Optimal. Leaf size=31 \[ -\frac{2 (c x)^{-n/2} \sqrt{a+b x^n}}{a c n} \]
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Rubi [A] time = 0.0086017, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {264} \[ -\frac{2 (c x)^{-n/2} \sqrt{a+b x^n}}{a c n} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{(c x)^{-1-\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx &=-\frac{2 (c x)^{-n/2} \sqrt{a+b x^n}}{a c n}\\ \end{align*}
Mathematica [A] time = 0.0100875, size = 31, normalized size = 1. \[ -\frac{2 x (c x)^{-\frac{n}{2}-1} \sqrt{a+b x^n}}{a n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-1-{\frac{n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \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{\left (c x\right )^{-\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.0092, size = 29, normalized size = 0.94 \begin{align*} - \frac{2 \sqrt{b} c^{- \frac{n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{a c n} \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{\left (c x\right )^{-\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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