Optimal. Leaf size=18 \[ \frac{x \left (c+d x^n\right )^{-1/n}}{c} \]
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Rubi [A] time = 0.0026634, 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} \[ \frac{x \left (c+d x^n\right )^{-1/n}}{c} \]
Antiderivative was successfully verified.
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Rule 191
Rubi steps
\begin{align*} \int \left (c+d x^n\right )^{-1-\frac{1}{n}} \, dx &=\frac{x \left (c+d x^n\right )^{-1/n}}{c}\\ \end{align*}
Mathematica [A] time = 0.0267099, size = 18, normalized size = 1. \[ \frac{x \left (c+d x^n\right )^{-1/n}}{c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 53, normalized size = 2.9 \begin{align*} x{{\rm e}^{ \left ( -1-{n}^{-1} \right ) \ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}+{\frac{dx{{\rm e}^{n\ln \left ( x \right ) }}}{c}{{\rm e}^{ \left ( -1-{n}^{-1} \right ) \ln \left ( c+d{{\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 (d x^{n} + c\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 = 1.58121, size = 61, normalized size = 3.39 \begin{align*} \frac{d x x^{n} + c x}{{\left (d x^{n} + c\right )}^{\frac{n + 1}{n}} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 35.6476, size = 211, normalized size = 11.72 \begin{align*} \begin{cases} - \frac{d^{- \frac{1}{n}} x x^{- n} \left (x^{n}\right )^{- \frac{1}{n}}}{d n} & \text{for}\: c = 0 \\0^{-1 - \frac{1}{n}} x & \text{for}\: c = - d x^{n} \\x \left (0^{n}\right )^{-1 - \frac{1}{n}} & \text{for}\: c = 0^{n} - d x^{n} \\\frac{c^{2} x}{c^{3} \left (c + d x^{n}\right )^{\frac{1}{n}} + 2 c^{2} d x^{n} \left (c + d x^{n}\right )^{\frac{1}{n}} + c d^{2} x^{2 n} \left (c + d x^{n}\right )^{\frac{1}{n}}} + \frac{c d x x^{n}}{c^{3} \left (c + d x^{n}\right )^{\frac{1}{n}} + 2 c^{2} d x^{n} \left (c + d x^{n}\right )^{\frac{1}{n}} + c d^{2} x^{2 n} \left (c + d x^{n}\right )^{\frac{1}{n}}} + \frac{d x x^{n}}{c^{2} \left (c + d x^{n}\right )^{\frac{1}{n}} + c d x^{n} \left (c + d 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 (d x^{n} + c\right )}^{-\frac{1}{n} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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