Integrand size = 19, antiderivative size = 69 \[ \int x^{-1+\frac {n}{3}} \left (a+b x^n\right )^3 \, dx=\frac {3 a^3 x^{n/3}}{n}+\frac {9 a^2 b x^{4 n/3}}{4 n}+\frac {9 a b^2 x^{7 n/3}}{7 n}+\frac {3 b^3 x^{10 n/3}}{10 n} \] Output:
3*a^3*x^(1/3*n)/n+9/4*a^2*b*x^(4/3*n)/n+9/7*a*b^2*x^(7/3*n)/n+3/10*b^3*x^( 10/3*n)/n
Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.72 \[ \int x^{-1+\frac {n}{3}} \left (a+b x^n\right )^3 \, dx=\frac {3 x^{n/3} \left (140 a^3+105 a^2 b x^n+60 a b^2 x^{2 n}+14 b^3 x^{3 n}\right )}{140 n} \] Input:
Integrate[x^(-1 + n/3)*(a + b*x^n)^3,x]
Output:
(3*x^(n/3)*(140*a^3 + 105*a^2*b*x^n + 60*a*b^2*x^(2*n) + 14*b^3*x^(3*n)))/ (140*n)
Time = 0.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
\(\displaystyle \int x^{\frac {n}{3}-1} \left (a+b x^n\right )^3 \, dx\) |
\(\Big \downarrow \) 802 |
\(\displaystyle \int \left (a^3 x^{\frac {n-3}{3}}+3 a^2 b x^{\frac {4 n}{3}-1}+3 a b^2 x^{\frac {7 n}{3}-1}+b^3 x^{\frac {10 n}{3}-1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 a^3 x^{n/3}}{n}+\frac {9 a^2 b x^{4 n/3}}{4 n}+\frac {9 a b^2 x^{7 n/3}}{7 n}+\frac {3 b^3 x^{10 n/3}}{10 n}\) |
Input:
Int[x^(-1 + n/3)*(a + b*x^n)^3,x]
Output:
(3*a^3*x^(n/3))/n + (9*a^2*b*x^((4*n)/3))/(4*n) + (9*a*b^2*x^((7*n)/3))/(7 *n) + (3*b^3*x^((10*n)/3))/(10*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 = 0.72 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {3 a^{3} x^{\frac {n}{3}}}{n}+\frac {9 a^{2} b \,x^{\frac {4 n}{3}}}{4 n}+\frac {9 a \,b^{2} x^{\frac {7 n}{3}}}{7 n}+\frac {3 b^{3} x^{\frac {10 n}{3}}}{10 n}\) | \(62\) |
parallelrisch | \(\frac {42 x \,x^{3 n} x^{-1+\frac {n}{3}} b^{3}+180 x \,x^{2 n} x^{-1+\frac {n}{3}} a \,b^{2}+315 x \,x^{n} x^{-1+\frac {n}{3}} a^{2} b +420 x \,x^{-1+\frac {n}{3}} a^{3}}{140 n}\) | \(74\) |
orering | \(\frac {3 x \left (418 n^{3}-477 n^{2}+198 n -27\right ) x^{-1+\frac {n}{3}} \left (a +b \,x^{n}\right )^{3}}{280 n^{4}}-\frac {27 x^{2} \left (53 n^{2}-66 n +21\right ) \left (\frac {x^{-1+\frac {n}{3}} \left (-1+\frac {n}{3}\right ) \left (a +b \,x^{n}\right )^{3}}{x}+\frac {3 x^{-1+\frac {n}{3}} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n}{x}\right )}{280 n^{4}}+\frac {27 x^{3} \left (11 n -9\right ) \left (\frac {x^{-1+\frac {n}{3}} \left (-1+\frac {n}{3}\right )^{2} \left (a +b \,x^{n}\right )^{3}}{x^{2}}-\frac {x^{-1+\frac {n}{3}} \left (-1+\frac {n}{3}\right ) \left (a +b \,x^{n}\right )^{3}}{x^{2}}+\frac {6 x^{-1+\frac {n}{3}} \left (-1+\frac {n}{3}\right ) \left (a +b \,x^{n}\right )^{2} b \,x^{n} n}{x^{2}}+\frac {6 x^{-1+\frac {n}{3}} x^{2 n} b^{2} n^{2} \left (a +b \,x^{n}\right )}{x^{2}}+\frac {3 x^{-1+\frac {n}{3}} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n^{2}}{x^{2}}-\frac {3 x^{-1+\frac {n}{3}} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n}{x^{2}}\right )}{140 n^{4}}-\frac {81 x^{4} \left (\frac {x^{-1+\frac {n}{3}} \left (-1+\frac {n}{3}\right )^{3} \left (a +b \,x^{n}\right )^{3}}{x^{3}}-\frac {3 x^{-1+\frac {n}{3}} \left (-1+\frac {n}{3}\right )^{2} \left (a +b \,x^{n}\right )^{3}}{x^{3}}+\frac {9 x^{-1+\frac {n}{3}} \left (-1+\frac {n}{3}\right )^{2} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n}{x^{3}}+\frac {2 x^{-1+\frac {n}{3}} \left (-1+\frac {n}{3}\right ) \left (a +b \,x^{n}\right )^{3}}{x^{3}}-\frac {18 x^{-1+\frac {n}{3}} \left (-1+\frac {n}{3}\right ) \left (a +b \,x^{n}\right )^{2} b \,x^{n} n}{x^{3}}+\frac {18 x^{-1+\frac {n}{3}} \left (-1+\frac {n}{3}\right ) \left (a +b \,x^{n}\right ) b^{2} x^{2 n} n^{2}}{x^{3}}+\frac {9 x^{-1+\frac {n}{3}} \left (-1+\frac {n}{3}\right ) \left (a +b \,x^{n}\right )^{2} b \,x^{n} n^{2}}{x^{3}}+\frac {18 x^{-1+\frac {n}{3}} x^{2 n} b^{2} n^{3} \left (a +b \,x^{n}\right )}{x^{3}}+\frac {6 x^{-1+\frac {n}{3}} x^{3 n} b^{3} n^{3}}{x^{3}}-\frac {18 x^{-1+\frac {n}{3}} x^{2 n} b^{2} n^{2} \left (a +b \,x^{n}\right )}{x^{3}}+\frac {3 x^{-1+\frac {n}{3}} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n^{3}}{x^{3}}-\frac {9 x^{-1+\frac {n}{3}} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n^{2}}{x^{3}}+\frac {6 x^{-1+\frac {n}{3}} \left (a +b \,x^{n}\right )^{2} b \,x^{n} n}{x^{3}}\right )}{280 n^{4}}\) | \(678\) |
Input:
int(x^(-1+1/3*n)*(a+b*x^n)^3,x,method=_RETURNVERBOSE)
Output:
3/10*b^3/n*(x^(1/3*n))^10+9/7*a*b^2/n*(x^(1/3*n))^7+9/4*a^2*b/n*(x^(1/3*n) )^4+3*a^3*x^(1/3*n)/n
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int x^{-1+\frac {n}{3}} \left (a+b x^n\right )^3 \, dx=\frac {3 \, {\left (14 \, b^{3} x^{10} x^{\frac {10}{3} \, n - 10} + 60 \, a b^{2} x^{7} x^{\frac {7}{3} \, n - 7} + 105 \, a^{2} b x^{4} x^{\frac {4}{3} \, n - 4} + 140 \, a^{3} x x^{\frac {1}{3} \, n - 1}\right )}}{140 \, n} \] Input:
integrate(x^(-1+1/3*n)*(a+b*x^n)^3,x, algorithm="fricas")
Output:
3/140*(14*b^3*x^10*x^(10/3*n - 10) + 60*a*b^2*x^7*x^(7/3*n - 7) + 105*a^2* b*x^4*x^(4/3*n - 4) + 140*a^3*x*x^(1/3*n - 1))/n
Time = 0.41 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.30 \[ \int x^{-1+\frac {n}{3}} \left (a+b x^n\right )^3 \, dx=\begin {cases} \frac {3 a^{3} x x^{\frac {n}{3} - 1}}{n} + \frac {9 a^{2} b x x^{n} x^{\frac {n}{3} - 1}}{4 n} + \frac {9 a b^{2} x x^{2 n} x^{\frac {n}{3} - 1}}{7 n} + \frac {3 b^{3} x x^{3 n} x^{\frac {n}{3} - 1}}{10 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{3} \log {\left (x \right )} & \text {otherwise} \end {cases} \] Input:
integrate(x**(-1+1/3*n)*(a+b*x**n)**3,x)
Output:
Piecewise((3*a**3*x*x**(n/3 - 1)/n + 9*a**2*b*x*x**n*x**(n/3 - 1)/(4*n) + 9*a*b**2*x*x**(2*n)*x**(n/3 - 1)/(7*n) + 3*b**3*x*x**(3*n)*x**(n/3 - 1)/(1 0*n), Ne(n, 0)), ((a + b)**3*log(x), True))
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80 \[ \int x^{-1+\frac {n}{3}} \left (a+b x^n\right )^3 \, dx=\frac {3 \, b^{3} x^{\frac {10}{3} \, n}}{10 \, n} + \frac {9 \, a b^{2} x^{\frac {7}{3} \, n}}{7 \, n} + \frac {9 \, a^{2} b x^{\frac {4}{3} \, n}}{4 \, n} + \frac {3 \, a^{3} x^{\frac {1}{3} \, n}}{n} \] Input:
integrate(x^(-1+1/3*n)*(a+b*x^n)^3,x, algorithm="maxima")
Output:
3/10*b^3*x^(10/3*n)/n + 9/7*a*b^2*x^(7/3*n)/n + 9/4*a^2*b*x^(4/3*n)/n + 3* a^3*x^(1/3*n)/n
Time = 0.14 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int x^{-1+\frac {n}{3}} \left (a+b x^n\right )^3 \, dx=\frac {3 \, {\left (14 \, b^{3} x^{3 \, n} {\left (x^{n}\right )}^{\frac {1}{3}} + 60 \, a b^{2} x^{2 \, n} {\left (x^{n}\right )}^{\frac {1}{3}} + 105 \, a^{2} b {\left (x^{n}\right )}^{\frac {4}{3}} + 140 \, a^{3} x^{\frac {1}{3} \, n}\right )}}{140 \, n} \] Input:
integrate(x^(-1+1/3*n)*(a+b*x^n)^3,x, algorithm="giac")
Output:
3/140*(14*b^3*x^(3*n)*(x^n)^(1/3) + 60*a*b^2*x^(2*n)*(x^n)^(1/3) + 105*a^2 *b*(x^n)^(4/3) + 140*a^3*x^(1/3*n))/n
Time = 0.66 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int x^{-1+\frac {n}{3}} \left (a+b x^n\right )^3 \, dx=x^{\frac {n}{3}-1}\,\left (\frac {3\,a^3\,x}{n}+\frac {3\,b^3\,x\,x^{3\,n}}{10\,n}+\frac {9\,a^2\,b\,x\,x^n}{4\,n}+\frac {9\,a\,b^2\,x\,x^{2\,n}}{7\,n}\right ) \] Input:
int(x^(n/3 - 1)*(a + b*x^n)^3,x)
Output:
x^(n/3 - 1)*((3*a^3*x)/n + (3*b^3*x*x^(3*n))/(10*n) + (9*a^2*b*x*x^n)/(4*n ) + (9*a*b^2*x*x^(2*n))/(7*n))
Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.67 \[ \int x^{-1+\frac {n}{3}} \left (a+b x^n\right )^3 \, dx=\frac {3 x^{\frac {n}{3}} \left (14 x^{3 n} b^{3}+60 x^{2 n} a \,b^{2}+105 x^{n} a^{2} b +140 a^{3}\right )}{140 n} \] Input:
int(x^(-1+1/3*n)*(a+b*x^n)^3,x)
Output:
(3*x**(n/3)*(14*x**(3*n)*b**3 + 60*x**(2*n)*a*b**2 + 105*x**n*a**2*b + 140 *a**3))/(140*n)