Integrand size = 17, antiderivative size = 132 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^4 \, dx=a c^4 x+\frac {c^3 (b c+4 a d) x^{1+n}}{1+n}+\frac {2 c^2 d (2 b c+3 a d) x^{1+2 n}}{1+2 n}+\frac {2 c d^2 (3 b c+2 a d) x^{1+3 n}}{1+3 n}+\frac {d^3 (4 b c+a d) x^{1+4 n}}{1+4 n}+\frac {b d^4 x^{1+5 n}}{1+5 n} \] Output:
a*c^4*x+c^3*(4*a*d+b*c)*x^(1+n)/(1+n)+2*c^2*d*(3*a*d+2*b*c)*x^(1+2*n)/(1+2 *n)+2*c*d^2*(2*a*d+3*b*c)*x^(1+3*n)/(1+3*n)+d^3*(a*d+4*b*c)*x^(1+4*n)/(1+4 *n)+b*d^4*x^(1+5*n)/(1+5*n)
Time = 0.69 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^4 \, dx=\frac {b x \left (c+d x^n\right )^5-(b c-a d (1+5 n)) x \left (c^4+\frac {4 c^3 d x^n}{1+n}+\frac {6 c^2 d^2 x^{2 n}}{1+2 n}+\frac {4 c d^3 x^{3 n}}{1+3 n}+\frac {d^4 x^{4 n}}{1+4 n}\right )}{d+5 d n} \] Input:
Integrate[(a + b*x^n)*(c + d*x^n)^4,x]
Output:
(b*x*(c + d*x^n)^5 - (b*c - a*d*(1 + 5*n))*x*(c^4 + (4*c^3*d*x^n)/(1 + n) + (6*c^2*d^2*x^(2*n))/(1 + 2*n) + (4*c*d^3*x^(3*n))/(1 + 3*n) + (d^4*x^(4* n))/(1 + 4*n)))/(d + 5*d*n)
Time = 0.53 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {897, 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.
| \(\displaystyle \int \left (a+b x^n\right ) \left (c+d x^n\right )^4 \, dx\) |
| \(\Big \downarrow \) 897 |
| \(\displaystyle \int \left (c^3 x^n (4 a d+b c)+2 c^2 d x^{2 n} (3 a d+2 b c)+d^3 x^{4 n} (a d+4 b c)+2 c d^2 x^{3 n} (2 a d+3 b c)+a c^4+b d^4 x^{5 n}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^3 x^{n+1} (4 a d+b c)}{n+1}+\frac {2 c^2 d x^{2 n+1} (3 a d+2 b c)}{2 n+1}+\frac {d^3 x^{4 n+1} (a d+4 b c)}{4 n+1}+\frac {2 c d^2 x^{3 n+1} (2 a d+3 b c)}{3 n+1}+a c^4 x+\frac {b d^4 x^{5 n+1}}{5 n+1}\) |
Input:
Int[(a + b*x^n)*(c + d*x^n)^4,x]
Output:
a*c^4*x + (c^3*(b*c + 4*a*d)*x^(1 + n))/(1 + n) + (2*c^2*d*(2*b*c + 3*a*d) *x^(1 + 2*n))/(1 + 2*n) + (2*c*d^2*(3*b*c + 2*a*d)*x^(1 + 3*n))/(1 + 3*n) + (d^3*(4*b*c + a*d)*x^(1 + 4*n))/(1 + 4*n) + (b*d^4*x^(1 + 5*n))/(1 + 5*n )
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b , c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Time = 1.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(a \,c^{4} x +\frac {b \,d^{4} x \,x^{5 n}}{5 n +1}+\frac {c^{3} \left (4 a d +b c \right ) x \,x^{n}}{1+n}+\frac {d^{3} \left (a d +4 b c \right ) x \,x^{4 n}}{1+4 n}+\frac {2 c \,d^{2} \left (2 a d +3 b c \right ) x \,x^{3 n}}{1+3 n}+\frac {2 c^{2} d \left (3 a d +2 b c \right ) x \,x^{2 n}}{1+2 n}\) | \(128\) |
| norman | \(a \,c^{4} x +\frac {b \,d^{4} x \,{\mathrm e}^{5 n \ln \left (x \right )}}{5 n +1}+\frac {c^{3} \left (4 a d +b c \right ) x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}+\frac {d^{3} \left (a d +4 b c \right ) x \,{\mathrm e}^{4 n \ln \left (x \right )}}{1+4 n}+\frac {2 c \,d^{2} \left (2 a d +3 b c \right ) x \,{\mathrm e}^{3 n \ln \left (x \right )}}{1+3 n}+\frac {2 c^{2} d \left (3 a d +2 b c \right ) x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}\) | \(138\) |
| parallelrisch | \(\frac {a \,c^{4} x +616 x \,x^{n} a \,c^{3} d \,n^{3}+480 x \,x^{n} a \,c^{3} d \,n^{4}+284 x \,x^{n} a \,c^{3} d \,n^{2}+56 x \,x^{n} a \,c^{3} d n +24 x \,x^{5 n} b \,d^{4} n^{4}+50 x \,x^{5 n} b \,d^{4} n^{3}+30 x \,x^{4 n} a \,d^{4} n^{4}+35 x \,x^{5 n} b \,d^{4} n^{2}+61 x \,x^{4 n} a \,d^{4} n^{3}+10 x \,x^{5 n} b \,d^{4} n +41 x \,x^{4 n} a \,d^{4} n^{2}+11 x \,x^{4 n} a \,d^{4} n +120 x \,x^{n} b \,c^{4} n^{4}+154 x \,x^{n} b \,c^{4} n^{3}+71 x \,x^{n} b \,c^{4} n^{2}+14 x \,x^{n} b \,c^{4} n +4 x \,x^{n} a \,c^{3} d +120 x a \,c^{4} n^{5}+274 x a \,c^{4} n^{4}+225 x a \,c^{4} n^{3}+85 x a \,c^{4} n^{2}+x \,x^{n} b \,c^{4}+15 x a \,c^{4} n +x \,x^{4 n} a \,d^{4}+48 x \,x^{3 n} a c \,d^{3} n +72 x \,x^{3 n} b \,c^{2} d^{2} n +160 x \,x^{3 n} a c \,d^{3} n^{4}+240 x \,x^{3 n} b \,c^{2} d^{2} n^{4}+164 x \,x^{4 n} b c \,d^{3} n^{2}+312 x \,x^{3 n} a c \,d^{3} n^{3}+468 x \,x^{3 n} b \,c^{2} d^{2} n^{3}+354 x \,x^{2 n} a \,c^{2} d^{2} n^{2}+236 x \,x^{2 n} b \,c^{3} d \,n^{2}+b \,d^{4} x \,x^{5 n}+78 x \,x^{2 n} a \,c^{2} d^{2} n +52 x \,x^{2 n} b \,c^{3} d n +360 x \,x^{2 n} a \,c^{2} d^{2} n^{4}+240 x \,x^{2 n} b \,c^{3} d \,n^{4}+44 x \,x^{4 n} b c \,d^{3} n +196 x \,x^{3 n} a c \,d^{3} n^{2}+294 x \,x^{3 n} b \,c^{2} d^{2} n^{2}+642 x \,x^{2 n} a \,c^{2} d^{2} n^{3}+428 x \,x^{2 n} b \,c^{3} d \,n^{3}+120 x \,x^{4 n} b c \,d^{3} n^{4}+244 x \,x^{4 n} b c \,d^{3} n^{3}+4 x \,x^{4 n} b c \,d^{3}+4 x \,x^{3 n} a c \,d^{3}+6 x \,x^{3 n} b \,c^{2} d^{2}+6 x \,x^{2 n} a \,c^{2} d^{2}+4 x \,x^{2 n} b \,c^{3} d}{\left (5 n +1\right ) \left (1+n \right ) \left (1+4 n \right ) \left (1+3 n \right ) \left (1+2 n \right )}\) | \(747\) |
| orering | \(\text {Expression too large to display}\) | \(1997\) |
Input:
int((a+b*x^n)*(c+d*x^n)^4,x,method=_RETURNVERBOSE)
Output:
a*c^4*x+b*d^4/(5*n+1)*x*(x^n)^5+c^3*(4*a*d+b*c)/(1+n)*x*x^n+d^3*(a*d+4*b*c )/(1+4*n)*x*(x^n)^4+2*c*d^2*(2*a*d+3*b*c)/(1+3*n)*x*(x^n)^3+2*c^2*d*(3*a*d +2*b*c)/(1+2*n)*x*(x^n)^2
Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (132) = 264\).
Time = 0.18 (sec) , antiderivative size = 527, normalized size of antiderivative = 3.99 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^4 \, dx=\frac {{\left (24 \, b d^{4} n^{4} + 50 \, b d^{4} n^{3} + 35 \, b d^{4} n^{2} + 10 \, b d^{4} n + b d^{4}\right )} x x^{5 \, n} + {\left (4 \, b c d^{3} + a d^{4} + 30 \, {\left (4 \, b c d^{3} + a d^{4}\right )} n^{4} + 61 \, {\left (4 \, b c d^{3} + a d^{4}\right )} n^{3} + 41 \, {\left (4 \, b c d^{3} + a d^{4}\right )} n^{2} + 11 \, {\left (4 \, b c d^{3} + a d^{4}\right )} n\right )} x x^{4 \, n} + 2 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3} + 40 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} n^{4} + 78 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} n^{3} + 49 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} n^{2} + 12 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} n\right )} x x^{3 \, n} + 2 \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2} + 60 \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} n^{4} + 107 \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} n^{3} + 59 \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} n^{2} + 13 \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} n\right )} x x^{2 \, n} + {\left (b c^{4} + 4 \, a c^{3} d + 120 \, {\left (b c^{4} + 4 \, a c^{3} d\right )} n^{4} + 154 \, {\left (b c^{4} + 4 \, a c^{3} d\right )} n^{3} + 71 \, {\left (b c^{4} + 4 \, a c^{3} d\right )} n^{2} + 14 \, {\left (b c^{4} + 4 \, a c^{3} d\right )} n\right )} x x^{n} + {\left (120 \, a c^{4} n^{5} + 274 \, a c^{4} n^{4} + 225 \, a c^{4} n^{3} + 85 \, a c^{4} n^{2} + 15 \, a c^{4} n + a c^{4}\right )} x}{120 \, n^{5} + 274 \, n^{4} + 225 \, n^{3} + 85 \, n^{2} + 15 \, n + 1} \] Input:
integrate((a+b*x^n)*(c+d*x^n)^4,x, algorithm="fricas")
Output:
((24*b*d^4*n^4 + 50*b*d^4*n^3 + 35*b*d^4*n^2 + 10*b*d^4*n + b*d^4)*x*x^(5* n) + (4*b*c*d^3 + a*d^4 + 30*(4*b*c*d^3 + a*d^4)*n^4 + 61*(4*b*c*d^3 + a*d ^4)*n^3 + 41*(4*b*c*d^3 + a*d^4)*n^2 + 11*(4*b*c*d^3 + a*d^4)*n)*x*x^(4*n) + 2*(3*b*c^2*d^2 + 2*a*c*d^3 + 40*(3*b*c^2*d^2 + 2*a*c*d^3)*n^4 + 78*(3*b *c^2*d^2 + 2*a*c*d^3)*n^3 + 49*(3*b*c^2*d^2 + 2*a*c*d^3)*n^2 + 12*(3*b*c^2 *d^2 + 2*a*c*d^3)*n)*x*x^(3*n) + 2*(2*b*c^3*d + 3*a*c^2*d^2 + 60*(2*b*c^3* d + 3*a*c^2*d^2)*n^4 + 107*(2*b*c^3*d + 3*a*c^2*d^2)*n^3 + 59*(2*b*c^3*d + 3*a*c^2*d^2)*n^2 + 13*(2*b*c^3*d + 3*a*c^2*d^2)*n)*x*x^(2*n) + (b*c^4 + 4 *a*c^3*d + 120*(b*c^4 + 4*a*c^3*d)*n^4 + 154*(b*c^4 + 4*a*c^3*d)*n^3 + 71* (b*c^4 + 4*a*c^3*d)*n^2 + 14*(b*c^4 + 4*a*c^3*d)*n)*x*x^n + (120*a*c^4*n^5 + 274*a*c^4*n^4 + 225*a*c^4*n^3 + 85*a*c^4*n^2 + 15*a*c^4*n + a*c^4)*x)/( 120*n^5 + 274*n^4 + 225*n^3 + 85*n^2 + 15*n + 1)
Leaf count of result is larger than twice the leaf count of optimal. 2744 vs. \(2 (124) = 248\).
Time = 0.82 (sec) , antiderivative size = 2744, normalized size of antiderivative = 20.79 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^4 \, dx=\text {Too large to display} \] Input:
integrate((a+b*x**n)*(c+d*x**n)**4,x)
Output:
Piecewise((a*c**4*x + 4*a*c**3*d*log(x) - 6*a*c**2*d**2/x - 2*a*c*d**3/x** 2 - a*d**4/(3*x**3) + b*c**4*log(x) - 4*b*c**3*d/x - 3*b*c**2*d**2/x**2 - 4*b*c*d**3/(3*x**3) - b*d**4/(4*x**4), Eq(n, -1)), (a*c**4*x + 8*a*c**3*d* sqrt(x) + 6*a*c**2*d**2*log(x) - 8*a*c*d**3/sqrt(x) - a*d**4/x + 2*b*c**4* sqrt(x) + 4*b*c**3*d*log(x) - 12*b*c**2*d**2/sqrt(x) - 4*b*c*d**3/x - 2*b* d**4/(3*x**(3/2)), Eq(n, -1/2)), (a*c**4*x + 6*a*c**3*d*x**(2/3) + 18*a*c* *2*d**2*x**(1/3) + 4*a*c*d**3*log(x) - 3*a*d**4/x**(1/3) + 3*b*c**4*x**(2/ 3)/2 + 12*b*c**3*d*x**(1/3) + 6*b*c**2*d**2*log(x) - 12*b*c*d**3/x**(1/3) - 3*b*d**4/(2*x**(2/3)), Eq(n, -1/3)), (a*c**4*x + 16*a*c**3*d*x**(3/4)/3 + 12*a*c**2*d**2*sqrt(x) + 16*a*c*d**3*x**(1/4) + a*d**4*log(x) + 4*b*c**4 *x**(3/4)/3 + 8*b*c**3*d*sqrt(x) + 24*b*c**2*d**2*x**(1/4) + 4*b*c*d**3*lo g(x) - 4*b*d**4/x**(1/4), Eq(n, -1/4)), (a*c**4*x + 5*a*c**3*d*x**(4/5) + 10*a*c**2*d**2*x**(3/5) + 10*a*c*d**3*x**(2/5) + 5*a*d**4*x**(1/5) + 5*b*c **4*x**(4/5)/4 + 20*b*c**3*d*x**(3/5)/3 + 15*b*c**2*d**2*x**(2/5) + 20*b*c *d**3*x**(1/5) + b*d**4*log(x), Eq(n, -1/5)), (120*a*c**4*n**5*x/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 274*a*c**4*n**4*x/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 225*a*c**4*n**3*x/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 85*a*c**4*n**2*x/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 15*a*c**4*n*x/(120*n**5 + 27 4*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + a*c**4*x/(120*n**5 + 274*n**4...
Time = 0.04 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.41 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^4 \, dx=a c^{4} x + \frac {b d^{4} x^{5 \, n + 1}}{5 \, n + 1} + \frac {4 \, b c d^{3} x^{4 \, n + 1}}{4 \, n + 1} + \frac {a d^{4} x^{4 \, n + 1}}{4 \, n + 1} + \frac {6 \, b c^{2} d^{2} x^{3 \, n + 1}}{3 \, n + 1} + \frac {4 \, a c d^{3} x^{3 \, n + 1}}{3 \, n + 1} + \frac {4 \, b c^{3} d x^{2 \, n + 1}}{2 \, n + 1} + \frac {6 \, a c^{2} d^{2} x^{2 \, n + 1}}{2 \, n + 1} + \frac {b c^{4} x^{n + 1}}{n + 1} + \frac {4 \, a c^{3} d x^{n + 1}}{n + 1} \] Input:
integrate((a+b*x^n)*(c+d*x^n)^4,x, algorithm="maxima")
Output:
a*c^4*x + b*d^4*x^(5*n + 1)/(5*n + 1) + 4*b*c*d^3*x^(4*n + 1)/(4*n + 1) + a*d^4*x^(4*n + 1)/(4*n + 1) + 6*b*c^2*d^2*x^(3*n + 1)/(3*n + 1) + 4*a*c*d^ 3*x^(3*n + 1)/(3*n + 1) + 4*b*c^3*d*x^(2*n + 1)/(2*n + 1) + 6*a*c^2*d^2*x^ (2*n + 1)/(2*n + 1) + b*c^4*x^(n + 1)/(n + 1) + 4*a*c^3*d*x^(n + 1)/(n + 1 )
Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (132) = 264\).
Time = 0.15 (sec) , antiderivative size = 740, normalized size of antiderivative = 5.61 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^4 \, dx=\frac {120 \, a c^{4} n^{5} x + 24 \, b d^{4} n^{4} x x^{5 \, n} + 120 \, b c d^{3} n^{4} x x^{4 \, n} + 30 \, a d^{4} n^{4} x x^{4 \, n} + 240 \, b c^{2} d^{2} n^{4} x x^{3 \, n} + 160 \, a c d^{3} n^{4} x x^{3 \, n} + 240 \, b c^{3} d n^{4} x x^{2 \, n} + 360 \, a c^{2} d^{2} n^{4} x x^{2 \, n} + 120 \, b c^{4} n^{4} x x^{n} + 480 \, a c^{3} d n^{4} x x^{n} + 274 \, a c^{4} n^{4} x + 50 \, b d^{4} n^{3} x x^{5 \, n} + 244 \, b c d^{3} n^{3} x x^{4 \, n} + 61 \, a d^{4} n^{3} x x^{4 \, n} + 468 \, b c^{2} d^{2} n^{3} x x^{3 \, n} + 312 \, a c d^{3} n^{3} x x^{3 \, n} + 428 \, b c^{3} d n^{3} x x^{2 \, n} + 642 \, a c^{2} d^{2} n^{3} x x^{2 \, n} + 154 \, b c^{4} n^{3} x x^{n} + 616 \, a c^{3} d n^{3} x x^{n} + 225 \, a c^{4} n^{3} x + 35 \, b d^{4} n^{2} x x^{5 \, n} + 164 \, b c d^{3} n^{2} x x^{4 \, n} + 41 \, a d^{4} n^{2} x x^{4 \, n} + 294 \, b c^{2} d^{2} n^{2} x x^{3 \, n} + 196 \, a c d^{3} n^{2} x x^{3 \, n} + 236 \, b c^{3} d n^{2} x x^{2 \, n} + 354 \, a c^{2} d^{2} n^{2} x x^{2 \, n} + 71 \, b c^{4} n^{2} x x^{n} + 284 \, a c^{3} d n^{2} x x^{n} + 85 \, a c^{4} n^{2} x + 10 \, b d^{4} n x x^{5 \, n} + 44 \, b c d^{3} n x x^{4 \, n} + 11 \, a d^{4} n x x^{4 \, n} + 72 \, b c^{2} d^{2} n x x^{3 \, n} + 48 \, a c d^{3} n x x^{3 \, n} + 52 \, b c^{3} d n x x^{2 \, n} + 78 \, a c^{2} d^{2} n x x^{2 \, n} + 14 \, b c^{4} n x x^{n} + 56 \, a c^{3} d n x x^{n} + 15 \, a c^{4} n x + b d^{4} x x^{5 \, n} + 4 \, b c d^{3} x x^{4 \, n} + a d^{4} x x^{4 \, n} + 6 \, b c^{2} d^{2} x x^{3 \, n} + 4 \, a c d^{3} x x^{3 \, n} + 4 \, b c^{3} d x x^{2 \, n} + 6 \, a c^{2} d^{2} x x^{2 \, n} + b c^{4} x x^{n} + 4 \, a c^{3} d x x^{n} + a c^{4} x}{120 \, n^{5} + 274 \, n^{4} + 225 \, n^{3} + 85 \, n^{2} + 15 \, n + 1} \] Input:
integrate((a+b*x^n)*(c+d*x^n)^4,x, algorithm="giac")
Output:
(120*a*c^4*n^5*x + 24*b*d^4*n^4*x*x^(5*n) + 120*b*c*d^3*n^4*x*x^(4*n) + 30 *a*d^4*n^4*x*x^(4*n) + 240*b*c^2*d^2*n^4*x*x^(3*n) + 160*a*c*d^3*n^4*x*x^( 3*n) + 240*b*c^3*d*n^4*x*x^(2*n) + 360*a*c^2*d^2*n^4*x*x^(2*n) + 120*b*c^4 *n^4*x*x^n + 480*a*c^3*d*n^4*x*x^n + 274*a*c^4*n^4*x + 50*b*d^4*n^3*x*x^(5 *n) + 244*b*c*d^3*n^3*x*x^(4*n) + 61*a*d^4*n^3*x*x^(4*n) + 468*b*c^2*d^2*n ^3*x*x^(3*n) + 312*a*c*d^3*n^3*x*x^(3*n) + 428*b*c^3*d*n^3*x*x^(2*n) + 642 *a*c^2*d^2*n^3*x*x^(2*n) + 154*b*c^4*n^3*x*x^n + 616*a*c^3*d*n^3*x*x^n + 2 25*a*c^4*n^3*x + 35*b*d^4*n^2*x*x^(5*n) + 164*b*c*d^3*n^2*x*x^(4*n) + 41*a *d^4*n^2*x*x^(4*n) + 294*b*c^2*d^2*n^2*x*x^(3*n) + 196*a*c*d^3*n^2*x*x^(3* n) + 236*b*c^3*d*n^2*x*x^(2*n) + 354*a*c^2*d^2*n^2*x*x^(2*n) + 71*b*c^4*n^ 2*x*x^n + 284*a*c^3*d*n^2*x*x^n + 85*a*c^4*n^2*x + 10*b*d^4*n*x*x^(5*n) + 44*b*c*d^3*n*x*x^(4*n) + 11*a*d^4*n*x*x^(4*n) + 72*b*c^2*d^2*n*x*x^(3*n) + 48*a*c*d^3*n*x*x^(3*n) + 52*b*c^3*d*n*x*x^(2*n) + 78*a*c^2*d^2*n*x*x^(2*n ) + 14*b*c^4*n*x*x^n + 56*a*c^3*d*n*x*x^n + 15*a*c^4*n*x + b*d^4*x*x^(5*n) + 4*b*c*d^3*x*x^(4*n) + a*d^4*x*x^(4*n) + 6*b*c^2*d^2*x*x^(3*n) + 4*a*c*d ^3*x*x^(3*n) + 4*b*c^3*d*x*x^(2*n) + 6*a*c^2*d^2*x*x^(2*n) + b*c^4*x*x^n + 4*a*c^3*d*x*x^n + a*c^4*x)/(120*n^5 + 274*n^4 + 225*n^3 + 85*n^2 + 15*n + 1)
Time = 0.92 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.99 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^4 \, dx=a\,c^4\,x+\frac {x\,x^n\,\left (b\,c^4+4\,a\,d\,c^3\right )}{n+1}+\frac {x\,x^{4\,n}\,\left (a\,d^4+4\,b\,c\,d^3\right )}{4\,n+1}+\frac {b\,d^4\,x\,x^{5\,n}}{5\,n+1}+\frac {2\,c^2\,d\,x\,x^{2\,n}\,\left (3\,a\,d+2\,b\,c\right )}{2\,n+1}+\frac {2\,c\,d^2\,x\,x^{3\,n}\,\left (2\,a\,d+3\,b\,c\right )}{3\,n+1} \] Input:
int((a + b*x^n)*(c + d*x^n)^4,x)
Output:
a*c^4*x + (x*x^n*(b*c^4 + 4*a*c^3*d))/(n + 1) + (x*x^(4*n)*(a*d^4 + 4*b*c* d^3))/(4*n + 1) + (b*d^4*x*x^(5*n))/(5*n + 1) + (2*c^2*d*x*x^(2*n)*(3*a*d + 2*b*c))/(2*n + 1) + (2*c*d^2*x*x^(3*n)*(2*a*d + 3*b*c))/(3*n + 1)
Time = 0.21 (sec) , antiderivative size = 690, normalized size of antiderivative = 5.23 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^4 \, dx=\frac {x \left (120 a \,c^{4} n^{5}+274 a \,c^{4} n^{4}+225 a \,c^{4} n^{3}+85 a \,c^{4} n^{2}+15 a \,c^{4} n +a \,c^{4}+24 x^{5 n} b \,d^{4} n^{4}+50 x^{5 n} b \,d^{4} n^{3}+35 x^{5 n} b \,d^{4} n^{2}+10 x^{5 n} b \,d^{4} n +30 x^{4 n} a \,d^{4} n^{4}+61 x^{4 n} a \,d^{4} n^{3}+41 x^{4 n} a \,d^{4} n^{2}+11 x^{4 n} a \,d^{4} n +4 x^{4 n} b c \,d^{3}+4 x^{3 n} a c \,d^{3}+6 x^{3 n} b \,c^{2} d^{2}+6 x^{2 n} a \,c^{2} d^{2}+4 x^{2 n} b \,c^{3} d +4 x^{n} a \,c^{3} d +120 x^{n} b \,c^{4} n^{4}+154 x^{n} b \,c^{4} n^{3}+71 x^{n} b \,c^{4} n^{2}+14 x^{n} b \,c^{4} n +x^{4 n} a \,d^{4}+x^{5 n} b \,d^{4}+x^{n} b \,c^{4}+284 x^{n} a \,c^{3} d \,n^{2}+56 x^{n} a \,c^{3} d n +120 x^{4 n} b c \,d^{3} n^{4}+244 x^{4 n} b c \,d^{3} n^{3}+164 x^{4 n} b c \,d^{3} n^{2}+44 x^{4 n} b c \,d^{3} n +160 x^{3 n} a c \,d^{3} n^{4}+312 x^{3 n} a c \,d^{3} n^{3}+196 x^{3 n} a c \,d^{3} n^{2}+48 x^{3 n} a c \,d^{3} n +240 x^{3 n} b \,c^{2} d^{2} n^{4}+468 x^{3 n} b \,c^{2} d^{2} n^{3}+294 x^{3 n} b \,c^{2} d^{2} n^{2}+72 x^{3 n} b \,c^{2} d^{2} n +360 x^{2 n} a \,c^{2} d^{2} n^{4}+642 x^{2 n} a \,c^{2} d^{2} n^{3}+354 x^{2 n} a \,c^{2} d^{2} n^{2}+78 x^{2 n} a \,c^{2} d^{2} n +240 x^{2 n} b \,c^{3} d \,n^{4}+428 x^{2 n} b \,c^{3} d \,n^{3}+236 x^{2 n} b \,c^{3} d \,n^{2}+52 x^{2 n} b \,c^{3} d n +480 x^{n} a \,c^{3} d \,n^{4}+616 x^{n} a \,c^{3} d \,n^{3}\right )}{120 n^{5}+274 n^{4}+225 n^{3}+85 n^{2}+15 n +1} \] Input:
int((a+b*x^n)*(c+d*x^n)^4,x)
Output:
(x*(24*x**(5*n)*b*d**4*n**4 + 50*x**(5*n)*b*d**4*n**3 + 35*x**(5*n)*b*d**4 *n**2 + 10*x**(5*n)*b*d**4*n + x**(5*n)*b*d**4 + 30*x**(4*n)*a*d**4*n**4 + 61*x**(4*n)*a*d**4*n**3 + 41*x**(4*n)*a*d**4*n**2 + 11*x**(4*n)*a*d**4*n + x**(4*n)*a*d**4 + 120*x**(4*n)*b*c*d**3*n**4 + 244*x**(4*n)*b*c*d**3*n** 3 + 164*x**(4*n)*b*c*d**3*n**2 + 44*x**(4*n)*b*c*d**3*n + 4*x**(4*n)*b*c*d **3 + 160*x**(3*n)*a*c*d**3*n**4 + 312*x**(3*n)*a*c*d**3*n**3 + 196*x**(3* n)*a*c*d**3*n**2 + 48*x**(3*n)*a*c*d**3*n + 4*x**(3*n)*a*c*d**3 + 240*x**( 3*n)*b*c**2*d**2*n**4 + 468*x**(3*n)*b*c**2*d**2*n**3 + 294*x**(3*n)*b*c** 2*d**2*n**2 + 72*x**(3*n)*b*c**2*d**2*n + 6*x**(3*n)*b*c**2*d**2 + 360*x** (2*n)*a*c**2*d**2*n**4 + 642*x**(2*n)*a*c**2*d**2*n**3 + 354*x**(2*n)*a*c* *2*d**2*n**2 + 78*x**(2*n)*a*c**2*d**2*n + 6*x**(2*n)*a*c**2*d**2 + 240*x* *(2*n)*b*c**3*d*n**4 + 428*x**(2*n)*b*c**3*d*n**3 + 236*x**(2*n)*b*c**3*d* n**2 + 52*x**(2*n)*b*c**3*d*n + 4*x**(2*n)*b*c**3*d + 480*x**n*a*c**3*d*n* *4 + 616*x**n*a*c**3*d*n**3 + 284*x**n*a*c**3*d*n**2 + 56*x**n*a*c**3*d*n + 4*x**n*a*c**3*d + 120*x**n*b*c**4*n**4 + 154*x**n*b*c**4*n**3 + 71*x**n* b*c**4*n**2 + 14*x**n*b*c**4*n + x**n*b*c**4 + 120*a*c**4*n**5 + 274*a*c** 4*n**4 + 225*a*c**4*n**3 + 85*a*c**4*n**2 + 15*a*c**4*n + a*c**4))/(120*n* *5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1)