Optimal. Leaf size=45 \[ -\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{1}{a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \]
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Rubi [A] time = 0.0202457, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {368, 44} \[ -\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{1}{a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \]
Antiderivative was successfully verified.
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Rule 368
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{1}{a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{\log (x)}{a^2}-\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0335881, size = 40, normalized size = 0.89 \[ \frac{\frac{a}{a+b \left (c x^n\right )^{\frac{1}{n}}}-\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )+\log (x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 54, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( \sqrt [n]{c{x}^{n}} \right ) }{{a}^{2}}}-{\frac{\ln \left ( a+b\sqrt [n]{c{x}^{n}} \right ) }{{a}^{2}}}+{\frac{1}{a \left ( a+b\sqrt [n]{c{x}^{n}} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985928, size = 82, normalized size = 1.82 \begin{align*} \frac{1}{a b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2}} + \frac{\log \left (x\right )}{a^{2}} - \frac{\log \left (\frac{b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a}{b c^{\left (\frac{1}{n}\right )}}\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54628, size = 138, normalized size = 3.07 \begin{align*} \frac{b c^{\left (\frac{1}{n}\right )} x \log \left (x\right ) -{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right ) + a \log \left (x\right ) + a}{a^{2} b c^{\left (\frac{1}{n}\right )} x + a^{3}} \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} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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