Integrand size = 16, antiderivative size = 126 \[ \int x (a+b x)^n \left (c+d x^3\right ) \, dx=-\frac {a \left (b^3 c-a^3 d\right ) (a+b x)^{1+n}}{b^5 (1+n)}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)^{2+n}}{b^5 (2+n)}+\frac {6 a^2 d (a+b x)^{3+n}}{b^5 (3+n)}-\frac {4 a d (a+b x)^{4+n}}{b^5 (4+n)}+\frac {d (a+b x)^{5+n}}{b^5 (5+n)} \] Output:
-a*(-a^3*d+b^3*c)*(b*x+a)^(1+n)/b^5/(1+n)+(-4*a^3*d+b^3*c)*(b*x+a)^(2+n)/b ^5/(2+n)+6*a^2*d*(b*x+a)^(3+n)/b^5/(3+n)-4*a*d*(b*x+a)^(4+n)/b^5/(4+n)+d*( b*x+a)^(5+n)/b^5/(5+n)
Time = 0.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.83 \[ \int x (a+b x)^n \left (c+d x^3\right ) \, dx=\frac {(a+b x)^{1+n} \left (\frac {a \left (-b^3 c+a^3 d\right )}{1+n}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)}{2+n}+\frac {6 a^2 d (a+b x)^2}{3+n}-\frac {4 a d (a+b x)^3}{4+n}+\frac {d (a+b x)^4}{5+n}\right )}{b^5} \] Input:
Integrate[x*(a + b*x)^n*(c + d*x^3),x]
Output:
((a + b*x)^(1 + n)*((a*(-(b^3*c) + a^3*d))/(1 + n) + ((b^3*c - 4*a^3*d)*(a + b*x))/(2 + n) + (6*a^2*d*(a + b*x)^2)/(3 + n) - (4*a*d*(a + b*x)^3)/(4 + n) + (d*(a + b*x)^4)/(5 + n)))/b^5
Time = 0.48 (sec) , antiderivative size = 126, 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 = {2123, 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.
| \(\displaystyle \int x \left (c+d x^3\right ) (a+b x)^n \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {a \left (a^3 d-b^3 c\right ) (a+b x)^n}{b^4}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)^{n+1}}{b^4}+\frac {6 a^2 d (a+b x)^{n+2}}{b^4}-\frac {4 a d (a+b x)^{n+3}}{b^4}+\frac {d (a+b x)^{n+4}}{b^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^5 (n+1)}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac {6 a^2 d (a+b x)^{n+3}}{b^5 (n+3)}-\frac {4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac {d (a+b x)^{n+5}}{b^5 (n+5)}\) |
Input:
Int[x*(a + b*x)^n*(c + d*x^3),x]
Output:
-((a*(b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^5*(1 + n))) + ((b^3*c - 4*a^3*d )*(a + b*x)^(2 + n))/(b^5*(2 + n)) + (6*a^2*d*(a + b*x)^(3 + n))/(b^5*(3 + n)) - (4*a*d*(a + b*x)^(4 + n))/(b^5*(4 + n)) + (d*(a + b*x)^(5 + n))/(b^ 5*(5 + n))
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Leaf count of result is larger than twice the leaf count of optimal. \(282\) vs. \(2(126)=252\).
Time = 0.19 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.25
| method | result | size |
| gosper | \(\frac {\left (b x +a \right )^{1+n} \left (b^{4} d \,n^{4} x^{4}+10 b^{4} d \,n^{3} x^{4}-4 a \,b^{3} d \,n^{3} x^{3}+35 b^{4} d \,n^{2} x^{4}-24 a \,b^{3} d \,n^{2} x^{3}+b^{4} c \,n^{4} x +50 b^{4} d n \,x^{4}+12 a^{2} b^{2} d \,n^{2} x^{2}-44 a \,b^{3} d n \,x^{3}+13 b^{4} c \,n^{3} x +24 x^{4} d \,b^{4}+36 a^{2} b^{2} d n \,x^{2}-a \,b^{3} c \,n^{3}-24 a d \,x^{3} b^{3}+59 b^{4} c \,n^{2} x -24 a^{3} b d n x +24 d \,a^{2} x^{2} b^{2}-12 a \,b^{3} c \,n^{2}+107 b^{4} c n x -24 a^{3} b d x -47 a \,b^{3} c n +60 b^{4} c x +24 a^{4} d -60 a \,b^{3} c \right )}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(283\) |
| orering | \(\frac {\left (b x +a \right ) \left (b^{4} d \,n^{4} x^{4}+10 b^{4} d \,n^{3} x^{4}-4 a \,b^{3} d \,n^{3} x^{3}+35 b^{4} d \,n^{2} x^{4}-24 a \,b^{3} d \,n^{2} x^{3}+b^{4} c \,n^{4} x +50 b^{4} d n \,x^{4}+12 a^{2} b^{2} d \,n^{2} x^{2}-44 a \,b^{3} d n \,x^{3}+13 b^{4} c \,n^{3} x +24 x^{4} d \,b^{4}+36 a^{2} b^{2} d n \,x^{2}-a \,b^{3} c \,n^{3}-24 a d \,x^{3} b^{3}+59 b^{4} c \,n^{2} x -24 a^{3} b d n x +24 d \,a^{2} x^{2} b^{2}-12 a \,b^{3} c \,n^{2}+107 b^{4} c n x -24 a^{3} b d x -47 a \,b^{3} c n +60 b^{4} c x +24 a^{4} d -60 a \,b^{3} c \right ) \left (b x +a \right )^{n}}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(286\) |
| norman | \(\frac {d \,x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{5+n}+\frac {a^{2} \left (-b^{3} c \,n^{3}-12 b^{3} c \,n^{2}-47 b^{3} c n +24 a^{3} d -60 b^{3} c \right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}+\frac {\left (b^{3} c \,n^{3}+12 b^{3} c \,n^{2}+12 a^{3} d n +47 b^{3} c n +60 b^{3} c \right ) x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+14 n^{3}+71 n^{2}+154 n +120\right )}+\frac {a d n \,x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+9 n +20\right )}-\frac {n a \left (-b^{3} c \,n^{3}-12 b^{3} c \,n^{2}-47 b^{3} c n +24 a^{3} d -60 b^{3} c \right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}-\frac {4 n d \,a^{2} x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+12 n^{2}+47 n +60\right )}\) | \(320\) |
| risch | \(\frac {\left (b^{5} d \,n^{4} x^{5}+a \,b^{4} d \,n^{4} x^{4}+10 b^{5} d \,n^{3} x^{5}+6 a \,b^{4} d \,n^{3} x^{4}+35 b^{5} d \,n^{2} x^{5}-4 a^{2} b^{3} d \,n^{3} x^{3}+11 a \,b^{4} d \,n^{2} x^{4}+b^{5} c \,n^{4} x^{2}+50 b^{5} d n \,x^{5}-12 a^{2} b^{3} d \,n^{2} x^{3}+a \,b^{4} c \,n^{4} x +6 a \,b^{4} d n \,x^{4}+13 b^{5} c \,n^{3} x^{2}+24 d \,x^{5} b^{5}+12 a^{3} b^{2} d \,n^{2} x^{2}-8 a^{2} b^{3} d n \,x^{3}+12 a \,b^{4} c \,n^{3} x +59 b^{5} c \,n^{2} x^{2}+12 a^{3} b^{2} d n \,x^{2}-a^{2} b^{3} c \,n^{3}+47 a \,b^{4} c \,n^{2} x +107 b^{5} c n \,x^{2}-24 a^{4} b d n x -12 a^{2} b^{3} c \,n^{2}+60 a \,b^{4} c n x +60 b^{5} c \,x^{2}-47 a^{2} b^{3} c n +24 a^{5} d -60 a^{2} b^{3} c \right ) \left (b x +a \right )^{n}}{\left (4+n \right ) \left (5+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{5}}\) | \(365\) |
| parallelrisch | \(\frac {x^{4} \left (b x +a \right )^{n} a \,b^{4} d \,n^{4}+6 x^{4} \left (b x +a \right )^{n} a \,b^{4} d \,n^{3}+11 x^{4} \left (b x +a \right )^{n} a \,b^{4} d \,n^{2}-4 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d \,n^{3}+6 x^{4} \left (b x +a \right )^{n} a \,b^{4} d n -12 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d \,n^{2}+x \left (b x +a \right )^{n} a \,b^{4} c \,n^{4}-8 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d n +12 x^{2} \left (b x +a \right )^{n} a^{3} b^{2} d \,n^{2}+12 x \left (b x +a \right )^{n} a \,b^{4} c \,n^{3}+x^{5} \left (b x +a \right )^{n} b^{5} d \,n^{4}+59 x^{2} \left (b x +a \right )^{n} b^{5} c \,n^{2}+107 x^{2} \left (b x +a \right )^{n} b^{5} c n -\left (b x +a \right )^{n} a^{2} b^{3} c \,n^{3}-12 \left (b x +a \right )^{n} a^{2} b^{3} c \,n^{2}-47 \left (b x +a \right )^{n} a^{2} b^{3} c n +24 x^{5} \left (b x +a \right )^{n} b^{5} d +60 x^{2} \left (b x +a \right )^{n} b^{5} c -60 \left (b x +a \right )^{n} a^{2} b^{3} c +12 x^{2} \left (b x +a \right )^{n} a^{3} b^{2} d n +47 x \left (b x +a \right )^{n} a \,b^{4} c \,n^{2}-24 x \left (b x +a \right )^{n} a^{4} b d n +60 x \left (b x +a \right )^{n} a \,b^{4} c n +10 x^{5} \left (b x +a \right )^{n} b^{5} d \,n^{3}+35 x^{5} \left (b x +a \right )^{n} b^{5} d \,n^{2}+50 x^{5} \left (b x +a \right )^{n} b^{5} d n +x^{2} \left (b x +a \right )^{n} b^{5} c \,n^{4}+13 x^{2} \left (b x +a \right )^{n} b^{5} c \,n^{3}+24 \left (b x +a \right )^{n} a^{5} d}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(561\) |
Input:
int(x*(b*x+a)^n*(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
1/b^5*(b*x+a)^(1+n)/(n^5+15*n^4+85*n^3+225*n^2+274*n+120)*(b^4*d*n^4*x^4+1 0*b^4*d*n^3*x^4-4*a*b^3*d*n^3*x^3+35*b^4*d*n^2*x^4-24*a*b^3*d*n^2*x^3+b^4* c*n^4*x+50*b^4*d*n*x^4+12*a^2*b^2*d*n^2*x^2-44*a*b^3*d*n*x^3+13*b^4*c*n^3* x+24*b^4*d*x^4+36*a^2*b^2*d*n*x^2-a*b^3*c*n^3-24*a*b^3*d*x^3+59*b^4*c*n^2* x-24*a^3*b*d*n*x+24*a^2*b^2*d*x^2-12*a*b^3*c*n^2+107*b^4*c*n*x-24*a^3*b*d* x-47*a*b^3*c*n+60*b^4*c*x+24*a^4*d-60*a*b^3*c)
Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (126) = 252\).
Time = 0.08 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.76 \[ \int x (a+b x)^n \left (c+d x^3\right ) \, dx=-\frac {{\left (a^{2} b^{3} c n^{3} + 12 \, a^{2} b^{3} c n^{2} + 47 \, a^{2} b^{3} c n + 60 \, a^{2} b^{3} c - 24 \, a^{5} d - {\left (b^{5} d n^{4} + 10 \, b^{5} d n^{3} + 35 \, b^{5} d n^{2} + 50 \, b^{5} d n + 24 \, b^{5} d\right )} x^{5} - {\left (a b^{4} d n^{4} + 6 \, a b^{4} d n^{3} + 11 \, a b^{4} d n^{2} + 6 \, a b^{4} d n\right )} x^{4} + 4 \, {\left (a^{2} b^{3} d n^{3} + 3 \, a^{2} b^{3} d n^{2} + 2 \, a^{2} b^{3} d n\right )} x^{3} - {\left (b^{5} c n^{4} + 13 \, b^{5} c n^{3} + 60 \, b^{5} c + {\left (59 \, b^{5} c + 12 \, a^{3} b^{2} d\right )} n^{2} + {\left (107 \, b^{5} c + 12 \, a^{3} b^{2} d\right )} n\right )} x^{2} - {\left (a b^{4} c n^{4} + 12 \, a b^{4} c n^{3} + 47 \, a b^{4} c n^{2} + 12 \, {\left (5 \, a b^{4} c - 2 \, a^{4} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \] Input:
integrate(x*(b*x+a)^n*(d*x^3+c),x, algorithm="fricas")
Output:
-(a^2*b^3*c*n^3 + 12*a^2*b^3*c*n^2 + 47*a^2*b^3*c*n + 60*a^2*b^3*c - 24*a^ 5*d - (b^5*d*n^4 + 10*b^5*d*n^3 + 35*b^5*d*n^2 + 50*b^5*d*n + 24*b^5*d)*x^ 5 - (a*b^4*d*n^4 + 6*a*b^4*d*n^3 + 11*a*b^4*d*n^2 + 6*a*b^4*d*n)*x^4 + 4*( a^2*b^3*d*n^3 + 3*a^2*b^3*d*n^2 + 2*a^2*b^3*d*n)*x^3 - (b^5*c*n^4 + 13*b^5 *c*n^3 + 60*b^5*c + (59*b^5*c + 12*a^3*b^2*d)*n^2 + (107*b^5*c + 12*a^3*b^ 2*d)*n)*x^2 - (a*b^4*c*n^4 + 12*a*b^4*c*n^3 + 47*a*b^4*c*n^2 + 12*(5*a*b^4 *c - 2*a^4*b*d)*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. 3704 vs. \(2 (112) = 224\).
Time = 1.60 (sec) , antiderivative size = 3704, normalized size of antiderivative = 29.40 \[ \int x (a+b x)^n \left (c+d x^3\right ) \, dx=\text {Too large to display} \] Input:
integrate(x*(b*x+a)**n*(d*x**3+c),x)
Output:
Piecewise((a**n*(c*x**2/2 + d*x**5/5), Eq(b, 0)), (12*a**4*d*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/(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**3*b*d*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) + 8 8*a**3*b*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) + 72*a**2*b**2*d*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*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/(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*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 + 12*b**9*x**4) + 48*a*b**3*d*x**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) - 4*b**4*c*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) + 12*b **4*d*x**4*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), Eq(n, -5)), (-24*a**4*d*log(a/b + x)/(6 *a**3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) - 44*a**4*d/(6 *a**3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) - 72*a**3*b*d* x*log(a/b + x)/(6*a**3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*...
Time = 0.04 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.46 \[ \int x (a+b x)^n \left (c+d x^3\right ) \, dx=\frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \] Input:
integrate(x*(b*x+a)^n*(d*x^3+c),x, algorithm="maxima")
Output:
(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c/((n^2 + 3*n + 2)*b^2) + (( 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/((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. 577 vs. \(2 (126) = 252\).
Time = 0.13 (sec) , antiderivative size = 577, normalized size of antiderivative = 4.58 \[ \int x (a+b x)^n \left (c+d x^3\right ) \, dx=\frac {{\left (b x + a\right )}^{n} b^{5} d n^{4} x^{5} + {\left (b x + a\right )}^{n} a b^{4} d n^{4} x^{4} + 10 \, {\left (b x + a\right )}^{n} b^{5} d n^{3} x^{5} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d n^{3} x^{4} + 35 \, {\left (b x + a\right )}^{n} b^{5} d n^{2} x^{5} + {\left (b x + a\right )}^{n} b^{5} c n^{4} x^{2} - 4 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n^{3} x^{3} + 11 \, {\left (b x + a\right )}^{n} a b^{4} d n^{2} x^{4} + 50 \, {\left (b x + a\right )}^{n} b^{5} d n x^{5} + {\left (b x + a\right )}^{n} a b^{4} c n^{4} x + 13 \, {\left (b x + a\right )}^{n} b^{5} c n^{3} x^{2} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n^{2} x^{3} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d n x^{4} + 24 \, {\left (b x + a\right )}^{n} b^{5} d x^{5} + 12 \, {\left (b x + a\right )}^{n} a b^{4} c n^{3} x + 59 \, {\left (b x + a\right )}^{n} b^{5} c n^{2} x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d n^{2} x^{2} - 8 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n x^{3} - {\left (b x + a\right )}^{n} a^{2} b^{3} c n^{3} + 47 \, {\left (b x + a\right )}^{n} a b^{4} c n^{2} x + 107 \, {\left (b x + a\right )}^{n} b^{5} c n x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d n x^{2} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c n^{2} + 60 \, {\left (b x + a\right )}^{n} a b^{4} c n x - 24 \, {\left (b x + a\right )}^{n} a^{4} b d n x + 60 \, {\left (b x + a\right )}^{n} b^{5} c x^{2} - 47 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c n - 60 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c + 24 \, {\left (b x + a\right )}^{n} a^{5} d}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \] Input:
integrate(x*(b*x+a)^n*(d*x^3+c),x, algorithm="giac")
Output:
((b*x + a)^n*b^5*d*n^4*x^5 + (b*x + a)^n*a*b^4*d*n^4*x^4 + 10*(b*x + a)^n* b^5*d*n^3*x^5 + 6*(b*x + a)^n*a*b^4*d*n^3*x^4 + 35*(b*x + a)^n*b^5*d*n^2*x ^5 + (b*x + a)^n*b^5*c*n^4*x^2 - 4*(b*x + a)^n*a^2*b^3*d*n^3*x^3 + 11*(b*x + a)^n*a*b^4*d*n^2*x^4 + 50*(b*x + a)^n*b^5*d*n*x^5 + (b*x + a)^n*a*b^4*c *n^4*x + 13*(b*x + a)^n*b^5*c*n^3*x^2 - 12*(b*x + a)^n*a^2*b^3*d*n^2*x^3 + 6*(b*x + a)^n*a*b^4*d*n*x^4 + 24*(b*x + a)^n*b^5*d*x^5 + 12*(b*x + a)^n*a *b^4*c*n^3*x + 59*(b*x + a)^n*b^5*c*n^2*x^2 + 12*(b*x + a)^n*a^3*b^2*d*n^2 *x^2 - 8*(b*x + a)^n*a^2*b^3*d*n*x^3 - (b*x + a)^n*a^2*b^3*c*n^3 + 47*(b*x + a)^n*a*b^4*c*n^2*x + 107*(b*x + a)^n*b^5*c*n*x^2 + 12*(b*x + a)^n*a^3*b ^2*d*n*x^2 - 12*(b*x + a)^n*a^2*b^3*c*n^2 + 60*(b*x + a)^n*a*b^4*c*n*x - 2 4*(b*x + a)^n*a^4*b*d*n*x + 60*(b*x + a)^n*b^5*c*x^2 - 47*(b*x + a)^n*a^2* b^3*c*n - 60*(b*x + a)^n*a^2*b^3*c + 24*(b*x + a)^n*a^5*d)/(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)
Time = 22.12 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.88 \[ \int x (a+b x)^n \left (c+d x^3\right ) \, dx={\left (a+b\,x\right )}^n\,\left (\frac {d\,x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}-\frac {a^2\,\left (-24\,d\,a^3+c\,b^3\,n^3+12\,c\,b^3\,n^2+47\,c\,b^3\,n+60\,c\,b^3\right )}{b^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {x^2\,\left (n+1\right )\,\left (12\,d\,a^3\,n+c\,b^3\,n^3+12\,c\,b^3\,n^2+47\,c\,b^3\,n+60\,c\,b^3\right )}{b^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,n\,x\,\left (-24\,d\,a^3+c\,b^3\,n^3+12\,c\,b^3\,n^2+47\,c\,b^3\,n+60\,c\,b^3\right )}{b^4\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,d\,n\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{b\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}-\frac {4\,a^2\,d\,n\,x^3\,\left (n^2+3\,n+2\right )}{b^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}\right ) \] Input:
int(x*(c + d*x^3)*(a + b*x)^n,x)
Output:
(a + b*x)^n*((d*x^5*(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*(60*b^3*c - 24*a^3*d + 12*b^3*c*n^2 + b^3*c*n^3 + 47*b^3*c*n))/(b^5*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (x^2*(n + 1)*(60*b^3*c + 12*b^3*c*n^2 + b^3*c*n^3 + 12*a^3*d*n + 47*b^3*c*n))/(b^3*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (a*n* x*(60*b^3*c - 24*a^3*d + 12*b^3*c*n^2 + b^3*c*n^3 + 47*b^3*c*n))/(b^4*(274 *n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (a*d*n*x^4*(11*n + 6*n^2 + n^3 + 6))/(b*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) - (4*a^2*d*n *x^3*(3*n + n^2 + 2))/(b^2*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120) ))
Time = 0.26 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.89 \[ \int x (a+b x)^n \left (c+d x^3\right ) \, dx=\frac {\left (b x +a \right )^{n} \left (b^{5} d \,n^{4} x^{5}+a \,b^{4} d \,n^{4} x^{4}+10 b^{5} d \,n^{3} x^{5}+6 a \,b^{4} d \,n^{3} x^{4}+35 b^{5} d \,n^{2} x^{5}-4 a^{2} b^{3} d \,n^{3} x^{3}+11 a \,b^{4} d \,n^{2} x^{4}+b^{5} c \,n^{4} x^{2}+50 b^{5} d n \,x^{5}-12 a^{2} b^{3} d \,n^{2} x^{3}+a \,b^{4} c \,n^{4} x +6 a \,b^{4} d n \,x^{4}+13 b^{5} c \,n^{3} x^{2}+24 b^{5} d \,x^{5}+12 a^{3} b^{2} d \,n^{2} x^{2}-8 a^{2} b^{3} d n \,x^{3}+12 a \,b^{4} c \,n^{3} x +59 b^{5} c \,n^{2} x^{2}+12 a^{3} b^{2} d n \,x^{2}-a^{2} b^{3} c \,n^{3}+47 a \,b^{4} c \,n^{2} x +107 b^{5} c n \,x^{2}-24 a^{4} b d n x -12 a^{2} b^{3} c \,n^{2}+60 a \,b^{4} c n x +60 b^{5} c \,x^{2}-47 a^{2} b^{3} c n +24 a^{5} d -60 a^{2} b^{3} c \right )}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )} \] Input:
int(x*(b*x+a)^n*(d*x^3+c),x)
Output:
((a + b*x)**n*(24*a**5*d - 24*a**4*b*d*n*x + 12*a**3*b**2*d*n**2*x**2 + 12 *a**3*b**2*d*n*x**2 - a**2*b**3*c*n**3 - 12*a**2*b**3*c*n**2 - 47*a**2*b** 3*c*n - 60*a**2*b**3*c - 4*a**2*b**3*d*n**3*x**3 - 12*a**2*b**3*d*n**2*x** 3 - 8*a**2*b**3*d*n*x**3 + a*b**4*c*n**4*x + 12*a*b**4*c*n**3*x + 47*a*b** 4*c*n**2*x + 60*a*b**4*c*n*x + a*b**4*d*n**4*x**4 + 6*a*b**4*d*n**3*x**4 + 11*a*b**4*d*n**2*x**4 + 6*a*b**4*d*n*x**4 + b**5*c*n**4*x**2 + 13*b**5*c* n**3*x**2 + 59*b**5*c*n**2*x**2 + 107*b**5*c*n*x**2 + 60*b**5*c*x**2 + b** 5*d*n**4*x**5 + 10*b**5*d*n**3*x**5 + 35*b**5*d*n**2*x**5 + 50*b**5*d*n*x* *5 + 24*b**5*d*x**5))/(b**5*(n**5 + 15*n**4 + 85*n**3 + 225*n**2 + 274*n + 120))