Optimal. Leaf size=32 \[ \frac{x \left (c x^n\right )^{\frac{1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]
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Rubi [A] time = 0.0129269, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {15, 368, 37} \[ \frac{x \left (c x^n\right )^{\frac{1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 368
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (c x^n\right )^{\frac{1}{n}}}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \int \frac{x}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \, dx}{x}\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x}{(a+b x)^3} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{x \left (c x^n\right )^{\frac{1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0078196, size = 32, normalized size = 1. \[ \frac{x \left (c x^n\right )^{\frac{1}{n}}}{2 a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 203, normalized size = 6.3 \begin{align*}{\frac{x}{2\,a}{{\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}}}} \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 ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03809, size = 81, normalized size = 2.53 \begin{align*} \frac{c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}}{2 \,{\left (a b^{2} c^{\frac{2}{n}}{\left (x^{n}\right )}^{\frac{2}{n}} + 2 \, a^{2} b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63759, size = 111, normalized size = 3.47 \begin{align*} -\frac{2 \, b c^{\left (\frac{1}{n}\right )} x + a}{2 \,{\left (b^{4} c^{\frac{3}{n}} x^{2} + 2 \, a b^{3} c^{\frac{2}{n}} x + a^{2} b^{2} c^{\left (\frac{1}{n}\right )}\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{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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