Optimal. Leaf size=24 \[ \frac{(c x)^n}{a c n \left (a+b x^n\right )} \]
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Rubi [A] time = 0.0056073, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ \frac{(c x)^n}{a c n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{(c x)^{-1+n}}{\left (a+b x^n\right )^2} \, dx &=\frac{(c x)^n}{a c n \left (a+b x^n\right )}\\ \end{align*}
Mathematica [A] time = 0.013752, size = 31, normalized size = 1.29 \[ -\frac{x^{1-n} (c x)^{n-1}}{b n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 99, normalized size = 4.1 \begin{align*}{\frac{x}{an \left ( a+b{x}^{n} \right ) }{{\rm e}^{{\frac{ \left ( -1+n \right ) \left ( -i \left ({\it csgn} \left ( icx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( icx \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \pi +i \left ({\it csgn} \left ( icx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i{\it csgn} \left ( icx \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ix \right ) \pi +2\,\ln \left ( x \right ) +2\,\ln \left ( c \right ) \right ) }{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.37117, size = 30, normalized size = 1.25 \begin{align*} -\frac{c^{n}}{b^{2} c n x^{n} + a b c n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.275, size = 43, normalized size = 1.79 \begin{align*} -\frac{c^{n - 1}}{b^{2} n x^{n} + a b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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\right )^{n - 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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