Integrand size = 16, antiderivative size = 154 \[ \int x (a+b x)^n (c+d x)^3 \, dx=-\frac {a (b c-a d)^3 (a+b x)^{1+n}}{b^5 (1+n)}+\frac {(b c-4 a d) (b c-a d)^2 (a+b x)^{2+n}}{b^5 (2+n)}+\frac {3 d (b c-2 a d) (b c-a d) (a+b x)^{3+n}}{b^5 (3+n)}+\frac {d^2 (3 b c-4 a d) (a+b x)^{4+n}}{b^5 (4+n)}+\frac {d^3 (a+b x)^{5+n}}{b^5 (5+n)} \]
-a*(-a*d+b*c)^3*(b*x+a)^(1+n)/b^5/(1+n)+(-4*a*d+b*c)*(-a*d+b*c)^2*(b*x+a)^ (2+n)/b^5/(2+n)+3*d*(-2*a*d+b*c)*(-a*d+b*c)*(b*x+a)^(3+n)/b^5/(3+n)+d^2*(- 4*a*d+3*b*c)*(b*x+a)^(4+n)/b^5/(4+n)+d^3*(b*x+a)^(5+n)/b^5/(5+n)
Time = 0.14 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.86 \[ \int x (a+b x)^n (c+d x)^3 \, dx=\frac {(a+b x)^{1+n} \left (\frac {a (-b c+a d)^3}{1+n}+\frac {(b c-4 a d) (b c-a d)^2 (a+b x)}{2+n}+\frac {3 d (b c-2 a d) (b c-a d) (a+b x)^2}{3+n}+\frac {d^2 (3 b c-4 a d) (a+b x)^3}{4+n}+\frac {d^3 (a+b x)^4}{5+n}\right )}{b^5} \]
((a + b*x)^(1 + n)*((a*(-(b*c) + a*d)^3)/(1 + n) + ((b*c - 4*a*d)*(b*c - a *d)^2*(a + b*x))/(2 + n) + (3*d*(b*c - 2*a*d)*(b*c - a*d)*(a + b*x)^2)/(3 + n) + (d^2*(3*b*c - 4*a*d)*(a + b*x)^3)/(4 + n) + (d^3*(a + b*x)^4)/(5 + n)))/b^5
Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {86, 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.
| \(\displaystyle \int x (c+d x)^3 (a+b x)^n \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {d^2 (3 b c-4 a d) (a+b x)^{n+3}}{b^4}+\frac {a (a d-b c)^3 (a+b x)^n}{b^4}+\frac {(b c-4 a d) (b c-a d)^2 (a+b x)^{n+1}}{b^4}+\frac {3 d (b c-2 a d) (b c-a d) (a+b x)^{n+2}}{b^4}+\frac {d^3 (a+b x)^{n+4}}{b^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 (3 b c-4 a d) (a+b x)^{n+4}}{b^5 (n+4)}-\frac {a (b c-a d)^3 (a+b x)^{n+1}}{b^5 (n+1)}+\frac {(b c-4 a d) (b c-a d)^2 (a+b x)^{n+2}}{b^5 (n+2)}+\frac {3 d (b c-2 a d) (b c-a d) (a+b x)^{n+3}}{b^5 (n+3)}+\frac {d^3 (a+b x)^{n+5}}{b^5 (n+5)}\) |
-((a*(b*c - a*d)^3*(a + b*x)^(1 + n))/(b^5*(1 + n))) + ((b*c - 4*a*d)*(b*c - a*d)^2*(a + b*x)^(2 + n))/(b^5*(2 + n)) + (3*d*(b*c - 2*a*d)*(b*c - a*d )*(a + b*x)^(3 + n))/(b^5*(3 + n)) + (d^2*(3*b*c - 4*a*d)*(a + b*x)^(4 + n ))/(b^5*(4 + n)) + (d^3*(a + b*x)^(5 + n))/(b^5*(5 + n))
3.10.31.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Leaf count of result is larger than twice the leaf count of optimal. \(586\) vs. \(2(154)=308\).
Time = 0.68 (sec) , antiderivative size = 587, normalized size of antiderivative = 3.81
| method | result | size |
| norman | \(\frac {d^{3} x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{5+n}+\frac {a^{2} \left (-b^{3} c^{3} n^{3}+6 a \,b^{2} c^{2} d \,n^{2}-12 b^{3} c^{3} n^{2}-18 a^{2} b c \,d^{2} n +54 a \,b^{2} c^{2} d n -47 b^{3} c^{3} n +24 a^{3} d^{3}-90 a^{2} b c \,d^{2}+120 a \,b^{2} c^{2} d -60 b^{3} c^{3}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}+\frac {\left (3 a \,b^{2} c^{2} d \,n^{3}+b^{3} c^{3} n^{3}-9 a^{2} b c \,d^{2} n^{2}+27 a \,b^{2} c^{2} d \,n^{2}+12 b^{3} c^{3} n^{2}+12 a^{3} d^{3} n -45 a^{2} b c \,d^{2} n +60 a \,b^{2} c^{2} d n +47 b^{3} c^{3} n +60 b^{3} c^{3}\right ) x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+14 n^{3}+71 n^{2}+154 n +120\right )}+\frac {\left (a d n +3 b c n +15 b c \right ) d^{2} x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+9 n +20\right )}-\frac {\left (-3 a b c d \,n^{2}-3 b^{2} c^{2} n^{2}+4 a^{2} d^{2} n -15 a b c d n -27 b^{2} c^{2} n -60 b^{2} c^{2}\right ) d \,x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+12 n^{2}+47 n +60\right )}-\frac {n a \left (-b^{3} c^{3} n^{3}+6 a \,b^{2} c^{2} d \,n^{2}-12 b^{3} c^{3} n^{2}-18 a^{2} b c \,d^{2} n +54 a \,b^{2} c^{2} d n -47 b^{3} c^{3} n +24 a^{3} d^{3}-90 a^{2} b c \,d^{2}+120 a \,b^{2} c^{2} d -60 b^{3} c^{3}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(587\) |
| gosper | \(\frac {\left (b x +a \right )^{1+n} \left (b^{4} d^{3} n^{4} x^{4}+3 b^{4} c \,d^{2} n^{4} x^{3}+10 b^{4} d^{3} n^{3} x^{4}-4 a \,b^{3} d^{3} n^{3} x^{3}+3 b^{4} c^{2} d \,n^{4} x^{2}+33 b^{4} c \,d^{2} n^{3} x^{3}+35 b^{4} d^{3} n^{2} x^{4}-9 a \,b^{3} c \,d^{2} n^{3} x^{2}-24 a \,b^{3} d^{3} n^{2} x^{3}+b^{4} c^{3} n^{4} x +36 b^{4} c^{2} d \,n^{3} x^{2}+123 b^{4} c \,d^{2} n^{2} x^{3}+50 b^{4} d^{3} n \,x^{4}+12 a^{2} b^{2} d^{3} n^{2} x^{2}-6 a \,b^{3} c^{2} d \,n^{3} x -72 a \,b^{3} c \,d^{2} n^{2} x^{2}-44 a \,b^{3} d^{3} n \,x^{3}+13 b^{4} c^{3} n^{3} x +147 b^{4} c^{2} d \,n^{2} x^{2}+183 b^{4} c \,d^{2} n \,x^{3}+24 d^{3} x^{4} b^{4}+18 a^{2} b^{2} c \,d^{2} n^{2} x +36 a^{2} b^{2} d^{3} n \,x^{2}-a \,b^{3} c^{3} n^{3}-60 a \,b^{3} c^{2} d \,n^{2} x -153 a \,b^{3} c \,d^{2} n \,x^{2}-24 x^{3} a \,b^{3} d^{3}+59 b^{4} c^{3} n^{2} x +234 b^{4} c^{2} d n \,x^{2}+90 x^{3} b^{4} c \,d^{2}-24 a^{3} b \,d^{3} n x +6 a^{2} b^{2} c^{2} d \,n^{2}+108 a^{2} b^{2} c \,d^{2} n x +24 x^{2} a^{2} b^{2} d^{3}-12 a \,b^{3} c^{3} n^{2}-174 a \,b^{3} c^{2} d n x -90 x^{2} a \,b^{3} c \,d^{2}+107 b^{4} c^{3} n x +120 x^{2} b^{4} c^{2} d -18 a^{3} b c \,d^{2} n -24 x \,a^{3} b \,d^{3}+54 a^{2} b^{2} c^{2} d n +90 x \,a^{2} b^{2} c \,d^{2}-47 a \,b^{3} c^{3} n -120 x a \,b^{3} c^{2} d +60 b^{4} c^{3} x +24 a^{4} d^{3}-90 a^{3} b c \,d^{2}+120 a^{2} b^{2} c^{2} d -60 a \,b^{3} c^{3}\right )}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(685\) |
| risch | \(\frac {\left (b^{5} d^{3} n^{4} x^{5}+a \,b^{4} d^{3} n^{4} x^{4}+3 b^{5} c \,d^{2} n^{4} x^{4}+10 b^{5} d^{3} n^{3} x^{5}+3 a \,b^{4} c \,d^{2} n^{4} x^{3}+6 a \,b^{4} d^{3} n^{3} x^{4}+3 b^{5} c^{2} d \,n^{4} x^{3}+33 b^{5} c \,d^{2} n^{3} x^{4}+35 b^{5} d^{3} n^{2} x^{5}-4 a^{2} b^{3} d^{3} n^{3} x^{3}+3 a \,b^{4} c^{2} d \,n^{4} x^{2}+24 a \,b^{4} c \,d^{2} n^{3} x^{3}+11 a \,b^{4} d^{3} n^{2} x^{4}+b^{5} c^{3} n^{4} x^{2}+36 b^{5} c^{2} d \,n^{3} x^{3}+123 b^{5} c \,d^{2} n^{2} x^{4}+50 b^{5} d^{3} n \,x^{5}-9 a^{2} b^{3} c \,d^{2} n^{3} x^{2}-12 a^{2} b^{3} d^{3} n^{2} x^{3}+a \,b^{4} c^{3} n^{4} x +30 a \,b^{4} c^{2} d \,n^{3} x^{2}+51 a \,b^{4} c \,d^{2} n^{2} x^{3}+6 a \,b^{4} d^{3} n \,x^{4}+13 b^{5} c^{3} n^{3} x^{2}+147 b^{5} c^{2} d \,n^{2} x^{3}+183 b^{5} c \,d^{2} n \,x^{4}+24 d^{3} x^{5} b^{5}+12 a^{3} b^{2} d^{3} n^{2} x^{2}-6 a^{2} b^{3} c^{2} d \,n^{3} x -54 a^{2} b^{3} c \,d^{2} n^{2} x^{2}-8 a^{2} b^{3} d^{3} n \,x^{3}+12 a \,b^{4} c^{3} n^{3} x +87 a \,b^{4} c^{2} d \,n^{2} x^{2}+30 a \,b^{4} c \,d^{2} n \,x^{3}+59 b^{5} c^{3} n^{2} x^{2}+234 b^{5} c^{2} d n \,x^{3}+90 x^{4} b^{5} c \,d^{2}+18 a^{3} b^{2} c \,d^{2} n^{2} x +12 a^{3} b^{2} d^{3} n \,x^{2}-a^{2} b^{3} c^{3} n^{3}-54 a^{2} b^{3} c^{2} d \,n^{2} x -45 a^{2} b^{3} c \,d^{2} n \,x^{2}+47 a \,b^{4} c^{3} n^{2} x +60 a \,b^{4} c^{2} d n \,x^{2}+107 b^{5} c^{3} n \,x^{2}+120 x^{3} b^{5} c^{2} d -24 a^{4} b \,d^{3} n x +6 a^{3} b^{2} c^{2} d \,n^{2}+90 a^{3} b^{2} c \,d^{2} n x -12 a^{2} b^{3} c^{3} n^{2}-120 a^{2} b^{3} c^{2} d n x +60 a \,b^{4} c^{3} n x +60 x^{2} b^{5} c^{3}-18 a^{4} b c \,d^{2} n +54 a^{3} b^{2} c^{2} d n -47 a^{2} b^{3} c^{3} n +24 a^{5} d^{3}-90 a^{4} b c \,d^{2}+120 a^{3} b^{2} c^{2} d -60 a^{2} b^{3} c^{3}\right ) \left (b x +a \right )^{n}}{\left (4+n \right ) \left (5+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{5}}\) | \(876\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1289\) |
d^3/(5+n)*x^5*exp(n*ln(b*x+a))+a^2*(-b^3*c^3*n^3+6*a*b^2*c^2*d*n^2-12*b^3* c^3*n^2-18*a^2*b*c*d^2*n+54*a*b^2*c^2*d*n-47*b^3*c^3*n+24*a^3*d^3-90*a^2*b *c*d^2+120*a*b^2*c^2*d-60*b^3*c^3)/b^5/(n^5+15*n^4+85*n^3+225*n^2+274*n+12 0)*exp(n*ln(b*x+a))+(3*a*b^2*c^2*d*n^3+b^3*c^3*n^3-9*a^2*b*c*d^2*n^2+27*a* b^2*c^2*d*n^2+12*b^3*c^3*n^2+12*a^3*d^3*n-45*a^2*b*c*d^2*n+60*a*b^2*c^2*d* n+47*b^3*c^3*n+60*b^3*c^3)/b^3/(n^4+14*n^3+71*n^2+154*n+120)*x^2*exp(n*ln( b*x+a))+(a*d*n+3*b*c*n+15*b*c)/b*d^2/(n^2+9*n+20)*x^4*exp(n*ln(b*x+a))-(-3 *a*b*c*d*n^2-3*b^2*c^2*n^2+4*a^2*d^2*n-15*a*b*c*d*n-27*b^2*c^2*n-60*b^2*c^ 2)*d/b^2/(n^3+12*n^2+47*n+60)*x^3*exp(n*ln(b*x+a))-1/b^4*n*a*(-b^3*c^3*n^3 +6*a*b^2*c^2*d*n^2-12*b^3*c^3*n^2-18*a^2*b*c*d^2*n+54*a*b^2*c^2*d*n-47*b^3 *c^3*n+24*a^3*d^3-90*a^2*b*c*d^2+120*a*b^2*c^2*d-60*b^3*c^3)/(n^5+15*n^4+8 5*n^3+225*n^2+274*n+120)*x*exp(n*ln(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (154) = 308\).
Time = 0.24 (sec) , antiderivative size = 766, normalized size of antiderivative = 4.97 \[ \int x (a+b x)^n (c+d x)^3 \, dx=-\frac {{\left (a^{2} b^{3} c^{3} n^{3} + 60 \, a^{2} b^{3} c^{3} - 120 \, a^{3} b^{2} c^{2} d + 90 \, a^{4} b c d^{2} - 24 \, a^{5} d^{3} - {\left (b^{5} d^{3} n^{4} + 10 \, b^{5} d^{3} n^{3} + 35 \, b^{5} d^{3} n^{2} + 50 \, b^{5} d^{3} n + 24 \, b^{5} d^{3}\right )} x^{5} - {\left (90 \, b^{5} c d^{2} + {\left (3 \, b^{5} c d^{2} + a b^{4} d^{3}\right )} n^{4} + 3 \, {\left (11 \, b^{5} c d^{2} + 2 \, a b^{4} d^{3}\right )} n^{3} + {\left (123 \, b^{5} c d^{2} + 11 \, a b^{4} d^{3}\right )} n^{2} + 3 \, {\left (61 \, b^{5} c d^{2} + 2 \, a b^{4} d^{3}\right )} n\right )} x^{4} - {\left (120 \, b^{5} c^{2} d + 3 \, {\left (b^{5} c^{2} d + a b^{4} c d^{2}\right )} n^{4} + 4 \, {\left (9 \, b^{5} c^{2} d + 6 \, a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )} n^{3} + 3 \, {\left (49 \, b^{5} c^{2} d + 17 \, a b^{4} c d^{2} - 4 \, a^{2} b^{3} d^{3}\right )} n^{2} + 2 \, {\left (117 \, b^{5} c^{2} d + 15 \, a b^{4} c d^{2} - 4 \, a^{2} b^{3} d^{3}\right )} n\right )} x^{3} + 6 \, {\left (2 \, a^{2} b^{3} c^{3} - a^{3} b^{2} c^{2} d\right )} n^{2} - {\left (60 \, b^{5} c^{3} + {\left (b^{5} c^{3} + 3 \, a b^{4} c^{2} d\right )} n^{4} + {\left (13 \, b^{5} c^{3} + 30 \, a b^{4} c^{2} d - 9 \, a^{2} b^{3} c d^{2}\right )} n^{3} + {\left (59 \, b^{5} c^{3} + 87 \, a b^{4} c^{2} d - 54 \, a^{2} b^{3} c d^{2} + 12 \, a^{3} b^{2} d^{3}\right )} n^{2} + {\left (107 \, b^{5} c^{3} + 60 \, a b^{4} c^{2} d - 45 \, a^{2} b^{3} c d^{2} + 12 \, a^{3} b^{2} d^{3}\right )} n\right )} x^{2} + {\left (47 \, a^{2} b^{3} c^{3} - 54 \, a^{3} b^{2} c^{2} d + 18 \, a^{4} b c d^{2}\right )} n - {\left (a b^{4} c^{3} n^{4} + 6 \, {\left (2 \, a b^{4} c^{3} - a^{2} b^{3} c^{2} d\right )} n^{3} + {\left (47 \, a b^{4} c^{3} - 54 \, a^{2} b^{3} c^{2} d + 18 \, a^{3} b^{2} c d^{2}\right )} n^{2} + 6 \, {\left (10 \, a b^{4} c^{3} - 20 \, a^{2} b^{3} c^{2} d + 15 \, a^{3} b^{2} c d^{2} - 4 \, a^{4} b d^{3}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \]
-(a^2*b^3*c^3*n^3 + 60*a^2*b^3*c^3 - 120*a^3*b^2*c^2*d + 90*a^4*b*c*d^2 - 24*a^5*d^3 - (b^5*d^3*n^4 + 10*b^5*d^3*n^3 + 35*b^5*d^3*n^2 + 50*b^5*d^3*n + 24*b^5*d^3)*x^5 - (90*b^5*c*d^2 + (3*b^5*c*d^2 + a*b^4*d^3)*n^4 + 3*(11 *b^5*c*d^2 + 2*a*b^4*d^3)*n^3 + (123*b^5*c*d^2 + 11*a*b^4*d^3)*n^2 + 3*(61 *b^5*c*d^2 + 2*a*b^4*d^3)*n)*x^4 - (120*b^5*c^2*d + 3*(b^5*c^2*d + a*b^4*c *d^2)*n^4 + 4*(9*b^5*c^2*d + 6*a*b^4*c*d^2 - a^2*b^3*d^3)*n^3 + 3*(49*b^5* c^2*d + 17*a*b^4*c*d^2 - 4*a^2*b^3*d^3)*n^2 + 2*(117*b^5*c^2*d + 15*a*b^4* c*d^2 - 4*a^2*b^3*d^3)*n)*x^3 + 6*(2*a^2*b^3*c^3 - a^3*b^2*c^2*d)*n^2 - (6 0*b^5*c^3 + (b^5*c^3 + 3*a*b^4*c^2*d)*n^4 + (13*b^5*c^3 + 30*a*b^4*c^2*d - 9*a^2*b^3*c*d^2)*n^3 + (59*b^5*c^3 + 87*a*b^4*c^2*d - 54*a^2*b^3*c*d^2 + 12*a^3*b^2*d^3)*n^2 + (107*b^5*c^3 + 60*a*b^4*c^2*d - 45*a^2*b^3*c*d^2 + 1 2*a^3*b^2*d^3)*n)*x^2 + (47*a^2*b^3*c^3 - 54*a^3*b^2*c^2*d + 18*a^4*b*c*d^ 2)*n - (a*b^4*c^3*n^4 + 6*(2*a*b^4*c^3 - a^2*b^3*c^2*d)*n^3 + (47*a*b^4*c^ 3 - 54*a^2*b^3*c^2*d + 18*a^3*b^2*c*d^2)*n^2 + 6*(10*a*b^4*c^3 - 20*a^2*b^ 3*c^2*d + 15*a^3*b^2*c*d^2 - 4*a^4*b*d^3)*n)*x)*(b*x + a)^n/(b^5*n^5 + 15* b^5*n^4 + 85*b^5*n^3 + 225*b^5*n^2 + 274*b^5*n + 120*b^5)
Leaf count of result is larger than twice the leaf count of optimal. 7803 vs. \(2 (138) = 276\).
Time = 1.84 (sec) , antiderivative size = 7803, normalized size of antiderivative = 50.67 \[ \int x (a+b x)^n (c+d x)^3 \, dx=\text {Too large to display} \]
Piecewise((a**n*(c**3*x**2/2 + c**2*d*x**3 + 3*c*d**2*x**4/4 + d**3*x**5/5 ), Eq(b, 0)), (12*a**4*d**3*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 25*a**4*d**3/(12*a**4 *b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4 ) - 9*a**3*b*c*d**2/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 4 8*a*b**8*x**3 + 12*b**9*x**4) + 48*a**3*b*d**3*x*log(a/b + x)/(12*a**4*b** 5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 88*a**3*b*d**3*x/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a *b**8*x**3 + 12*b**9*x**4) - 3*a**2*b**2*c**2*d/(12*a**4*b**5 + 48*a**3*b* *6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 36*a**2*b**2*c *d**2*x/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x** 3 + 12*b**9*x**4) + 72*a**2*b**2*d**3*x**2*log(a/b + x)/(12*a**4*b**5 + 48 *a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 108*a* *2*b**2*d**3*x**2/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48* a*b**8*x**3 + 12*b**9*x**4) - a*b**3*c**3/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 12*a*b**3*c**2*d*x/( 12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b* *9*x**4) - 54*a*b**3*c*d**2*x**2/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2* b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 48*a*b**3*d**3*x**3*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3...
Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (154) = 308\).
Time = 0.23 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.38 \[ \int x (a+b x)^n (c+d x)^3 \, dx=\frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c^{3}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {3 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c^{2} d}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {3 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} d^{3}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \]
(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^3/((n^2 + 3*n + 2)*b^2) + 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^2*d/((n^3 + 6*n^2 + 11*n + 6)*b^3) + 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*c*d^2/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b ^4) + ((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*a*b^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*a^2*b^3*x^3 + 12*(n^2 + n)*a^3*b ^2*x^2 - 24*a^4*b*n*x + 24*a^5)*(b*x + a)^n*d^3/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^5)
Leaf count of result is larger than twice the leaf count of optimal. 1305 vs. \(2 (154) = 308\).
Time = 0.28 (sec) , antiderivative size = 1305, normalized size of antiderivative = 8.47 \[ \int x (a+b x)^n (c+d x)^3 \, dx=\text {Too large to display} \]
((b*x + a)^n*b^5*d^3*n^4*x^5 + 3*(b*x + a)^n*b^5*c*d^2*n^4*x^4 + (b*x + a) ^n*a*b^4*d^3*n^4*x^4 + 10*(b*x + a)^n*b^5*d^3*n^3*x^5 + 3*(b*x + a)^n*b^5* c^2*d*n^4*x^3 + 3*(b*x + a)^n*a*b^4*c*d^2*n^4*x^3 + 33*(b*x + a)^n*b^5*c*d ^2*n^3*x^4 + 6*(b*x + a)^n*a*b^4*d^3*n^3*x^4 + 35*(b*x + a)^n*b^5*d^3*n^2* x^5 + (b*x + a)^n*b^5*c^3*n^4*x^2 + 3*(b*x + a)^n*a*b^4*c^2*d*n^4*x^2 + 36 *(b*x + a)^n*b^5*c^2*d*n^3*x^3 + 24*(b*x + a)^n*a*b^4*c*d^2*n^3*x^3 - 4*(b *x + a)^n*a^2*b^3*d^3*n^3*x^3 + 123*(b*x + a)^n*b^5*c*d^2*n^2*x^4 + 11*(b* x + a)^n*a*b^4*d^3*n^2*x^4 + 50*(b*x + a)^n*b^5*d^3*n*x^5 + (b*x + a)^n*a* b^4*c^3*n^4*x + 13*(b*x + a)^n*b^5*c^3*n^3*x^2 + 30*(b*x + a)^n*a*b^4*c^2* d*n^3*x^2 - 9*(b*x + a)^n*a^2*b^3*c*d^2*n^3*x^2 + 147*(b*x + a)^n*b^5*c^2* d*n^2*x^3 + 51*(b*x + a)^n*a*b^4*c*d^2*n^2*x^3 - 12*(b*x + a)^n*a^2*b^3*d^ 3*n^2*x^3 + 183*(b*x + a)^n*b^5*c*d^2*n*x^4 + 6*(b*x + a)^n*a*b^4*d^3*n*x^ 4 + 24*(b*x + a)^n*b^5*d^3*x^5 + 12*(b*x + a)^n*a*b^4*c^3*n^3*x - 6*(b*x + a)^n*a^2*b^3*c^2*d*n^3*x + 59*(b*x + a)^n*b^5*c^3*n^2*x^2 + 87*(b*x + a)^ n*a*b^4*c^2*d*n^2*x^2 - 54*(b*x + a)^n*a^2*b^3*c*d^2*n^2*x^2 + 12*(b*x + a )^n*a^3*b^2*d^3*n^2*x^2 + 234*(b*x + a)^n*b^5*c^2*d*n*x^3 + 30*(b*x + a)^n *a*b^4*c*d^2*n*x^3 - 8*(b*x + a)^n*a^2*b^3*d^3*n*x^3 + 90*(b*x + a)^n*b^5* c*d^2*x^4 - (b*x + a)^n*a^2*b^3*c^3*n^3 + 47*(b*x + a)^n*a*b^4*c^3*n^2*x - 54*(b*x + a)^n*a^2*b^3*c^2*d*n^2*x + 18*(b*x + a)^n*a^3*b^2*c*d^2*n^2*x + 107*(b*x + a)^n*b^5*c^3*n*x^2 + 60*(b*x + a)^n*a*b^4*c^2*d*n*x^2 - 45*...
Time = 1.46 (sec) , antiderivative size = 663, normalized size of antiderivative = 4.31 \[ \int x (a+b x)^n (c+d x)^3 \, dx=\frac {d^3\,x^5\,{\left (a+b\,x\right )}^n\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}-\frac {a^2\,{\left (a+b\,x\right )}^n\,\left (-24\,a^3\,d^3+18\,a^2\,b\,c\,d^2\,n+90\,a^2\,b\,c\,d^2-6\,a\,b^2\,c^2\,d\,n^2-54\,a\,b^2\,c^2\,d\,n-120\,a\,b^2\,c^2\,d+b^3\,c^3\,n^3+12\,b^3\,c^3\,n^2+47\,b^3\,c^3\,n+60\,b^3\,c^3\right )}{b^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (12\,a^3\,d^3\,n-9\,a^2\,b\,c\,d^2\,n^2-45\,a^2\,b\,c\,d^2\,n+3\,a\,b^2\,c^2\,d\,n^3+27\,a\,b^2\,c^2\,d\,n^2+60\,a\,b^2\,c^2\,d\,n+b^3\,c^3\,n^3+12\,b^3\,c^3\,n^2+47\,b^3\,c^3\,n+60\,b^3\,c^3\right )}{b^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {d^2\,x^4\,{\left (a+b\,x\right )}^n\,\left (15\,b\,c+a\,d\,n+3\,b\,c\,n\right )\,\left (n^3+6\,n^2+11\,n+6\right )}{b\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {d\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (-4\,a^2\,d^2\,n+3\,a\,b\,c\,d\,n^2+15\,a\,b\,c\,d\,n+3\,b^2\,c^2\,n^2+27\,b^2\,c^2\,n+60\,b^2\,c^2\right )}{b^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,n\,x\,{\left (a+b\,x\right )}^n\,\left (-24\,a^3\,d^3+18\,a^2\,b\,c\,d^2\,n+90\,a^2\,b\,c\,d^2-6\,a\,b^2\,c^2\,d\,n^2-54\,a\,b^2\,c^2\,d\,n-120\,a\,b^2\,c^2\,d+b^3\,c^3\,n^3+12\,b^3\,c^3\,n^2+47\,b^3\,c^3\,n+60\,b^3\,c^3\right )}{b^4\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )} \]
(d^3*x^5*(a + b*x)^n*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120) - (a^2*(a + b*x)^n*(60*b^3*c^3 - 24*a^3*d^ 3 + 47*b^3*c^3*n + 12*b^3*c^3*n^2 + b^3*c^3*n^3 - 120*a*b^2*c^2*d + 90*a^2 *b*c*d^2 - 54*a*b^2*c^2*d*n + 18*a^2*b*c*d^2*n - 6*a*b^2*c^2*d*n^2))/(b^5* (274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (x^2*(n + 1)*(a + b*x)^ n*(60*b^3*c^3 + 12*a^3*d^3*n + 47*b^3*c^3*n + 12*b^3*c^3*n^2 + b^3*c^3*n^3 + 60*a*b^2*c^2*d*n - 45*a^2*b*c*d^2*n + 27*a*b^2*c^2*d*n^2 - 9*a^2*b*c*d^ 2*n^2 + 3*a*b^2*c^2*d*n^3))/(b^3*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (d^2*x^4*(a + b*x)^n*(15*b*c + a*d*n + 3*b*c*n)*(11*n + 6*n^2 + n^3 + 6))/(b*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (d*x^3*(a + b*x)^n*(3*n + n^2 + 2)*(60*b^2*c^2 - 4*a^2*d^2*n + 27*b^2*c^2*n + 3*b^2* c^2*n^2 + 15*a*b*c*d*n + 3*a*b*c*d*n^2))/(b^2*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (a*n*x*(a + b*x)^n*(60*b^3*c^3 - 24*a^3*d^3 + 47*b^ 3*c^3*n + 12*b^3*c^3*n^2 + b^3*c^3*n^3 - 120*a*b^2*c^2*d + 90*a^2*b*c*d^2 - 54*a*b^2*c^2*d*n + 18*a^2*b*c*d^2*n - 6*a*b^2*c^2*d*n^2))/(b^4*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))