Optimal. Leaf size=48 \[ \frac{x^{2 n}}{2 a n \left (a+b x^n\right ) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A] time = 0.0268692, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1355, 264} \[ \frac{x^{2 n}}{2 a n \left (a+b x^n\right ) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 264
Rubi steps
\begin{align*} \int \frac{x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^n\right )\right ) \int \frac{x^{-1+2 n}}{\left (a b+b^2 x^n\right )^3} \, dx}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac{x^{2 n}}{2 a n \left (a+b x^n\right ) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end{align*}
Mathematica [A] time = 0.0165093, size = 35, normalized size = 0.73 \[ \frac{x^{2 n} \left (a+b x^n\right )}{2 a n \left (\left (a+b x^n\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 37, normalized size = 0.8 \begin{align*} -{\frac{2\,b{x}^{n}+a}{2\, \left ( a+b{x}^{n} \right ) ^{3}{b}^{2}n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990497, size = 55, normalized size = 1.15 \begin{align*} -\frac{2 \, b x^{n} + a}{2 \,{\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55798, size = 86, normalized size = 1.79 \begin{align*} -\frac{2 \, b x^{n} + a}{2 \,{\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2 \, n - 1}}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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