Integrand size = 37, antiderivative size = 54 \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=\frac {(1-a x)^{-\frac {1}{2} n (1+n)} (1+a x)^{\frac {1}{2} (1-n) n} (1-a n x)}{a^3 n \left (1-n^2\right )} \] Output:
(a*x+1)^(1/2*(1-n)*n)*(-a*n*x+1)/a^3/n/(-n^2+1)/((-a*x+1)^(1/2*n*(1+n)))
Time = 0.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=\frac {(1-a x)^{-\frac {1}{2} n (1+n)} (1+a x)^{-\frac {1}{2} (-1+n) n} (-1+a n x)}{a^3 n \left (-1+n^2\right )} \] Input:
Integrate[x^2*(1 - a*x)^(-1 - (n*(1 + n))/2)*(1 + a*x)^(-1 - ((-1 + n)*n)/ 2),x]
Output:
(-1 + a*n*x)/(a^3*n*(-1 + n^2)*(1 - a*x)^((n*(1 + n))/2)*(1 + a*x)^(((-1 + n)*n)/2))
Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {91}
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.
\(\displaystyle \int x^2 (1-a x)^{-\frac {1}{2} n (n+1)-1} (a x+1)^{-\frac {1}{2} (n-1) n-1} \, dx\) |
\(\Big \downarrow \) 91 |
\(\displaystyle \frac {(1-a x)^{-\frac {1}{2} n (n+1)} (a x+1)^{\frac {1}{2} (1-n) n} (1-a n x)}{a^3 n \left (1-n^2\right )}\) |
Input:
Int[x^2*(1 - a*x)^(-1 - (n*(1 + n))/2)*(1 + a*x)^(-1 - ((-1 + n)*n)/2),x]
Output:
((1 + a*x)^(((1 - n)*n)/2)*(1 - a*n*x))/(a^3*n*(1 - n^2)*(1 - a*x)^((n*(1 + n))/2))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)*((2*a*d*f*(n + p + 3 ) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x)/(d^2*f^2*(n + p + 2)*(n + p + 3))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2 , 0] && NeQ[n + p + 3, 0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*( b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1) + c*f*(p + 1))*(a* d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]
Time = 0.51 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.96
method | result | size |
gosper | \(\frac {\left (a x +1\right )^{-\frac {1}{2} n^{2}+\frac {1}{2} n} \left (n x a -1\right ) \left (-a x +1\right )^{-\frac {1}{2} n^{2}-\frac {1}{2} n}}{a^{3} n \left (n^{2}-1\right )}\) | \(52\) |
orering | \(-\frac {\left (n x a -1\right ) \left (a x -1\right ) \left (a x +1\right ) \left (-a x +1\right )^{-1-\frac {n \left (1+n \right )}{2}} \left (a x +1\right )^{-1-\frac {\left (-1+n \right ) n}{2}}}{a^{3} n \left (n^{2}-1\right )}\) | \(61\) |
risch | \(-\frac {\left (-a x +1\right )^{-1-\frac {1}{2} n^{2}-\frac {1}{2} n} \left (a^{3} x^{3} n -a^{2} x^{2}-n x a +1\right ) \left (a x +1\right )^{-1-\frac {1}{2} n^{2}+\frac {1}{2} n}}{n \left (n^{2}-1\right ) a^{3}}\) | \(72\) |
parallelrisch | \(-\frac {\left (a x +1\right )^{-1-\frac {1}{2} n^{2}+\frac {1}{2} n} \left (-a x +1\right )^{-1-\frac {1}{2} n^{2}-\frac {1}{2} n} x^{3} a^{3} n -x^{2} \left (-a x +1\right )^{-1-\frac {1}{2} n^{2}-\frac {1}{2} n} \left (a x +1\right )^{-1-\frac {1}{2} n^{2}+\frac {1}{2} n} a^{2}-\left (a x +1\right )^{-1-\frac {1}{2} n^{2}+\frac {1}{2} n} x \left (-a x +1\right )^{-1-\frac {1}{2} n^{2}-\frac {1}{2} n} a n +\left (-a x +1\right )^{-1-\frac {1}{2} n^{2}-\frac {1}{2} n} \left (a x +1\right )^{-1-\frac {1}{2} n^{2}+\frac {1}{2} n}}{a^{3} n \left (n^{2}-1\right )}\) | \(171\) |
Input:
int(x^2*(-a*x+1)^(-1-1/2*n*(1+n))*(a*x+1)^(-1-1/2*(-1+n)*n),x,method=_RETU RNVERBOSE)
Output:
1/a^3/n/(n^2-1)*(a*x+1)^(-1/2*n^2+1/2*n)*(a*n*x-1)*(-a*x+1)^(-1/2*n^2-1/2* n)
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.37 \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=-\frac {{\left (a^{3} n x^{3} - a^{2} x^{2} - a n x + 1\right )} {\left (a x + 1\right )}^{-\frac {1}{2} \, n^{2} + \frac {1}{2} \, n - 1} {\left (-a x + 1\right )}^{-\frac {1}{2} \, n^{2} - \frac {1}{2} \, n - 1}}{a^{3} n^{3} - a^{3} n} \] Input:
integrate(x^2*(-a*x+1)^(-1-1/2*n*(1+n))*(a*x+1)^(-1-1/2*(-1+n)*n),x, algor ithm="fricas")
Output:
-(a^3*n*x^3 - a^2*x^2 - a*n*x + 1)*(a*x + 1)^(-1/2*n^2 + 1/2*n - 1)*(-a*x + 1)^(-1/2*n^2 - 1/2*n - 1)/(a^3*n^3 - a^3*n)
Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (41) = 82\).
Time = 71.21 (sec) , antiderivative size = 425, normalized size of antiderivative = 7.87 \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=\begin {cases} \frac {x^{3}}{3} & \text {for}\: a = 0 \\- \frac {a x \log {\left (x - \frac {1}{a} \right )}}{4 a^{4} x + 4 a^{3}} - \frac {3 a x \log {\left (x + \frac {1}{a} \right )}}{4 a^{4} x + 4 a^{3}} - \frac {\log {\left (x - \frac {1}{a} \right )}}{4 a^{4} x + 4 a^{3}} - \frac {3 \log {\left (x + \frac {1}{a} \right )}}{4 a^{4} x + 4 a^{3}} - \frac {2}{4 a^{4} x + 4 a^{3}} & \text {for}\: n = -1 \\- \frac {x}{a^{2}} - \frac {\log {\left (x - \frac {1}{a} \right )}}{2 a^{3}} + \frac {\log {\left (x + \frac {1}{a} \right )}}{2 a^{3}} & \text {for}\: n = 0 \\\frac {3 a x \log {\left (x - \frac {1}{a} \right )}}{4 a^{4} x - 4 a^{3}} + \frac {a x \log {\left (x + \frac {1}{a} \right )}}{4 a^{4} x - 4 a^{3}} - \frac {3 \log {\left (x - \frac {1}{a} \right )}}{4 a^{4} x - 4 a^{3}} - \frac {\log {\left (x + \frac {1}{a} \right )}}{4 a^{4} x - 4 a^{3}} - \frac {2}{4 a^{4} x - 4 a^{3}} & \text {for}\: n = 1 \\- \frac {a^{3} n x^{3} \left (- a x + 1\right )^{- \frac {n^{2}}{2} - \frac {n}{2} - 1} \left (a x + 1\right )^{- \frac {n^{2}}{2} + \frac {n}{2} - 1}}{a^{3} n^{3} - a^{3} n} + \frac {a^{2} x^{2} \left (- a x + 1\right )^{- \frac {n^{2}}{2} - \frac {n}{2} - 1} \left (a x + 1\right )^{- \frac {n^{2}}{2} + \frac {n}{2} - 1}}{a^{3} n^{3} - a^{3} n} + \frac {a n x \left (- a x + 1\right )^{- \frac {n^{2}}{2} - \frac {n}{2} - 1} \left (a x + 1\right )^{- \frac {n^{2}}{2} + \frac {n}{2} - 1}}{a^{3} n^{3} - a^{3} n} - \frac {\left (- a x + 1\right )^{- \frac {n^{2}}{2} - \frac {n}{2} - 1} \left (a x + 1\right )^{- \frac {n^{2}}{2} + \frac {n}{2} - 1}}{a^{3} n^{3} - a^{3} n} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(-a*x+1)**(-1-1/2*n*(1+n))*(a*x+1)**(-1-1/2*(-1+n)*n),x)
Output:
Piecewise((x**3/3, Eq(a, 0)), (-a*x*log(x - 1/a)/(4*a**4*x + 4*a**3) - 3*a *x*log(x + 1/a)/(4*a**4*x + 4*a**3) - log(x - 1/a)/(4*a**4*x + 4*a**3) - 3 *log(x + 1/a)/(4*a**4*x + 4*a**3) - 2/(4*a**4*x + 4*a**3), Eq(n, -1)), (-x /a**2 - log(x - 1/a)/(2*a**3) + log(x + 1/a)/(2*a**3), Eq(n, 0)), (3*a*x*l og(x - 1/a)/(4*a**4*x - 4*a**3) + a*x*log(x + 1/a)/(4*a**4*x - 4*a**3) - 3 *log(x - 1/a)/(4*a**4*x - 4*a**3) - log(x + 1/a)/(4*a**4*x - 4*a**3) - 2/( 4*a**4*x - 4*a**3), Eq(n, 1)), (-a**3*n*x**3*(-a*x + 1)**(-n**2/2 - n/2 - 1)*(a*x + 1)**(-n**2/2 + n/2 - 1)/(a**3*n**3 - a**3*n) + a**2*x**2*(-a*x + 1)**(-n**2/2 - n/2 - 1)*(a*x + 1)**(-n**2/2 + n/2 - 1)/(a**3*n**3 - a**3* n) + a*n*x*(-a*x + 1)**(-n**2/2 - n/2 - 1)*(a*x + 1)**(-n**2/2 + n/2 - 1)/ (a**3*n**3 - a**3*n) - (-a*x + 1)**(-n**2/2 - n/2 - 1)*(a*x + 1)**(-n**2/2 + n/2 - 1)/(a**3*n**3 - a**3*n), True))
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=\frac {{\left (a n x - 1\right )} e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + 1\right ) - \frac {1}{2} \, n^{2} \log \left (-a x + 1\right ) + \frac {1}{2} \, n \log \left (a x + 1\right ) - \frac {1}{2} \, n \log \left (-a x + 1\right )\right )}}{{\left (n^{3} - n\right )} a^{3}} \] Input:
integrate(x^2*(-a*x+1)^(-1-1/2*n*(1+n))*(a*x+1)^(-1-1/2*(-1+n)*n),x, algor ithm="maxima")
Output:
(a*n*x - 1)*e^(-1/2*n^2*log(a*x + 1) - 1/2*n^2*log(-a*x + 1) + 1/2*n*log(a *x + 1) - 1/2*n*log(-a*x + 1))/((n^3 - n)*a^3)
\[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=\int { {\left (a x + 1\right )}^{-\frac {1}{2} \, {\left (n - 1\right )} n - 1} {\left (-a x + 1\right )}^{-\frac {1}{2} \, {\left (n + 1\right )} n - 1} x^{2} \,d x } \] Input:
integrate(x^2*(-a*x+1)^(-1-1/2*n*(1+n))*(a*x+1)^(-1-1/2*(-1+n)*n),x, algor ithm="giac")
Output:
integrate((a*x + 1)^(-1/2*(n - 1)*n - 1)*(-a*x + 1)^(-1/2*(n + 1)*n - 1)*x ^2, x)
Time = 0.66 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.59 \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=-\frac {\frac {x^3}{\left (n^2-1\right )\,{\left (a\,x+1\right )}^{\frac {n\,\left (n-1\right )}{2}+1}}-\frac {x}{a^2\,\left (n^2-1\right )\,{\left (a\,x+1\right )}^{\frac {n\,\left (n-1\right )}{2}+1}}+\frac {1}{a^3\,n\,\left (n^2-1\right )\,{\left (a\,x+1\right )}^{\frac {n\,\left (n-1\right )}{2}+1}}-\frac {x^2}{a\,n\,\left (n^2-1\right )\,{\left (a\,x+1\right )}^{\frac {n\,\left (n-1\right )}{2}+1}}}{{\left (1-a\,x\right )}^{\frac {n\,\left (n+1\right )}{2}+1}} \] Input:
int(x^2/((1 - a*x)^((n*(n + 1))/2 + 1)*(a*x + 1)^((n*(n - 1))/2 + 1)),x)
Output:
-(x^3/((n^2 - 1)*(a*x + 1)^((n*(n - 1))/2 + 1)) - x/(a^2*(n^2 - 1)*(a*x + 1)^((n*(n - 1))/2 + 1)) + 1/(a^3*n*(n^2 - 1)*(a*x + 1)^((n*(n - 1))/2 + 1) ) - x^2/(a*n*(n^2 - 1)*(a*x + 1)^((n*(n - 1))/2 + 1)))/(1 - a*x)^((n*(n + 1))/2 + 1)
\[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=-\left (\int \frac {\left (a x +1\right )^{\frac {n}{2}} x^{2}}{\left (a x +1\right )^{\frac {n^{2}}{2}} \left (-a x +1\right )^{\frac {1}{2} n^{2}+\frac {1}{2} n} a^{2} x^{2}-\left (a x +1\right )^{\frac {n^{2}}{2}} \left (-a x +1\right )^{\frac {1}{2} n^{2}+\frac {1}{2} n}}d x \right ) \] Input:
int(x^2*(-a*x+1)^(-1-1/2*n*(1+n))*(a*x+1)^(-1-1/2*(-1+n)*n),x)
Output:
- int(((a*x + 1)**(n/2)*x**2)/((a*x + 1)**(n**2/2)*( - a*x + 1)**((n**2 + n)/2)*a**2*x**2 - (a*x + 1)**(n**2/2)*( - a*x + 1)**((n**2 + n)/2)),x)