Optimal. Leaf size=18 \[ \frac{x \left (a+b x^n\right )^{-1/n}}{a} \]
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Rubi [A] time = 0.0029852, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {191} \[ \frac{x \left (a+b x^n\right )^{-1/n}}{a} \]
Antiderivative was successfully verified.
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Rule 191
Rubi steps
\begin{align*} \int \left (a+b x^n\right )^{-\frac{1+n}{n}} \, dx &=\frac{x \left (a+b x^n\right )^{-1/n}}{a}\\ \end{align*}
Mathematica [A] time = 0.0205476, size = 18, normalized size = 1. \[ \frac{x \left (a+b x^n\right )^{-1/n}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 35, normalized size = 1.9 \begin{align*}{ \left ( x+{\frac{bx{{\rm e}^{n\ln \left ( x \right ) }}}{a}} \right ) \left ({{\rm e}^{{\frac{ \left ( 1+n \right ) \ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n}}}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{n + 1}{n}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37482, size = 61, normalized size = 3.39 \begin{align*} \frac{b x x^{n} + a x}{{\left (b x^{n} + a\right )}^{\frac{n + 1}{n}} a} \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{1}{{\left (b x^{n} + a\right )}^{\frac{n + 1}{n}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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