\(\int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx\) [1005]

Optimal result

Integrand size = 37, antiderivative size = 54 \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=\frac {(1-a x)^{-\frac {1}{2} n (1+n)} (1+a x)^{\frac {1}{2} (1-n) n} (1-a n x)}{a^3 n \left (1-n^2\right )} \]

[Out]

(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)))

Rubi [A] (verified)

Time = 0.01 (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 = {82} \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=\frac {(1-a x)^{-\frac {1}{2} n (n+1)} (a x+1)^{\frac {1}{2} (1-n) n} (1-a n x)}{a^3 n \left (1-n^2\right )} \]

[In]

Int[x^2*(1 - a*x)^(-1 - (n*(1 + n))/2)*(1 + a*x)^(-1 - ((-1 + n)*n)/2),x]

[Out]

((1 + a*x)^(((1 - n)*n)/2)*(1 - a*n*x))/(a^3*n*(1 - n^2)*(1 - a*x)^((n*(1 + n))/2))

Rule 82

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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]

Rubi steps \begin{align*} \text {integral}& = \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 )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=\frac {(1-a x)^{-\frac {1}{2} n (1+n)} (1+a x)^{-\frac {1}{2} (-1+n) n} (-1+a n x)}{a^3 n \left (-1+n^2\right )} \]

[In]

Integrate[x^2*(1 - a*x)^(-1 - (n*(1 + n))/2)*(1 + a*x)^(-1 - ((-1 + n)*n)/2),x]

[Out]

(-1 + a*n*x)/(a^3*n*(-1 + n^2)*(1 - a*x)^((n*(1 + n))/2)*(1 + a*x)^(((-1 + n)*n)/2))

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.96

method	result	size
gosper	\(\frac {\left (a x +1\right )^{-\frac {1}{2} n^{2}+\frac {1}{2} n} \left (a n x -1\right ) \left (-a x +1\right )^{-\frac {1}{2} n^{2}-\frac {1}{2} n}}{a^{3} n \left (n^{2}-1\right )}\)	\(52\)
risch	\(-\frac {\left (-a x +1\right )^{-1-\frac {1}{2} n^{2}-\frac {1}{2} n} \left (a^{3} x^{3} n -a^{2} x^{2}-a n x +1\right ) \left (a x +1\right )^{-1-\frac {1}{2} n^{2}+\frac {1}{2} n}}{n \left (n^{2}-1\right ) a^{3}}\)	\(72\)
parallelrisch	\(-\frac {\left (a x +1\right )^{-1-\frac {1}{2} n^{2}+\frac {1}{2} n} \left (-a x +1\right )^{-1-\frac {1}{2} n^{2}-\frac {1}{2} n} x^{3} a^{3} n -x^{2} \left (-a x +1\right )^{-1-\frac {1}{2} n^{2}-\frac {1}{2} n} \left (a x +1\right )^{-1-\frac {1}{2} n^{2}+\frac {1}{2} n} a^{2}-\left (a x +1\right )^{-1-\frac {1}{2} n^{2}+\frac {1}{2} n} x \left (-a x +1\right )^{-1-\frac {1}{2} n^{2}-\frac {1}{2} n} a n +\left (-a x +1\right )^{-1-\frac {1}{2} n^{2}-\frac {1}{2} n} \left (a x +1\right )^{-1-\frac {1}{2} n^{2}+\frac {1}{2} n}}{a^{3} n \left (n^{2}-1\right )}\)	\(171\)

[In]

int(x^2*(-a*x+1)^(-1-1/2*n*(1+n))*(a*x+1)^(-1-1/2*(-1+n)*n),x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.37 \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=-\frac {{\left (a^{3} n x^{3} - a^{2} x^{2} - a n x + 1\right )} {\left (a x + 1\right )}^{-\frac {1}{2} \, n^{2} + \frac {1}{2} \, n - 1} {\left (-a x + 1\right )}^{-\frac {1}{2} \, n^{2} - \frac {1}{2} \, n - 1}}{a^{3} n^{3} - a^{3} n} \]

[In]

integrate(x^2*(-a*x+1)^(-1-1/2*n*(1+n))*(a*x+1)^(-1-1/2*(-1+n)*n),x, algorithm="fricas")

[Out]

-(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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (41) = 82\).

Time = 65.02 (sec) , antiderivative size = 425, normalized size of antiderivative = 7.87 \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=\begin {cases} \frac {x^{3}}{3} & \text {for}\: a = 0 \\- \frac {a x \log {\left (x - \frac {1}{a} \right )}}{4 a^{4} x + 4 a^{3}} - \frac {3 a x \log {\left (x + \frac {1}{a} \right )}}{4 a^{4} x + 4 a^{3}} - \frac {\log {\left (x - \frac {1}{a} \right )}}{4 a^{4} x + 4 a^{3}} - \frac {3 \log {\left (x + \frac {1}{a} \right )}}{4 a^{4} x + 4 a^{3}} - \frac {2}{4 a^{4} x + 4 a^{3}} & \text {for}\: n = -1 \\- \frac {x}{a^{2}} - \frac {\log {\left (x - \frac {1}{a} \right )}}{2 a^{3}} + \frac {\log {\left (x + \frac {1}{a} \right )}}{2 a^{3}} & \text {for}\: n = 0 \\\frac {3 a x \log {\left (x - \frac {1}{a} \right )}}{4 a^{4} x - 4 a^{3}} + \frac {a x \log {\left (x + \frac {1}{a} \right )}}{4 a^{4} x - 4 a^{3}} - \frac {3 \log {\left (x - \frac {1}{a} \right )}}{4 a^{4} x - 4 a^{3}} - \frac {\log {\left (x + \frac {1}{a} \right )}}{4 a^{4} x - 4 a^{3}} - \frac {2}{4 a^{4} x - 4 a^{3}} & \text {for}\: n = 1 \\- \frac {a^{3} n x^{3} \left (- a x + 1\right )^{- \frac {n^{2}}{2} - \frac {n}{2} - 1} \left (a x + 1\right )^{- \frac {n^{2}}{2} + \frac {n}{2} - 1}}{a^{3} n^{3} - a^{3} n} + \frac {a^{2} x^{2} \left (- a x + 1\right )^{- \frac {n^{2}}{2} - \frac {n}{2} - 1} \left (a x + 1\right )^{- \frac {n^{2}}{2} + \frac {n}{2} - 1}}{a^{3} n^{3} - a^{3} n} + \frac {a n x \left (- a x + 1\right )^{- \frac {n^{2}}{2} - \frac {n}{2} - 1} \left (a x + 1\right )^{- \frac {n^{2}}{2} + \frac {n}{2} - 1}}{a^{3} n^{3} - a^{3} n} - \frac {\left (- a x + 1\right )^{- \frac {n^{2}}{2} - \frac {n}{2} - 1} \left (a x + 1\right )^{- \frac {n^{2}}{2} + \frac {n}{2} - 1}}{a^{3} n^{3} - a^{3} n} & \text {otherwise} \end {cases} \]

[In]

integrate(x**2*(-a*x+1)**(-1-1/2*n*(1+n))*(a*x+1)**(-1-1/2*(-1+n)*n),x)

[Out]

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*log(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))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=\frac {{\left (a n x - 1\right )} e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + 1\right ) - \frac {1}{2} \, n^{2} \log \left (-a x + 1\right ) + \frac {1}{2} \, n \log \left (a x + 1\right ) - \frac {1}{2} \, n \log \left (-a x + 1\right )\right )}}{{\left (n^{3} - n\right )} a^{3}} \]

[In]

integrate(x^2*(-a*x+1)^(-1-1/2*n*(1+n))*(a*x+1)^(-1-1/2*(-1+n)*n),x, algorithm="maxima")

[Out]

(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)

Giac [F]

\[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=\int { {\left (a x + 1\right )}^{-\frac {1}{2} \, {\left (n - 1\right )} n - 1} {\left (-a x + 1\right )}^{-\frac {1}{2} \, {\left (n + 1\right )} n - 1} x^{2} \,d x } \]

[In]

integrate(x^2*(-a*x+1)^(-1-1/2*n*(1+n))*(a*x+1)^(-1-1/2*(-1+n)*n),x, algorithm="giac")

[Out]

integrate((a*x + 1)^(-1/2*(n - 1)*n - 1)*(-a*x + 1)^(-1/2*(n + 1)*n - 1)*x^2, x)

Mupad [B] (verification not implemented)

Time = 1.63 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.59 \[ \int x^2 (1-a x)^{-1-\frac {1}{2} n (1+n)} (1+a x)^{-1-\frac {1}{2} (-1+n) n} \, dx=-\frac {\frac {x^3}{\left (n^2-1\right )\,{\left (a\,x+1\right )}^{\frac {n\,\left (n-1\right )}{2}+1}}-\frac {x}{a^2\,\left (n^2-1\right )\,{\left (a\,x+1\right )}^{\frac {n\,\left (n-1\right )}{2}+1}}+\frac {1}{a^3\,n\,\left (n^2-1\right )\,{\left (a\,x+1\right )}^{\frac {n\,\left (n-1\right )}{2}+1}}-\frac {x^2}{a\,n\,\left (n^2-1\right )\,{\left (a\,x+1\right )}^{\frac {n\,\left (n-1\right )}{2}+1}}}{{\left (1-a\,x\right )}^{\frac {n\,\left (n+1\right )}{2}+1}} \]

[In]

int(x^2/((1 - a*x)^((n*(n + 1))/2 + 1)*(a*x + 1)^((n*(n - 1))/2 + 1)),x)

[Out]

-(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)