Optimal. Leaf size=53 \[ \frac{x^2 \left (c x^n\right )^{-1/n}}{b}-\frac{a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]
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Rubi [A] time = 0.0178758, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {368, 43} \[ \frac{x^2 \left (c x^n\right )^{-1/n}}{b}-\frac{a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 368
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx &=\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{x}{a+b x} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{x^2 \left (c x^n\right )^{-1/n}}{b}-\frac{a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0151165, size = 49, normalized size = 0.92 \[ x^2 \left (c x^n\right )^{-2/n} \left (\frac{\left (c x^n\right )^{\frac{1}{n}}}{b}-\frac{a \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.089, size = 324, normalized size = 6.1 \begin{align*}{\frac{x}{b\sqrt [n]{c}}{{\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{a}{{b}^{2} \left ( \sqrt [n]{c} \right ) ^{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 ) }{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{x}{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35676, size = 73, normalized size = 1.38 \begin{align*} \frac{b c^{\left (\frac{1}{n}\right )} x - a \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{2} c^{\frac{2}{n}}} \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{x}{a + b \left (c x^{n}\right )^{\frac{1}{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{x}{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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