3.1005 \(\int x^2 (1-a x)^{-1-\frac{1}{2} n (1+n)} (1+a x)^{-1-\frac{1}{2} (-1+n) n} \, dx\)

Optimal. Leaf size=54 \[ \frac{(1-a x)^{-\frac{1}{2} n (n+1)} (a x+1)^{\frac{1}{2} (1-n) n} (1-a n x)}{a^3 n \left (1-n^2\right )} \]

[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))

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Rubi [A]  time = 0.0157026, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027, Rules used = {81} \[ \frac{(1-a x)^{-\frac{1}{2} n (n+1)} (a x+1)^{\frac{1}{2} (1-n) n} (1-a n x)}{a^3 n \left (1-n^2\right )} \]

Antiderivative was successfully verified.

[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 81

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*} \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 )}\\ \end{align*}

Mathematica [A]  time = 0.0688265, size = 49, normalized size = 0.91 \[ \frac{(1-a x)^{-\frac{1}{2} n (n+1)} (a x+1)^{-\frac{1}{2} (n-1) n} (a n x-1)}{a^3 n \left (n^2-1\right )} \]

Antiderivative was successfully verified.

[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))

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Maple [A]  time = 0.005, size = 52, normalized size = 1. \begin{align*}{\frac{anx-1}{{a}^{3}n \left ({n}^{2}-1 \right ) } \left ( ax+1 \right ) ^{-{\frac{{n}^{2}}{2}}+{\frac{n}{2}}} \left ( -ax+1 \right ) ^{-{\frac{{n}^{2}}{2}}-{\frac{n}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.41582, size = 85, normalized size = 1.57 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.15941, size = 163, normalized size = 3.02 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.73961, size = 383, normalized size = 7.09 \begin{align*} -\frac{a^{3} n x^{3} 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 ) - \log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )} - a^{2} x^{2} 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 ) - \log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )} - a n x 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 ) - \log \left (a x + 1\right ) - \log \left (-a x + 1\right )\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 ) - \log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )}}{a^{3} n^{3} - a^{3} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

-(a^3*n*x^3*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) - log(
a*x + 1) - log(-a*x + 1)) - a^2*x^2*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) - log(a*x + 1) - log(-a*x + 1)) - a*n*x*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) - log(a*x + 1) - log(-a*x + 1)) + e^(-1/2*n^2*log(a*x + 1) - 1/2*n^2*l
og(-a*x + 1) + 1/2*n*log(a*x + 1) - 1/2*n*log(-a*x + 1) - log(a*x + 1) - log(-a*x + 1)))/(a^3*n^3 - a^3*n)