Optimal. Leaf size=20 \[ \frac{x}{a^2+a b \left (c x^n\right )^{\frac{1}{n}}} \]
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Rubi [A] time = 0.0083064, antiderivative size = 32, normalized size of antiderivative = 1.6, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {254, 32} \[ -\frac{x \left (c x^n\right )^{-1/n}}{b \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \]
Antiderivative was successfully verified.
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Rule 254
Rule 32
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, 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}}{b \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}\\ \end{align*}
Mathematica [A] time = 0.0158559, size = 33, normalized size = 1.65 \[ -\frac{x \left (c x^n\right )^{-1/n}}{a b+b^2 \left (c x^n\right )^{\frac{1}{n}}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 107, normalized size = 5.4 \begin{align*}{\frac{x}{a} \left ( a+b{{\rm e}^{-{\frac{i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) +i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00954, size = 31, normalized size = 1.55 \begin{align*} \frac{x}{a b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49023, size = 46, normalized size = 2.3 \begin{align*} -\frac{1}{b^{2} c^{\frac{2}{n}} x + a b c^{\left (\frac{1}{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )^{\left (\frac{1}{n}\right )} b + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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