Optimal. Leaf size=60 \[ -\frac{b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^2 x}+\frac{b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2 x}-\frac{1}{a x} \]
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Rubi [A] time = 0.0232409, antiderivative size = 60, 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} \[ -\frac{b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^2 x}+\frac{b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2 x}-\frac{1}{a x} \]
Antiderivative was successfully verified.
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Rule 368
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{x}\\ &=\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{x}\\ &=-\frac{1}{a x}-\frac{b \left (c x^n\right )^{\frac{1}{n}} \log (x)}{a^2 x}+\frac{b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2 x}\\ \end{align*}
Mathematica [A] time = 0.0238341, size = 49, normalized size = 0.82 \[ -\frac{-b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )+a+b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^2 x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.097, size = 331, normalized size = 5.5 \begin{align*} -{\frac{1}{ax}}+{\frac{b\sqrt [n]{c}}{{a}^{2}}\ln \left ( b{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -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 ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}x+a \right ){{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -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 ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}}-{\frac{b\ln \left ( x \right ) \sqrt [n]{c}}{{a}^{2}}{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -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 ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}} \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 (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53864, size = 93, normalized size = 1.55 \begin{align*} \frac{b c^{\left (\frac{1}{n}\right )} x \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right ) - b c^{\left (\frac{1}{n}\right )} x \log \left (x\right ) - a}{a^{2} x} \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}{x^{2} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )}\, 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 )^{\left (\frac{1}{n}\right )} b + a\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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