{3} b^{2} c \,d^{2} m +36 x \left (b x +a \right )^{m} a^{2} b^{3} c^{2} d m -36 \left (b x +a \right )^{m} a^{3} b^{2} c^{2} d -6 \left (b x +a \right )^{m} a^{5} d^{3}+3 x^{3} \left (b x +a \right )^{m} a \,b^{4} c \,d^{2} m^{3}+21 x^{3} \left (b x +a \right )^{m} a \,b^{4} c \,d^{2} m^{2}+3 x^{2} \left (b x +a \right )^{m} a^{2} b^{3} c \,d^{2} m^{3}+3 x^{2} \left (b x +a \right )^{m} a \,b^{4} c^{2} d \,m^{3}+42 x^{3} \left (b x +a \right )^{m} a \,b^{4} c \,d^{2} m +15 x^{2} \left (b x +a \right )^{m} a^{2} b^{3} c \,d^{2} m^{2}+24 x^{2} \left (b x +a \right )^{m} a \,b^{4} c^{2} d \,m^{2}+3 x \left (b x +a \right )^{m} a^{2} b^{3} c^{2} d \,m^{3}+12 x^{2} \left (b x +a \right )^{m} a^{2} b^{3} c \,d^{2} m +57 x^{2} \left (b x +a \right )^{m} a \,b^{4} c^{2} d m -6 x \left (b x +a \right )^{m} a^{3} b^{2} c \,d^{2} m^{2}}{\left (m^{3}+9 m^{2}+26 m +24\right ) a \left (1+m \right ) b^{4}}\) | \(865\) |
-1/b^4*(b*x+a)^(1+m)/(m^4+10*m^3+35*m^2+50*m+24)*(-b^3*d^3*m^3*x^3-3*b^3*c *d^2*m^3*x^2-6*b^3*d^3*m^2*x^3+3*a*b^2*d^3*m^2*x^2-3*b^3*c^2*d*m^3*x-21*b^ 3*c*d^2*m^2*x^2-11*b^3*d^3*m*x^3+6*a*b^2*c*d^2*m^2*x+9*a*b^2*d^3*m*x^2-b^3 *c^3*m^3-24*b^3*c^2*d*m^2*x-42*b^3*c*d^2*m*x^2-6*b^3*d^3*x^3-6*a^2*b*d^3*m *x+3*a*b^2*c^2*d*m^2+30*a*b^2*c*d^2*m*x+6*a*b^2*d^3*x^2-9*b^3*c^3*m^2-57*b ^3*c^2*d*m*x-24*b^3*c*d^2*x^2-6*a^2*b*c*d^2*m-6*a^2*b*d^3*x+21*a*b^2*c^2*d *m+24*a*b^2*c*d^2*x-26*b^3*c^3*m-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)
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (110) = 220\).
Time = 0.23 (sec) , antiderivative size = 497, normalized size of antiderivative = 4.52 \[ \int (a+b x)^m (c+d x)^3 \, dx=\frac {{\left (a b^{3} c^{3} m^{3} + 24 \, a b^{3} c^{3} - 36 \, a^{2} b^{2} c^{2} d + 24 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} + {\left (b^{4} d^{3} m^{3} + 6 \, b^{4} d^{3} m^{2} + 11 \, b^{4} d^{3} m + 6 \, b^{4} d^{3}\right )} x^{4} + {\left (24 \, b^{4} c d^{2} + {\left (3 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} m^{3} + 3 \, {\left (7 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} m^{2} + 2 \, {\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} m\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )} m^{2} + 3 \, {\left (12 \, b^{4} c^{2} d + {\left (b^{4} c^{2} d + a b^{3} c d^{2}\right )} m^{3} + {\left (8 \, b^{4} c^{2} d + 5 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} m^{2} + {\left (19 \, b^{4} c^{2} d + 4 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} m\right )} x^{2} + {\left (26 \, a b^{3} c^{3} - 21 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2}\right )} m + {\left (24 \, b^{4} c^{3} + {\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d\right )} m^{3} + 3 \, {\left (3 \, b^{4} c^{3} + 7 \, a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2}\right )} m^{2} + 2 \, {\left (13 \, b^{4} c^{3} + 18 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 3 \, a^{3} b d^{3}\right )} m\right )} x\right )} {\left (b x + a\right )}^{m}}{b^{4} m^{4} + 10 \, b^{4} m^{3} + 35 \, b^{4} m^{2} + 50 \, b^{4} m + 24 \, b^{4}} \]
(a*b^3*c^3*m^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*m^3 + 6*b^4*d^3*m^2 + 11*b^4*d^3*m + 6*b^4*d^3)*x^4 + (24*b ^4*c*d^2 + (3*b^4*c*d^2 + a*b^3*d^3)*m^3 + 3*(7*b^4*c*d^2 + a*b^3*d^3)*m^2 + 2*(21*b^4*c*d^2 + a*b^3*d^3)*m)*x^3 + 3*(3*a*b^3*c^3 - a^2*b^2*c^2*d)*m ^2 + 3*(12*b^4*c^2*d + (b^4*c^2*d + a*b^3*c*d^2)*m^3 + (8*b^4*c^2*d + 5*a* b^3*c*d^2 - a^2*b^2*d^3)*m^2 + (19*b^4*c^2*d + 4*a*b^3*c*d^2 - a^2*b^2*d^3 )*m)*x^2 + (26*a*b^3*c^3 - 21*a^2*b^2*c^2*d + 6*a^3*b*c*d^2)*m + (24*b^4*c ^3 + (b^4*c^3 + 3*a*b^3*c^2*d)*m^3 + 3*(3*b^4*c^3 + 7*a*b^3*c^2*d - 2*a^2* b^2*c*d^2)*m^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)*m)*x)*(b*x + a)^m/(b^4*m^4 + 10*b^4*m^3 + 35*b^4*m^2 + 50*b^4*m + 24*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 4058 vs. \(2 (95) = 190\).
Time = 1.13 (sec) , antiderivative size = 4058, normalized size of antiderivative = 36.89 \[ \int (a+b x)^m (c+d x)^3 \, dx=\text {Too large to display} \]
Piecewise((a**m*(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 (m, -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 + ...
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (110) = 220\).
Time = 0.21 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.24 \[ \int (a+b x)^m (c+d x)^3 \, dx=\frac {3 \, {\left (b^{2} {\left (m + 1\right )} x^{2} + a b m x - a^{2}\right )} {\left (b x + a\right )}^{m} c^{2} d}{{\left (m^{2} + 3 \, m + 2\right )} b^{2}} + \frac {{\left (b x + a\right )}^{m + 1} c^{3}}{b {\left (m + 1\right )}} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} x^{3} + {\left (m^{2} + m\right )} a b^{2} x^{2} - 2 \, a^{2} b m x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{m} c d^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a b^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b m x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{m} d^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} b^{4}} \]
3*(b^2*(m + 1)*x^2 + a*b*m*x - a^2)*(b*x + a)^m*c^2*d/((m^2 + 3*m + 2)*b^2 ) + (b*x + a)^(m + 1)*c^3/(b*(m + 1)) + 3*((m^2 + 3*m + 2)*b^3*x^3 + (m^2 + m)*a*b^2*x^2 - 2*a^2*b*m*x + 2*a^3)*(b*x + a)^m*c*d^2/((m^3 + 6*m^2 + 11 *m + 6)*b^3) + ((m^3 + 6*m^2 + 11*m + 6)*b^4*x^4 + (m^3 + 3*m^2 + 2*m)*a*b ^3*x^3 - 3*(m^2 + m)*a^2*b^2*x^2 + 6*a^3*b*m*x - 6*a^4)*(b*x + a)^m*d^3/(( m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 833 vs. \(2 (110) = 220\).
Time = 0.31 (sec) , antiderivative size = 833, normalized size of antiderivative = 7.57 \[ \int (a+b x)^m (c+d x)^3 \, dx =\text {Too large to display} \]
((b*x + a)^m*b^4*d^3*m^3*x^4 + 3*(b*x + a)^m*b^4*c*d^2*m^3*x^3 + (b*x + a) ^m*a*b^3*d^3*m^3*x^3 + 6*(b*x + a)^m*b^4*d^3*m^2*x^4 + 3*(b*x + a)^m*b^4*c ^2*d*m^3*x^2 + 3*(b*x + a)^m*a*b^3*c*d^2*m^3*x^2 + 21*(b*x + a)^m*b^4*c*d^ 2*m^2*x^3 + 3*(b*x + a)^m*a*b^3*d^3*m^2*x^3 + 11*(b*x + a)^m*b^4*d^3*m*x^4 + (b*x + a)^m*b^4*c^3*m^3*x + 3*(b*x + a)^m*a*b^3*c^2*d*m^3*x + 24*(b*x + a)^m*b^4*c^2*d*m^2*x^2 + 15*(b*x + a)^m*a*b^3*c*d^2*m^2*x^2 - 3*(b*x + a) ^m*a^2*b^2*d^3*m^2*x^2 + 42*(b*x + a)^m*b^4*c*d^2*m*x^3 + 2*(b*x + a)^m*a* b^3*d^3*m*x^3 + 6*(b*x + a)^m*b^4*d^3*x^4 + (b*x + a)^m*a*b^3*c^3*m^3 + 9* (b*x + a)^m*b^4*c^3*m^2*x + 21*(b*x + a)^m*a*b^3*c^2*d*m^2*x - 6*(b*x + a) ^m*a^2*b^2*c*d^2*m^2*x + 57*(b*x + a)^m*b^4*c^2*d*m*x^2 + 12*(b*x + a)^m*a *b^3*c*d^2*m*x^2 - 3*(b*x + a)^m*a^2*b^2*d^3*m*x^2 + 24*(b*x + a)^m*b^4*c* d^2*x^3 + 9*(b*x + a)^m*a*b^3*c^3*m^2 - 3*(b*x + a)^m*a^2*b^2*c^2*d*m^2 + 26*(b*x + a)^m*b^4*c^3*m*x + 36*(b*x + a)^m*a*b^3*c^2*d*m*x - 24*(b*x + a) ^m*a^2*b^2*c*d^2*m*x + 6*(b*x + a)^m*a^3*b*d^3*m*x + 36*(b*x + a)^m*b^4*c^ 2*d*x^2 + 26*(b*x + a)^m*a*b^3*c^3*m - 21*(b*x + a)^m*a^2*b^2*c^2*d*m + 6* (b*x + a)^m*a^3*b*c*d^2*m + 24*(b*x + a)^m*b^4*c^3*x + 24*(b*x + a)^m*a*b^ 3*c^3 - 36*(b*x + a)^m*a^2*b^2*c^2*d + 24*(b*x + a)^m*a^3*b*c*d^2 - 6*(b*x + a)^m*a^4*d^3)/(b^4*m^4 + 10*b^4*m^3 + 35*b^4*m^2 + 50*b^4*m + 24*b^4)
Time = 0.99 (sec) , antiderivative size = 478, normalized size of antiderivative = 4.35 \[ \int (a+b x)^m (c+d x)^3 \, dx=\frac {d^3\,x^4\,{\left (a+b\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {a\,{\left (a+b\,x\right )}^m\,\left (-6\,a^3\,d^3+6\,a^2\,b\,c\,d^2\,m+24\,a^2\,b\,c\,d^2-3\,a\,b^2\,c^2\,d\,m^2-21\,a\,b^2\,c^2\,d\,m-36\,a\,b^2\,c^2\,d+b^3\,c^3\,m^3+9\,b^3\,c^3\,m^2+26\,b^3\,c^3\,m+24\,b^3\,c^3\right )}{b^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x\,{\left (a+b\,x\right )}^m\,\left (6\,a^3\,b\,d^3\,m-6\,a^2\,b^2\,c\,d^2\,m^2-24\,a^2\,b^2\,c\,d^2\,m+3\,a\,b^3\,c^2\,d\,m^3+21\,a\,b^3\,c^2\,d\,m^2+36\,a\,b^3\,c^2\,d\,m+b^4\,c^3\,m^3+9\,b^4\,c^3\,m^2+26\,b^4\,c^3\,m+24\,b^4\,c^3\right )}{b^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {3\,d\,x^2\,\left (m+1\right )\,{\left (a+b\,x\right )}^m\,\left (-a^2\,d^2\,m+a\,b\,c\,d\,m^2+4\,a\,b\,c\,d\,m+b^2\,c^2\,m^2+7\,b^2\,c^2\,m+12\,b^2\,c^2\right )}{b^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {d^2\,x^3\,{\left (a+b\,x\right )}^m\,\left (12\,b\,c+a\,d\,m+3\,b\,c\,m\right )\,\left (m^2+3\,m+2\right )}{b\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \]
(d^3*x^4*(a + b*x)^m*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m ^4 + 24) + (a*(a + b*x)^m*(24*b^3*c^3 - 6*a^3*d^3 + 26*b^3*c^3*m + 9*b^3*c ^3*m^2 + b^3*c^3*m^3 - 36*a*b^2*c^2*d + 24*a^2*b*c*d^2 - 21*a*b^2*c^2*d*m + 6*a^2*b*c*d^2*m - 3*a*b^2*c^2*d*m^2))/(b^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x*(a + b*x)^m*(24*b^4*c^3 + 26*b^4*c^3*m + 9*b^4*c^3*m^2 + b^4* c^3*m^3 + 6*a^3*b*d^3*m + 36*a*b^3*c^2*d*m - 24*a^2*b^2*c*d^2*m + 21*a*b^3 *c^2*d*m^2 + 3*a*b^3*c^2*d*m^3 - 6*a^2*b^2*c*d^2*m^2))/(b^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (3*d*x^2*(m + 1)*(a + b*x)^m*(12*b^2*c^2 - a^2*d^ 2*m + 7*b^2*c^2*m + b^2*c^2*m^2 + 4*a*b*c*d*m + a*b*c*d*m^2))/(b^2*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (d^2*x^3*(a + b*x)^m*(12*b*c + a*d*m + 3*b *c*m)*(3*m + m^2 + 2))/(b*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))
Time = 0.02 (sec) , antiderivative size = 546, normalized size of antiderivative = 4.96 \[ \int (a+b x)^m (c+d x)^3 \, dx=\frac {\left (b x +a \right )^{m} \left (b^{4} d^{3} m^{3} x^{4}+a \,b^{3} d^{3} m^{3} x^{3}+3 b^{4} c \,d^{2} m^{3} x^{3}+6 b^{4} d^{3} m^{2} x^{4}+3 a \,b^{3} c \,d^{2} m^{3} x^{2}+3 a \,b^{3} d^{3} m^{2} x^{3}+3 b^{4} c^{2} d \,m^{3} x^{2}+21 b^{4} c \,d^{2} m^{2} x^{3}+11 b^{4} d^{3} m \,x^{4}-3 a^{2} b^{2} d^{3} m^{2} x^{2}+3 a \,b^{3} c^{2} d \,m^{3} x +15 a \,b^{3} c \,d^{2} m^{2} x^{2}+2 a \,b^{3} d^{3} m \,x^{3}+b^{4} c^{3} m^{3} x +24 b^{4} c^{2} d \,m^{2} x^{2}+42 b^{4} c \,d^{2} m \,x^{3}+6 b^{4} d^{3} x^{4}-6 a^{2} b^{2} c \,d^{2} m^{2} x -3 a^{2} b^{2} d^{3} m \,x^{2}+a \,b^{3} c^{3} m^{3}+21 a \,b^{3} c^{2} d \,m^{2} x +12 a \,b^{3} c \,d^{2} m \,x^{2}+9 b^{4} c^{3} m^{2} x +57 b^{4} c^{2} d m \,x^{2}+24 b^{4} c \,d^{2} x^{3}+6 a^{3} b \,d^{3} m x -3 a^{2} b^{2} c^{2} d \,m^{2}-24 a^{2} b^{2} c \,d^{2} m x +9 a \,b^{3} c^{3} m^{2}+36 a \,b^{3} c^{2} d m x +26 b^{4} c^{3} m x +36 b^{4} c^{2} d \,x^{2}+6 a^{3} b c \,d^{2} m -21 a^{2} b^{2} c^{2} d m +26 a \,b^{3} c^{3} m +24 b^{4} c^{3} x -6 a^{4} d^{3}+24 a^{3} b c \,d^{2}-36 a^{2} b^{2} c^{2} d +24 a \,b^{3} c^{3}\right )}{b^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )} \]
((a + b*x)**m*( - 6*a**4*d**3 + 6*a**3*b*c*d**2*m + 24*a**3*b*c*d**2 + 6*a **3*b*d**3*m*x - 3*a**2*b**2*c**2*d*m**2 - 21*a**2*b**2*c**2*d*m - 36*a**2 *b**2*c**2*d - 6*a**2*b**2*c*d**2*m**2*x - 24*a**2*b**2*c*d**2*m*x - 3*a** 2*b**2*d**3*m**2*x**2 - 3*a**2*b**2*d**3*m*x**2 + a*b**3*c**3*m**3 + 9*a*b **3*c**3*m**2 + 26*a*b**3*c**3*m + 24*a*b**3*c**3 + 3*a*b**3*c**2*d*m**3*x + 21*a*b**3*c**2*d*m**2*x + 36*a*b**3*c**2*d*m*x + 3*a*b**3*c*d**2*m**3*x **2 + 15*a*b**3*c*d**2*m**2*x**2 + 12*a*b**3*c*d**2*m*x**2 + a*b**3*d**3*m **3*x**3 + 3*a*b**3*d**3*m**2*x**3 + 2*a*b**3*d**3*m*x**3 + b**4*c**3*m**3 *x + 9*b**4*c**3*m**2*x + 26*b**4*c**3*m*x + 24*b**4*c**3*x + 3*b**4*c**2* d*m**3*x**2 + 24*b**4*c**2*d*m**2*x**2 + 57*b**4*c**2*d*m*x**2 + 36*b**4*c **2*d*x**2 + 3*b**4*c*d**2*m**3*x**3 + 21*b**4*c*d**2*m**2*x**3 + 42*b**4* c*d**2*m*x**3 + 24*b**4*c*d**2*x**3 + b**4*d**3*m**3*x**4 + 6*b**4*d**3*m* *2*x**4 + 11*b**4*d**3*m*x**4 + 6*b**4*d**3*x**4))/(b**4*(m**4 + 10*m**3 + 35*m**2 + 50*m + 24))