\(\int x^m (a+b x^n)^3 \, dx\) [2667]
Optimal result
Integrand size = 13, antiderivative size = 75 \[
\int x^m \left (a+b x^n\right )^3 \, dx=\frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{1+m+n}}{1+m+n}+\frac {3 a b^2 x^{1+m+2 n}}{1+m+2 n}+\frac {b^3 x^{1+m+3 n}}{1+m+3 n}
\]
[Out]
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)
Rubi [A] (verified)
Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276}
\[
\int x^m \left (a+b x^n\right )^3 \, dx=\frac {a^3 x^{m+1}}{m+1}+\frac {3 a^2 b x^{m+n+1}}{m+n+1}+\frac {3 a b^2 x^{m+2 n+1}}{m+2 n+1}+\frac {b^3 x^{m+3 n+1}}{m+3 n+1}
\]
[In]
Int[x^m*(a + b*x^n)^3,x]
[Out]
(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)
Rule 276
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Rubi steps \begin{align*}
\text {integral}& = \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 \\ & = \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} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89
\[
\int x^m \left (a+b x^n\right )^3 \, dx=x^{1+m} \left (\frac {a^3}{1+m}+\frac {3 a^2 b x^n}{1+m+n}+\frac {3 a b^2 x^{2 n}}{1+m+2 n}+\frac {b^3 x^{3 n}}{1+m+3 n}\right )
\]
[In]
Integrate[x^m*(a + b*x^n)^3,x]
[Out]
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))
Maple [A] (verified)
Time = 3.69 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.04
| | |
| method | result | size |
| | |
| risch |
\(\frac {a^{3} x \,x^{m}}{1+m}+\frac {b^{3} x \,x^{m} x^{3 n}}{1+m +3 n}+\frac {3 a \,b^{2} x \,x^{m} x^{2 n}}{1+m +2 n}+\frac {3 a^{2} b x \,x^{m} x^{n}}{1+m +n}\) |
\(78\) |
| norman |
\(\frac {a^{3} x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b^{3} x \,{\mathrm e}^{m \ln \left (x \right )} {\mathrm e}^{3 n \ln \left (x \right )}}{1+m +3 n}+\frac {3 a \,b^{2} x \,{\mathrm e}^{m \ln \left (x \right )} {\mathrm e}^{2 n \ln \left (x \right )}}{1+m +2 n}+\frac {3 a^{2} b x \,{\mathrm e}^{m \ln \left (x \right )} {\mathrm e}^{n \ln \left (x \right )}}{1+m +n}\) |
\(92\) |
| parallelrisch |
\(\frac {18 x \,x^{m} x^{n} a^{2} b m \,n^{2}+30 x \,x^{m} x^{n} a^{2} b m n +18 x \,x^{m} x^{n} a^{2} b \,n^{2}+9 x \,x^{m} x^{n} a^{2} b m +15 x \,x^{m} x^{n} a^{2} b n +15 x \,x^{m} x^{n} a^{2} b \,m^{2} n +3 a \,b^{2} x \,x^{m} x^{2 n}+b^{3} x \,x^{m} x^{3 n}+x \,x^{m} x^{3 n} b^{3} m^{3}+3 x \,x^{m} x^{3 n} b^{3} m^{2}+2 x \,x^{m} x^{3 n} b^{3} n^{2}+3 x \,x^{m} x^{3 n} b^{3} m +3 x \,x^{m} x^{3 n} b^{3} n +3 x \,x^{m} x^{n} a^{2} b \,m^{3}+9 x \,x^{m} x^{n} a^{2} b \,m^{2}+3 x \,x^{m} x^{3 n} b^{3} m^{2} n +9 x \,x^{m} x^{2 n} a \,b^{2} m^{2}+9 x \,x^{m} x^{2 n} a \,b^{2} n^{2}+9 x \,x^{m} x^{2 n} a \,b^{2} m +12 x \,x^{m} x^{2 n} a \,b^{2} n +2 x \,x^{m} x^{3 n} b^{3} m \,n^{2}+6 x \,x^{m} x^{3 n} b^{3} m n +3 x \,x^{m} x^{2 n} a \,b^{2} m^{3}+12 x \,x^{m} x^{2 n} a \,b^{2} m^{2} n +9 x \,x^{m} x^{2 n} a \,b^{2} m \,n^{2}+24 x \,x^{m} x^{2 n} a \,b^{2} m n +x \,x^{m} a^{3} m^{3}+6 x \,x^{m} a^{3} n^{3}+3 x \,x^{m} a^{3} m^{2}+11 x \,x^{m} a^{3} n^{2}+3 x \,x^{m} a^{3} m +6 x \,x^{m} a^{3} n +6 x \,x^{m} a^{3} m^{2} n +11 x \,x^{m} a^{3} m \,n^{2}+12 x \,x^{m} a^{3} m n +3 x \,x^{m} x^{n} a^{2} b +x \,x^{m} a^{3}}{\left (1+m \right ) \left (1+m +3 n \right ) \left (1+m +2 n \right ) \left (1+m +n \right )}\) |
\(581\) |
| | |
|
|
|
|
[In]
int(x^m*(a+b*x^n)^3,x,method=_RETURNVERBOSE)
[Out]
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+m+n)*x*x^m*x^n
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (75) = 150\).
Time = 0.28 (sec) , antiderivative size = 362, normalized size of antiderivative = 4.83
\[
\int x^m \left (a+b x^n\right )^3 \, dx=\frac {{\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3} + 2 \, {\left (b^{3} m + b^{3}\right )} n^{2} + 3 \, {\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n\right )} x x^{m} x^{3 \, n} + 3 \, {\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2} + 3 \, {\left (a b^{2} m + a b^{2}\right )} n^{2} + 4 \, {\left (a b^{2} m^{2} + 2 \, a b^{2} m + a b^{2}\right )} n\right )} x x^{m} x^{2 \, n} + 3 \, {\left (a^{2} b m^{3} + 3 \, a^{2} b m^{2} + 3 \, a^{2} b m + a^{2} b + 6 \, {\left (a^{2} b m + a^{2} b\right )} n^{2} + 5 \, {\left (a^{2} b m^{2} + 2 \, a^{2} b m + a^{2} b\right )} n\right )} x x^{m} x^{n} + {\left (a^{3} m^{3} + 6 \, a^{3} n^{3} + 3 \, a^{3} m^{2} + 3 \, a^{3} m + a^{3} + 11 \, {\left (a^{3} m + a^{3}\right )} n^{2} + 6 \, {\left (a^{3} m^{2} + 2 \, a^{3} m + a^{3}\right )} n\right )} x x^{m}}{m^{4} + 6 \, {\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \, {\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \, {\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1}
\]
[In]
integrate(x^m*(a+b*x^n)^3,x, algorithm="fricas")
[Out]
((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)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 3536 vs. \(2 (70) = 140\).
Time = 2.65 (sec) , antiderivative size = 3536, normalized size of antiderivative = 47.15
\[
\int x^m \left (a+b x^n\right )^3 \, dx=\text {Too large to display}
\]
[In]
integrate(x**m*(a+b*x**n)**3,x)
[Out]
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*
log(x), 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) + b**3*x**n/n, Eq(m, -2*
n - 1)), (-a**3/(n*x**n) + 3*a**2*b*log(x) + 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 + 11*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 + 11*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**3 + 11*n**2 + 6*n + 1) + 11*a**3*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) + 6*a**3*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) + a**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) + 3*a**2*b*m**3*x*x**m*x**n/(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) +
15*a**2*b*m**2*n*x*x**m*x**n/(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) + 9*a**2*b*m**2*x*x**m*x**n/(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) + 18*a**2*b*m
*n**2*x*x**m*x**n/(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) + 30*a**2*b*m*n*x*x**m*x**n/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 1
8*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) + 9*a**2*b*m*x*x**m*x**n
/(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) + 18*a**2*b*n**2*x*x**m*x**n/(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) + 15*a**2*b*n*x*x**m*x**n/(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**2*b*x*x**m*x**n/(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*b**2*m**3*x*x**m*x**(2*n)/(m**4 + 6*m**3*n + 4*m**3 + 1
1*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*b*
*2*m**2*n*x*x**m*x**(2*n)/(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) + 9*a*b**2*m**2*x*x**m*x**(2*n)/(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) + 9*a*b**2*m
*n**2*x*x**m*x**(2*n)/(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 + 1
8*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 24*a*b**2*m*n*x*x**m*x**(2*n)/(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) + 9*a*b**2*m*x*x
**m*x**(2*n)/(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) + 9*a*b**2*n**2*x*x**m*x**(2*n)/(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*b**2*n*x*x**m*x**(
2*n)/(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*b**2*x*x**m*x**(2*n)/(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) + b**3*m**3*x*x**m*x**(3*n)/(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*b**3*m**2*n*x*x**m*x**(3*n)/(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*b**3*m**2*x*x**m*x**(3*n)/(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
) + 2*b**3*m*n**2*x*x**m*x**(3*n)/(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 + 2
2*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1) + 6*b**3*m*n*x*x**m*x**(3*n)/(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*b**
3*m*x*x**m*x**(3*n)/(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) + 2*b**3*n**2*x*x**m*x**(3*n)/(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*b**3*n*x*x**m*x*
*(3*n)/(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) + b**3*x*x**m*x**(3*n)/(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), True))
Maxima [A] (verification not implemented)
none
Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00
\[
\int x^m \left (a+b x^n\right )^3 \, dx=\frac {b^{3} x^{m + 3 \, n + 1}}{m + 3 \, n + 1} + \frac {3 \, a b^{2} x^{m + 2 \, n + 1}}{m + 2 \, n + 1} + \frac {3 \, a^{2} b x^{m + n + 1}}{m + n + 1} + \frac {a^{3} x^{m + 1}}{m + 1}
\]
[In]
integrate(x^m*(a+b*x^n)^3,x, algorithm="maxima")
[Out]
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)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (75) = 150\).
Time = 0.29 (sec) , antiderivative size = 622, normalized size of antiderivative = 8.29
\[
\int x^m \left (a+b x^n\right )^3 \, dx=\frac {b^{3} m^{3} x x^{m} x^{3 \, n} + 3 \, b^{3} m^{2} n x x^{m} x^{3 \, n} + 2 \, b^{3} m n^{2} x x^{m} x^{3 \, n} + 3 \, a b^{2} m^{3} x x^{m} x^{2 \, n} + 12 \, a b^{2} m^{2} n x x^{m} x^{2 \, n} + 9 \, a b^{2} m n^{2} x x^{m} x^{2 \, n} + 3 \, a^{2} b m^{3} x x^{m} x^{n} + 15 \, a^{2} b m^{2} n x x^{m} x^{n} + 18 \, a^{2} b m n^{2} x x^{m} x^{n} + a^{3} m^{3} x x^{m} + 6 \, a^{3} m^{2} n x x^{m} + 11 \, a^{3} m n^{2} x x^{m} + 6 \, a^{3} n^{3} x x^{m} + 3 \, b^{3} m^{2} x x^{m} x^{3 \, n} + 6 \, b^{3} m n x x^{m} x^{3 \, n} + 2 \, b^{3} n^{2} x x^{m} x^{3 \, n} + 9 \, a b^{2} m^{2} x x^{m} x^{2 \, n} + 24 \, a b^{2} m n x x^{m} x^{2 \, n} + 9 \, a b^{2} n^{2} x x^{m} x^{2 \, n} + 9 \, a^{2} b m^{2} x x^{m} x^{n} + 30 \, a^{2} b m n x x^{m} x^{n} + 18 \, a^{2} b n^{2} x x^{m} x^{n} + 3 \, a^{3} m^{2} x x^{m} + 12 \, a^{3} m n x x^{m} + 11 \, a^{3} n^{2} x x^{m} + 3 \, b^{3} m x x^{m} x^{3 \, n} + 3 \, b^{3} n x x^{m} x^{3 \, n} + 9 \, a b^{2} m x x^{m} x^{2 \, n} + 12 \, a b^{2} n x x^{m} x^{2 \, n} + 9 \, a^{2} b m x x^{m} x^{n} + 15 \, a^{2} b n x x^{m} x^{n} + 3 \, a^{3} m x x^{m} + 6 \, a^{3} n x x^{m} + b^{3} x x^{m} x^{3 \, n} + 3 \, a b^{2} x x^{m} x^{2 \, n} + 3 \, a^{2} b x x^{m} x^{n} + a^{3} x x^{m}}{m^{4} + 6 \, m^{3} n + 11 \, m^{2} n^{2} + 6 \, m n^{3} + 4 \, m^{3} + 18 \, m^{2} n + 22 \, m n^{2} + 6 \, n^{3} + 6 \, m^{2} + 18 \, m n + 11 \, n^{2} + 4 \, m + 6 \, n + 1}
\]
[In]
integrate(x^m*(a+b*x^n)^3,x, algorithm="giac")
[Out]
(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) + 1
2*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)
Mupad [B] (verification not implemented)
Time = 5.98 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03
\[
\int x^m \left (a+b x^n\right )^3 \, dx=\frac {a^3\,x\,x^m}{m+1}+\frac {b^3\,x\,x^m\,x^{3\,n}}{m+3\,n+1}+\frac {3\,a^2\,b\,x\,x^m\,x^n}{m+n+1}+\frac {3\,a\,b^2\,x\,x^m\,x^{2\,n}}{m+2\,n+1}
\]
[In]
int(x^m*(a + b*x^n)^3,x)
[Out]
(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)