Optimal. Leaf size=24 \[ \frac {x^{2 n}}{2 a n \left (a+b x^n\right )^2} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {264} \[ \frac {x^{2 n}}{2 a n \left (a+b x^n\right )^2} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3} \, dx &=\frac {x^{2 n}}{2 a n \left (a+b x^n\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ \frac {x^{2 n}}{2 a n \left (a+b x^n\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 41, normalized size = 1.71 \[ -\frac {2 \, b x^{n} + a}{2 \, {\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 36, normalized size = 1.50 \[ \frac {-\frac {{\mathrm e}^{n \ln \relax (x )}}{b n}-\frac {a}{2 b^{2} n}}{\left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 41, normalized size = 1.71 \[ -\frac {2 \, b x^{n} + a}{2 \, {\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.38, size = 37, normalized size = 1.54 \[ \frac {x^{2\,n}}{2\,\left (a^3\,n+2\,a^2\,b\,n\,x^n+a\,b^2\,n\,x^{2\,n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 158.69, size = 61, normalized size = 2.54 \[ \begin {cases} \frac {\log {\relax (x )}}{b^{3}} & \text {for}\: a = 0 \wedge n = 0 \\- \frac {x^{- n}}{b^{3} n} & \text {for}\: a = 0 \\\frac {\log {\relax (x )}}{\left (a + b\right )^{3}} & \text {for}\: n = 0 \\\frac {x^{2 n}}{2 a^{3} n + 4 a^{2} b n x^{n} + 2 a b^{2} n x^{2 n}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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