Optimal. Leaf size=54 \[ \frac {(1-a x)^{-\frac {1}{2} n (n+1)} (a x+1)^{\frac {1}{2} (1-n) n} (1-a n x)}{a^3 n \left (1-n^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, 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.
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Rule 81
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*}
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Mathematica [A] time = 0.07, 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.
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fricas [A] time = 1.21, size = 74, normalized size = 1.37 \[ -\frac {{\left (a^{3} n x^{3} - a^{2} x^{2} - a n x + 1\right )} {\left (a x + 1\right )}^{-\frac {1}{2} \, n^{2} + \frac {1}{2} \, n - 1} {\left (-a x + 1\right )}^{-\frac {1}{2} \, n^{2} - \frac {1}{2} \, n - 1}}{a^{3} n^{3} - a^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.38, size = 284, normalized size = 5.26 \[ -\frac {a^{3} n x^{3} e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + 1\right ) - \frac {1}{2} \, n^{2} \log \left (-a x + 1\right ) + \frac {1}{2} \, n \log \left (a x + 1\right ) - \frac {1}{2} \, n \log \left (-a x + 1\right ) - \log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )} - a^{2} x^{2} e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + 1\right ) - \frac {1}{2} \, n^{2} \log \left (-a x + 1\right ) + \frac {1}{2} \, n \log \left (a x + 1\right ) - \frac {1}{2} \, n \log \left (-a x + 1\right ) - \log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )} - a n x e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + 1\right ) - \frac {1}{2} \, n^{2} \log \left (-a x + 1\right ) + \frac {1}{2} \, n \log \left (a x + 1\right ) - \frac {1}{2} \, n \log \left (-a x + 1\right ) - \log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )} + e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + 1\right ) - \frac {1}{2} \, n^{2} \log \left (-a x + 1\right ) + \frac {1}{2} \, n \log \left (a x + 1\right ) - \frac {1}{2} \, n \log \left (-a x + 1\right ) - \log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )}}{a^{3} n^{3} - a^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 52, normalized size = 0.96 \[ \frac {\left (n a x -1\right ) \left (-a x +1\right )^{-\frac {1}{2} n^{2}-\frac {1}{2} n} \left (a x +1\right )^{-\frac {1}{2} n^{2}+\frac {1}{2} n}}{\left (n^{2}-1\right ) a^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 63, normalized size = 1.17 \[ \frac {{\left (a n x - 1\right )} e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + 1\right ) - \frac {1}{2} \, n^{2} \log \left (-a x + 1\right ) + \frac {1}{2} \, n \log \left (a x + 1\right ) - \frac {1}{2} \, n \log \left (-a x + 1\right )\right )}}{{\left (n^{3} - n\right )} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 140, normalized size = 2.59 \[ -\frac {\frac {x^3}{\left (n^2-1\right )\,{\left (a\,x+1\right )}^{\frac {n\,\left (n-1\right )}{2}+1}}-\frac {x}{a^2\,\left (n^2-1\right )\,{\left (a\,x+1\right )}^{\frac {n\,\left (n-1\right )}{2}+1}}+\frac {1}{a^3\,n\,\left (n^2-1\right )\,{\left (a\,x+1\right )}^{\frac {n\,\left (n-1\right )}{2}+1}}-\frac {x^2}{a\,n\,\left (n^2-1\right )\,{\left (a\,x+1\right )}^{\frac {n\,\left (n-1\right )}{2}+1}}}{{\left (1-a\,x\right )}^{\frac {n\,\left (n+1\right )}{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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