Integrand size = 13, antiderivative size = 75 \[ \int x^m \left (a+b x^n\right )^3 \, dx=\frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{1+m+n}}{1+m+n}+\frac {3 a b^2 x^{1+m+2 n}}{1+m+2 n}+\frac {b^3 x^{1+m+3 n}}{1+m+3 n} \] Output:
a^3*x^(1+m)/(1+m)+3*a^2*b*x^(1+m+n)/(1+m+n)+3*a*b^2*x^(1+m+2*n)/(1+m+2*n)+ b^3*x^(1+m+3*n)/(1+m+3*n)
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89 \[ \int x^m \left (a+b x^n\right )^3 \, dx=x^{1+m} \left (\frac {a^3}{1+m}+\frac {3 a^2 b x^n}{1+m+n}+\frac {3 a b^2 x^{2 n}}{1+m+2 n}+\frac {b^3 x^{3 n}}{1+m+3 n}\right ) \] Input:
Integrate[x^m*(a + b*x^n)^3,x]
Output:
x^(1 + m)*(a^3/(1 + m) + (3*a^2*b*x^n)/(1 + m + n) + (3*a*b^2*x^(2*n))/(1 + m + 2*n) + (b^3*x^(3*n))/(1 + m + 3*n))
Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {802, 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.
| \(\displaystyle \int x^m \left (a+b x^n\right )^3 \, dx\) |
| \(\Big \downarrow \) 802 |
| \(\displaystyle \int \left (a^3 x^m+3 a^2 b x^{m+n}+3 a b^2 x^{m+2 n}+b^3 x^{m+3 n}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 x^{m+1}}{m+1}+\frac {3 a^2 b x^{m+n+1}}{m+n+1}+\frac {3 a b^2 x^{m+2 n+1}}{m+2 n+1}+\frac {b^3 x^{m+3 n+1}}{m+3 n+1}\) |
Input:
Int[x^m*(a + b*x^n)^3,x]
Output:
(a^3*x^(1 + m))/(1 + m) + (3*a^2*b*x^(1 + m + n))/(1 + m + n) + (3*a*b^2*x ^(1 + m + 2*n))/(1 + m + 2*n) + (b^3*x^(1 + m + 3*n))/(1 + m + 3*n)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Time = 0.73 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(\frac {a^{3} x \,x^{m}}{1+m}+\frac {b^{3} x \,x^{m} x^{3 n}}{1+m +3 n}+\frac {3 a \,b^{2} x \,x^{m} x^{2 n}}{1+m +2 n}+\frac {3 a^{2} b x \,x^{m} x^{n}}{1+n +m}\) | \(78\) |
| norman | \(\frac {a^{3} x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b^{3} x \,{\mathrm e}^{m \ln \left (x \right )} {\mathrm e}^{3 n \ln \left (x \right )}}{1+m +3 n}+\frac {3 a \,b^{2} x \,{\mathrm e}^{m \ln \left (x \right )} {\mathrm e}^{2 n \ln \left (x \right )}}{1+m +2 n}+\frac {3 a^{2} b x \,{\mathrm e}^{m \ln \left (x \right )} {\mathrm e}^{n \ln \left (x \right )}}{1+n +m}\) | \(92\) |
| parallelrisch | \(\frac {b^{3} x \,x^{m} x^{3 n}+x \,x^{m} a^{3} m^{3}+6 x \,x^{m} a^{3} n^{3}+3 x \,x^{m} a^{3} m^{2}+12 x \,x^{m} x^{2 n} a \,b^{2} m^{2} n +9 x \,x^{m} x^{2 n} a \,b^{2} m \,n^{2}+24 x \,x^{m} x^{2 n} a \,b^{2} m n +3 x \,x^{m} x^{n} a^{2} b \,m^{3}+9 x \,x^{m} x^{n} a^{2} b \,m^{2}+18 x \,x^{m} x^{n} a^{2} b \,n^{2}+9 x \,x^{m} x^{n} a^{2} b m +15 x \,x^{m} x^{n} a^{2} b n +11 x \,x^{m} a^{3} n^{2}+3 x \,x^{m} a^{3} m +6 x \,x^{m} a^{3} n +15 x \,x^{m} x^{n} a^{2} b \,m^{2} n +18 x \,x^{m} x^{n} a^{2} b m \,n^{2}+30 x \,x^{m} x^{n} a^{2} b m n +3 x \,x^{m} x^{n} a^{2} b +12 x \,x^{m} x^{2 n} a \,b^{2} n +x \,x^{m} a^{3}+9 x \,x^{m} x^{2 n} a \,b^{2} n^{2}+9 x \,x^{m} x^{2 n} a \,b^{2} m +6 x \,x^{m} a^{3} m^{2} n +11 x \,x^{m} a^{3} m \,n^{2}+12 x \,x^{m} a^{3} m n +x \,x^{m} x^{3 n} b^{3} m^{3}+3 x \,x^{m} x^{3 n} b^{3} m^{2}+2 x \,x^{m} x^{3 n} b^{3} n^{2}+3 x \,x^{m} x^{3 n} b^{3} m +3 x \,x^{m} x^{3 n} b^{3} n +3 a \,b^{2} x \,x^{m} x^{2 n}+3 x \,x^{m} x^{2 n} a \,b^{2} m^{3}+9 x \,x^{m} x^{2 n} a \,b^{2} m^{2}+3 x \,x^{m} x^{3 n} b^{3} m^{2} n +2 x \,x^{m} x^{3 n} b^{3} m \,n^{2}+6 x \,x^{m} x^{3 n} b^{3} m n}{\left (1+m \right ) \left (1+m +3 n \right ) \left (1+m +2 n \right ) \left (1+n +m \right )}\) | \(581\) |
| orering | \(\frac {x \left (4 m^{3}+18 m^{2} n +22 m \,n^{2}+6 n^{3}+6 m^{2}+18 m n +11 n^{2}+4 m +6 n +1\right ) x^{m} \left (a +b \,x^{n}\right )^{3}}{m^{4}+6 m^{3} n +11 m^{2} n^{2}+6 m \,n^{3}+4 m^{3}+18 m^{2} n +22 m \,n^{2}+6 n^{3}+6 m^{2}+18 m n +11 n^{2}+4 m +6 n +1}-\frac {x^{2} \left (6 m^{2}+18 m n +11 n^{2}+1\right ) \left (\frac {x^{m} m \left (a +b \,x^{n}\right )^{3}}{x}+\frac {3 x^{m} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n}{x}\right )}{m^{4}+6 m^{3} n +11 m^{2} n^{2}+6 m \,n^{3}+4 m^{3}+18 m^{2} n +22 m \,n^{2}+6 n^{3}+6 m^{2}+18 m n +11 n^{2}+4 m +6 n +1}+\frac {2 x^{3} \left (2 m +3 n -1\right ) \left (\frac {x^{m} m^{2} \left (a +b \,x^{n}\right )^{3}}{x^{2}}-\frac {x^{m} m \left (a +b \,x^{n}\right )^{3}}{x^{2}}+\frac {6 x^{m} m \left (a +b \,x^{n}\right )^{2} b \,x^{n} n}{x^{2}}+\frac {6 x^{m} x^{2 n} b^{2} n^{2} \left (a +b \,x^{n}\right )}{x^{2}}+\frac {3 x^{m} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n^{2}}{x^{2}}-\frac {3 x^{m} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n}{x^{2}}\right )}{m^{4}+6 m^{3} n +11 m^{2} n^{2}+6 m \,n^{3}+4 m^{3}+18 m^{2} n +22 m \,n^{2}+6 n^{3}+6 m^{2}+18 m n +11 n^{2}+4 m +6 n +1}-\frac {x^{4} \left (\frac {x^{m} m^{3} \left (a +b \,x^{n}\right )^{3}}{x^{3}}-\frac {3 x^{m} m^{2} \left (a +b \,x^{n}\right )^{3}}{x^{3}}+\frac {9 x^{m} m^{2} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n}{x^{3}}+\frac {2 x^{m} m \left (a +b \,x^{n}\right )^{3}}{x^{3}}-\frac {18 x^{m} m \left (a +b \,x^{n}\right )^{2} b \,x^{n} n}{x^{3}}+\frac {18 x^{m} m \left (a +b \,x^{n}\right ) b^{2} x^{2 n} n^{2}}{x^{3}}+\frac {9 x^{m} m \left (a +b \,x^{n}\right )^{2} b \,x^{n} n^{2}}{x^{3}}+\frac {18 x^{m} x^{2 n} b^{2} n^{3} \left (a +b \,x^{n}\right )}{x^{3}}+\frac {6 x^{m} x^{3 n} b^{3} n^{3}}{x^{3}}-\frac {18 x^{m} x^{2 n} b^{2} n^{2} \left (a +b \,x^{n}\right )}{x^{3}}+\frac {3 x^{m} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n^{3}}{x^{3}}-\frac {9 x^{m} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n^{2}}{x^{3}}+\frac {6 x^{m} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n}{x^{3}}\right )}{m^{4}+6 m^{3} n +11 m^{2} n^{2}+6 m \,n^{3}+4 m^{3}+18 m^{2} n +22 m \,n^{2}+6 n^{3}+6 m^{2}+18 m n +11 n^{2}+4 m +6 n +1}\) | \(847\) |
Input:
int(x^m*(a+b*x^n)^3,x,method=_RETURNVERBOSE)
Output:
a^3/(1+m)*x*x^m+b^3/(1+m+3*n)*x*x^m*(x^n)^3+3*a*b^2/(1+m+2*n)*x*x^m*(x^n)^ 2+3*a^2*b/(1+n+m)*x*x^m*x^n
Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (75) = 150\).
Time = 0.08 (sec) , antiderivative size = 362, normalized size of antiderivative = 4.83 \[ \int x^m \left (a+b x^n\right )^3 \, dx=\frac {{\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3} + 2 \, {\left (b^{3} m + b^{3}\right )} n^{2} + 3 \, {\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n\right )} x x^{m} x^{3 \, n} + 3 \, {\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2} + 3 \, {\left (a b^{2} m + a b^{2}\right )} n^{2} + 4 \, {\left (a b^{2} m^{2} + 2 \, a b^{2} m + a b^{2}\right )} n\right )} x x^{m} x^{2 \, n} + 3 \, {\left (a^{2} b m^{3} + 3 \, a^{2} b m^{2} + 3 \, a^{2} b m + a^{2} b + 6 \, {\left (a^{2} b m + a^{2} b\right )} n^{2} + 5 \, {\left (a^{2} b m^{2} + 2 \, a^{2} b m + a^{2} b\right )} n\right )} x x^{m} x^{n} + {\left (a^{3} m^{3} + 6 \, a^{3} n^{3} + 3 \, a^{3} m^{2} + 3 \, a^{3} m + a^{3} + 11 \, {\left (a^{3} m + a^{3}\right )} n^{2} + 6 \, {\left (a^{3} m^{2} + 2 \, a^{3} m + a^{3}\right )} n\right )} x x^{m}}{m^{4} + 6 \, {\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \, {\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \, {\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1} \] Input:
integrate(x^m*(a+b*x^n)^3,x, algorithm="fricas")
Output:
((b^3*m^3 + 3*b^3*m^2 + 3*b^3*m + b^3 + 2*(b^3*m + b^3)*n^2 + 3*(b^3*m^2 + 2*b^3*m + b^3)*n)*x*x^m*x^(3*n) + 3*(a*b^2*m^3 + 3*a*b^2*m^2 + 3*a*b^2*m + a*b^2 + 3*(a*b^2*m + a*b^2)*n^2 + 4*(a*b^2*m^2 + 2*a*b^2*m + a*b^2)*n)*x *x^m*x^(2*n) + 3*(a^2*b*m^3 + 3*a^2*b*m^2 + 3*a^2*b*m + a^2*b + 6*(a^2*b*m + a^2*b)*n^2 + 5*(a^2*b*m^2 + 2*a^2*b*m + a^2*b)*n)*x*x^m*x^n + (a^3*m^3 + 6*a^3*n^3 + 3*a^3*m^2 + 3*a^3*m + a^3 + 11*(a^3*m + a^3)*n^2 + 6*(a^3*m^ 2 + 2*a^3*m + a^3)*n)*x*x^m)/(m^4 + 6*(m + 1)*n^3 + 4*m^3 + 11*(m^2 + 2*m + 1)*n^2 + 6*m^2 + 6*(m^3 + 3*m^2 + 3*m + 1)*n + 4*m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 3546 vs. \(2 (70) = 140\).
Time = 3.03 (sec) , antiderivative size = 3546, normalized size of antiderivative = 47.28 \[ \int x^m \left (a+b x^n\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(x**m*(a+b*x**n)**3,x)
Output:
Piecewise(((a + b)**3*log(x), Eq(m, -1) & Eq(n, 0)), (a**3*log(x) + 3*a**2 *b*x**n/n + 3*a*b**2*x**(2*n)/(2*n) + b**3*x**(3*n)/(3*n), Eq(m, -1)), (-a **3/(3*n*x**(3*n)) - 3*a**2*b/(2*n*x**(2*n)) - 3*a*b**2/(n*x**n) + b**3*lo g(x**n)/n, Eq(m, -3*n - 1)), (-a**3/(2*n*x**(2*n)) - 3*a**2*b/(n*x**n) + 3 *a*b**2*log(x**n)/n + b**3*x**n/n, Eq(m, -2*n - 1)), (-a**3/(n*x**n) + 3*a **2*b*log(x**n)/n + 3*a*b**2*x**n/n + b**3*x**(2*n)/(2*n), Eq(m, -n - 1)), (a**3*m**3*x*x**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*a**3*m**2*n*x*x**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2 *n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 3*a**3*m**2*x*x**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m **2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 11*a**3*m*n**2*x*x**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n **2 + 6*n + 1) + 12*a**3*m*n*x*x**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n* *2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 1 1*n**2 + 6*n + 1) + 3*a**3*m*x*x**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n* *2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 1 1*n**2 + 6*n + 1) + 6*a**3*n**3*x*x**m/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2 *n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n*...
Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int x^m \left (a+b x^n\right )^3 \, dx=\frac {b^{3} x^{m + 3 \, n + 1}}{m + 3 \, n + 1} + \frac {3 \, a b^{2} x^{m + 2 \, n + 1}}{m + 2 \, n + 1} + \frac {3 \, a^{2} b x^{m + n + 1}}{m + n + 1} + \frac {a^{3} x^{m + 1}}{m + 1} \] Input:
integrate(x^m*(a+b*x^n)^3,x, algorithm="maxima")
Output:
b^3*x^(m + 3*n + 1)/(m + 3*n + 1) + 3*a*b^2*x^(m + 2*n + 1)/(m + 2*n + 1) + 3*a^2*b*x^(m + n + 1)/(m + n + 1) + a^3*x^(m + 1)/(m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (75) = 150\).
Time = 0.13 (sec) , antiderivative size = 622, normalized size of antiderivative = 8.29 \[ \int x^m \left (a+b x^n\right )^3 \, dx=\frac {b^{3} m^{3} x x^{m} x^{3 \, n} + 3 \, b^{3} m^{2} n x x^{m} x^{3 \, n} + 2 \, b^{3} m n^{2} x x^{m} x^{3 \, n} + 3 \, a b^{2} m^{3} x x^{m} x^{2 \, n} + 12 \, a b^{2} m^{2} n x x^{m} x^{2 \, n} + 9 \, a b^{2} m n^{2} x x^{m} x^{2 \, n} + 3 \, a^{2} b m^{3} x x^{m} x^{n} + 15 \, a^{2} b m^{2} n x x^{m} x^{n} + 18 \, a^{2} b m n^{2} x x^{m} x^{n} + a^{3} m^{3} x x^{m} + 6 \, a^{3} m^{2} n x x^{m} + 11 \, a^{3} m n^{2} x x^{m} + 6 \, a^{3} n^{3} x x^{m} + 3 \, b^{3} m^{2} x x^{m} x^{3 \, n} + 6 \, b^{3} m n x x^{m} x^{3 \, n} + 2 \, b^{3} n^{2} x x^{m} x^{3 \, n} + 9 \, a b^{2} m^{2} x x^{m} x^{2 \, n} + 24 \, a b^{2} m n x x^{m} x^{2 \, n} + 9 \, a b^{2} n^{2} x x^{m} x^{2 \, n} + 9 \, a^{2} b m^{2} x x^{m} x^{n} + 30 \, a^{2} b m n x x^{m} x^{n} + 18 \, a^{2} b n^{2} x x^{m} x^{n} + 3 \, a^{3} m^{2} x x^{m} + 12 \, a^{3} m n x x^{m} + 11 \, a^{3} n^{2} x x^{m} + 3 \, b^{3} m x x^{m} x^{3 \, n} + 3 \, b^{3} n x x^{m} x^{3 \, n} + 9 \, a b^{2} m x x^{m} x^{2 \, n} + 12 \, a b^{2} n x x^{m} x^{2 \, n} + 9 \, a^{2} b m x x^{m} x^{n} + 15 \, a^{2} b n x x^{m} x^{n} + 3 \, a^{3} m x x^{m} + 6 \, a^{3} n x x^{m} + b^{3} x x^{m} x^{3 \, n} + 3 \, a b^{2} x x^{m} x^{2 \, n} + 3 \, a^{2} b x x^{m} x^{n} + a^{3} x x^{m}}{m^{4} + 6 \, m^{3} n + 11 \, m^{2} n^{2} + 6 \, m n^{3} + 4 \, m^{3} + 18 \, m^{2} n + 22 \, m n^{2} + 6 \, n^{3} + 6 \, m^{2} + 18 \, m n + 11 \, n^{2} + 4 \, m + 6 \, n + 1} \] Input:
integrate(x^m*(a+b*x^n)^3,x, algorithm="giac")
Output:
(b^3*m^3*x*x^m*x^(3*n) + 3*b^3*m^2*n*x*x^m*x^(3*n) + 2*b^3*m*n^2*x*x^m*x^( 3*n) + 3*a*b^2*m^3*x*x^m*x^(2*n) + 12*a*b^2*m^2*n*x*x^m*x^(2*n) + 9*a*b^2* m*n^2*x*x^m*x^(2*n) + 3*a^2*b*m^3*x*x^m*x^n + 15*a^2*b*m^2*n*x*x^m*x^n + 1 8*a^2*b*m*n^2*x*x^m*x^n + a^3*m^3*x*x^m + 6*a^3*m^2*n*x*x^m + 11*a^3*m*n^2 *x*x^m + 6*a^3*n^3*x*x^m + 3*b^3*m^2*x*x^m*x^(3*n) + 6*b^3*m*n*x*x^m*x^(3* n) + 2*b^3*n^2*x*x^m*x^(3*n) + 9*a*b^2*m^2*x*x^m*x^(2*n) + 24*a*b^2*m*n*x* x^m*x^(2*n) + 9*a*b^2*n^2*x*x^m*x^(2*n) + 9*a^2*b*m^2*x*x^m*x^n + 30*a^2*b *m*n*x*x^m*x^n + 18*a^2*b*n^2*x*x^m*x^n + 3*a^3*m^2*x*x^m + 12*a^3*m*n*x*x ^m + 11*a^3*n^2*x*x^m + 3*b^3*m*x*x^m*x^(3*n) + 3*b^3*n*x*x^m*x^(3*n) + 9* a*b^2*m*x*x^m*x^(2*n) + 12*a*b^2*n*x*x^m*x^(2*n) + 9*a^2*b*m*x*x^m*x^n + 1 5*a^2*b*n*x*x^m*x^n + 3*a^3*m*x*x^m + 6*a^3*n*x*x^m + b^3*x*x^m*x^(3*n) + 3*a*b^2*x*x^m*x^(2*n) + 3*a^2*b*x*x^m*x^n + a^3*x*x^m)/(m^4 + 6*m^3*n + 11 *m^2*n^2 + 6*m*n^3 + 4*m^3 + 18*m^2*n + 22*m*n^2 + 6*n^3 + 6*m^2 + 18*m*n + 11*n^2 + 4*m + 6*n + 1)
Time = 0.87 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int x^m \left (a+b x^n\right )^3 \, dx=\frac {a^3\,x\,x^m}{m+1}+\frac {b^3\,x\,x^m\,x^{3\,n}}{m+3\,n+1}+\frac {3\,a^2\,b\,x\,x^m\,x^n}{m+n+1}+\frac {3\,a\,b^2\,x\,x^m\,x^{2\,n}}{m+2\,n+1} \] Input:
int(x^m*(a + b*x^n)^3,x)
Output:
(a^3*x*x^m)/(m + 1) + (b^3*x*x^m*x^(3*n))/(m + 3*n + 1) + (3*a^2*b*x*x^m*x ^n)/(m + n + 1) + (3*a*b^2*x*x^m*x^(2*n))/(m + 2*n + 1)
Time = 0.24 (sec) , antiderivative size = 477, normalized size of antiderivative = 6.36 \[ \int x^m \left (a+b x^n\right )^3 \, dx=\frac {x^{m} x \left (a^{3}+2 x^{3 n} b^{3} n^{2}+3 x^{3 n} b^{3} n +3 x^{2 n} a \,b^{2}+3 x^{n} a^{2} b +9 x^{2 n} a \,b^{2} n^{2}+12 x^{2 n} a \,b^{2} n +18 x^{n} a^{2} b \,n^{2}+15 x^{n} a^{2} b n +x^{3 n} b^{3}+6 a^{3} n^{3}+11 a^{3} n^{2}+6 a^{3} n +3 x^{3 n} b^{3} m^{2}+3 x^{3 n} b^{3} m +6 a^{3} m^{2} n +11 a^{3} m \,n^{2}+12 a^{3} m n +12 x^{2 n} a \,b^{2} m^{2} n +9 x^{2 n} a \,b^{2} m \,n^{2}+24 x^{2 n} a \,b^{2} m n +15 x^{n} a^{2} b \,m^{2} n +18 x^{n} a^{2} b m \,n^{2}+30 x^{n} a^{2} b m n +a^{3} m^{3}+3 a^{3} m^{2}+3 a^{3} m +x^{3 n} b^{3} m^{3}+3 x^{3 n} b^{3} m^{2} n +2 x^{3 n} b^{3} m \,n^{2}+6 x^{3 n} b^{3} m n +3 x^{2 n} a \,b^{2} m^{3}+9 x^{2 n} a \,b^{2} m^{2}+9 x^{2 n} a \,b^{2} m +3 x^{n} a^{2} b \,m^{3}+9 x^{n} a^{2} b \,m^{2}+9 x^{n} a^{2} b m \right )}{m^{4}+6 m^{3} n +11 m^{2} n^{2}+6 m \,n^{3}+4 m^{3}+18 m^{2} n +22 m \,n^{2}+6 n^{3}+6 m^{2}+18 m n +11 n^{2}+4 m +6 n +1} \] Input:
int(x^m*(a+b*x^n)^3,x)
Output:
(x**m*x*(x**(3*n)*b**3*m**3 + 3*x**(3*n)*b**3*m**2*n + 3*x**(3*n)*b**3*m** 2 + 2*x**(3*n)*b**3*m*n**2 + 6*x**(3*n)*b**3*m*n + 3*x**(3*n)*b**3*m + 2*x **(3*n)*b**3*n**2 + 3*x**(3*n)*b**3*n + x**(3*n)*b**3 + 3*x**(2*n)*a*b**2* m**3 + 12*x**(2*n)*a*b**2*m**2*n + 9*x**(2*n)*a*b**2*m**2 + 9*x**(2*n)*a*b **2*m*n**2 + 24*x**(2*n)*a*b**2*m*n + 9*x**(2*n)*a*b**2*m + 9*x**(2*n)*a*b **2*n**2 + 12*x**(2*n)*a*b**2*n + 3*x**(2*n)*a*b**2 + 3*x**n*a**2*b*m**3 + 15*x**n*a**2*b*m**2*n + 9*x**n*a**2*b*m**2 + 18*x**n*a**2*b*m*n**2 + 30*x **n*a**2*b*m*n + 9*x**n*a**2*b*m + 18*x**n*a**2*b*n**2 + 15*x**n*a**2*b*n + 3*x**n*a**2*b + a**3*m**3 + 6*a**3*m**2*n + 3*a**3*m**2 + 11*a**3*m*n**2 + 12*a**3*m*n + 3*a**3*m + 6*a**3*n**3 + 11*a**3*n**2 + 6*a**3*n + a**3)) /(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m*n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1)