Optimal. Leaf size=43 \[ \frac {x \left (a+b x^n\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac {n+1}{2 n}}}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1343, 191} \[ \frac {x \left (a+b x^n\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac {n+1}{2 n}}}{a} \]
Antiderivative was successfully verified.
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Rule 191
Rule 1343
Rubi steps
\begin {align*} \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac {1+n}{2 n}} \, dx &=\left (\left (2 a b+2 b^2 x^n\right )^{\frac {1+n}{n}} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac {1+n}{2 n}}\right ) \int \left (2 a b+2 b^2 x^n\right )^{-\frac {1+n}{n}} \, dx\\ &=\frac {x \left (a+b x^n\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac {1+n}{2 n}}}{a}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 32, normalized size = 0.74 \[ \frac {x \left (a+b x^n\right ) \left (\left (a+b x^n\right )^2\right )^{-\frac {n+1}{2 n}}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 45, normalized size = 1.05 \[ \frac {b x x^{n} + a x}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {n + 1}{2 \, n}} a} \]
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 \[ \int \frac {1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {n + 1}{2 \, n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 51, normalized size = 1.19 \[ \left (\frac {b x \,{\mathrm e}^{n \ln \relax (x )}}{a}+x \right ) {\mathrm e}^{-\frac {\left (n +1\right ) \ln \left (2 a b \,{\mathrm e}^{n \ln \relax (x )}+b^{2} {\mathrm e}^{2 n \ln \relax (x )}+a^{2}\right )}{2 n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {n + 1}{2 \, n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{\frac {\frac {n}{2}+\frac {1}{2}}{n}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{- \frac {\frac {n}{2} + \frac {1}{2}}{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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