Optimal. Leaf size=73 \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )} \]
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Rubi [A] time = 0.0240274, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {254, 199, 205} \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )} \]
Antiderivative was successfully verified.
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Rule 254
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \left (c x^n\right )^{2/n}\right )^2} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{x}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac{\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 )}{2 a}\\ &=\frac{x}{2 a \left (a+b \left (c x^n\right )^{2/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 )}{2 a^{3/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0447284, size = 73, normalized size = 1. \[ \frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.371, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( c{x}^{n} \right ) ^{2\,{n}^{-1}} \right ) ^{-2}\, 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*} \frac{x}{2 \,{\left (a b c^{\frac{2}{n}}{\left (x^{n}\right )}^{\frac{2}{n}} + a^{2}\right )}} + \int \frac{1}{2 \,{\left (a b c^{\frac{2}{n}}{\left (x^{n}\right )}^{\frac{2}{n}} + a^{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49737, size = 412, normalized size = 5.64 \begin{align*} \left [\frac{2 \, a b c^{\frac{2}{n}} x -{\left (b c^{\frac{2}{n}} x^{2} + a\right )} \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 )}{4 \,{\left (a^{2} b^{2} c^{\frac{4}{n}} x^{2} + a^{3} b c^{\frac{2}{n}}\right )}}, \frac{a b c^{\frac{2}{n}} x +{\left (b c^{\frac{2}{n}} x^{2} + a\right )} \sqrt{a b c^{\frac{2}{n}}} \arctan \left (\frac{\sqrt{a b c^{\frac{2}{n}}} x}{a}\right )}{2 \,{\left (a^{2} b^{2} c^{\frac{4}{n}} x^{2} + a^{3} b c^{\frac{2}{n}}\right )}}\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}{\left (a + b \left (c x^{n}\right )^{\frac{2}{n}}\right )^{2}}\, 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 (\left (c x^{n}\right )^{\frac{2}{n}} b + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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