Optimal. Leaf size=44 \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
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Rubi [A] time = 0.0123477, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {254, 205} \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 254
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{a+b \left (c x^n\right )^{2/n}} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.008214, size = 44, normalized size = 1. \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.675, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( c{x}^{n} \right ) ^{2\,{n}^{-1}} \right ) ^{-1}\, 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{1}{\left (c x^{n}\right )^{\frac{2}{n}} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3884, size = 238, normalized size = 5.41 \begin{align*} \left [-\frac{\sqrt{-a b c^{\frac{2}{n}}} \log \left (\frac{b c^{\frac{2}{n}} x^{2} - 2 \, \sqrt{-a b c^{\frac{2}{n}}} x - a}{b c^{\frac{2}{n}} x^{2} + a}\right )}{2 \, a b c^{\frac{2}{n}}}, \frac{\sqrt{a b c^{\frac{2}{n}}} \arctan \left (\frac{\sqrt{a b c^{\frac{2}{n}}} x}{a}\right )}{a b c^{\frac{2}{n}}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \left (c x^{n}\right )^{\frac{2}{n}}}\, dx \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 (c x^{n}\right )^{\frac{2}{n}} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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