Optimal. Leaf size=18 \[ \frac{x \left (a+b x^n\right )^{-1/n}}{a} \]
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Rubi [A] time = 0.002757, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, 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 )^{-1-\frac{1}{n}} \, dx &=\frac{x \left (a+b x^n\right )^{-1/n}}{a}\\ \end{align*}
Mathematica [A] time = 0.0247102, 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 [B] time = 0.021, size = 53, normalized size = 2.9 \begin{align*} x{{\rm e}^{ \left ( -1-{n}^{-1} \right ) \ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}+{\frac{bx{{\rm e}^{n\ln \left ( x \right ) }}}{a}{{\rm e}^{ \left ( -1-{n}^{-1} \right ) \ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}} \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{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33565, 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 [A] time = 37.2195, size = 211, normalized size = 11.72 \begin{align*} \begin{cases} - \frac{b^{- \frac{1}{n}} x x^{- n} \left (x^{n}\right )^{- \frac{1}{n}}}{b n} & \text{for}\: a = 0 \\0^{-1 - \frac{1}{n}} x & \text{for}\: a = - b x^{n} \\x \left (0^{n}\right )^{-1 - \frac{1}{n}} & \text{for}\: a = 0^{n} - b x^{n} \\\frac{a^{2} x}{a^{3} \left (a + b x^{n}\right )^{\frac{1}{n}} + 2 a^{2} b x^{n} \left (a + b x^{n}\right )^{\frac{1}{n}} + a b^{2} x^{2 n} \left (a + b x^{n}\right )^{\frac{1}{n}}} + \frac{a b x x^{n}}{a^{3} \left (a + b x^{n}\right )^{\frac{1}{n}} + 2 a^{2} b x^{n} \left (a + b x^{n}\right )^{\frac{1}{n}} + a b^{2} x^{2 n} \left (a + b x^{n}\right )^{\frac{1}{n}}} + \frac{b x x^{n}}{a^{2} \left (a + b x^{n}\right )^{\frac{1}{n}} + a b x^{n} \left (a + b x^{n}\right )^{\frac{1}{n}}} & \text{otherwise} \end{cases} \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{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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