Optimal. Leaf size=33 \[ \frac{a}{b^2 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^2 n} \]
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Rubi [A] time = 0.0230451, antiderivative size = 33, 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 = {266, 43} \[ \frac{a}{b^2 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{-1+2 n}}{\left (a+b x^n\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{(a+b x)^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^2}+\frac{1}{b (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{a}{b^2 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^2 n}\\ \end{align*}
Mathematica [A] time = 0.0242485, size = 27, normalized size = 0.82 \[ \frac{\frac{a}{a+b x^n}+\log \left (a+b x^n\right )}{b^2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 38, normalized size = 1.2 \begin{align*}{\frac{a}{{b}^{2}n \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}+{\frac{\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{2}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971322, size = 53, normalized size = 1.61 \begin{align*} \frac{a}{b^{3} n x^{n} + a b^{2} n} + \frac{\log \left (\frac{b x^{n} + a}{b}\right )}{b^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.0092, size = 76, normalized size = 2.3 \begin{align*} \frac{{\left (b x^{n} + a\right )} \log \left (b x^{n} + a\right ) + a}{b^{3} n x^{n} + a b^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 81.0982, size = 99, normalized size = 3. \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \wedge n = 0 \\\frac{x^{2 n}}{2 a^{2} n} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{\left (a + b\right )^{2}} & \text{for}\: n = 0 \\\frac{a \log{\left (\frac{a}{b} + x^{n} \right )}}{a b^{2} n + b^{3} n x^{n}} + \frac{b x^{n} \log{\left (\frac{a}{b} + x^{n} \right )}}{a b^{2} n + b^{3} n x^{n}} - \frac{b x^{n}}{a b^{2} n + b^{3} n x^{n}} & \text{otherwise} \end{cases} \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^{2 \, 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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