Optimal. Leaf size=39 \[ -\frac{\log \left (a+b x^n\right )}{a^2 n}+\frac{\log (x)}{a^2}+\frac{1}{a n \left (a+b x^n\right )} \]
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Rubi [A] time = 0.0236343, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ -\frac{\log \left (a+b x^n\right )}{a^2 n}+\frac{\log (x)}{a^2}+\frac{1}{a n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b x^n\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{1}{a n \left (a+b x^n\right )}+\frac{\log (x)}{a^2}-\frac{\log \left (a+b x^n\right )}{a^2 n}\\ \end{align*}
Mathematica [A] time = 0.0347675, size = 33, normalized size = 0.85 \[ \frac{\frac{a}{a+b x^n}-\log \left (a+b x^n\right )+n \log (x)}{a^2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 45, normalized size = 1.2 \begin{align*}{\frac{\ln \left ({x}^{n} \right ) }{{a}^{2}n}}-{\frac{\ln \left ( a+b{x}^{n} \right ) }{{a}^{2}n}}+{\frac{1}{an \left ( a+b{x}^{n} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969175, size = 58, normalized size = 1.49 \begin{align*} \frac{1}{a b n x^{n} + a^{2} n} - \frac{\log \left (b x^{n} + a\right )}{a^{2} n} + \frac{\log \left (x^{n}\right )}{a^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.022, size = 116, normalized size = 2.97 \begin{align*} \frac{b n x^{n} \log \left (x\right ) + a n \log \left (x\right ) -{\left (b x^{n} + a\right )} \log \left (b x^{n} + a\right ) + a}{a^{2} b n x^{n} + a^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.16899, size = 163, normalized size = 4.18 \begin{align*} \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac{x^{- 2 n}}{2 b^{2} n} & \text{for}\: a = 0 \\\tilde{\infty } \log{\left (x \right )} & \text{for}\: b = - a x^{- n} \\\frac{\log{\left (x \right )}}{\left (a + b\right )^{2}} & \text{for}\: n = 0 \\\frac{\log{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\\frac{a n \log{\left (x \right )}}{a^{3} n + a^{2} b n x^{n}} - \frac{a \log{\left (\frac{a}{b} + x^{n} \right )}}{a^{3} n + a^{2} b n x^{n}} + \frac{b n x^{n} \log{\left (x \right )}}{a^{3} n + a^{2} b n x^{n}} - \frac{b x^{n} \log{\left (\frac{a}{b} + x^{n} \right )}}{a^{3} n + a^{2} b n x^{n}} - \frac{b x^{n}}{a^{3} n + a^{2} b 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{1}{{\left (b x^{n} + a\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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