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 = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 44
Rule 266
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*}
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Mathematica [A] time = 0.06, 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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fricas [A] time = 0.66, size = 50, normalized size = 1.28 \[ \frac {b n x^{n} \log \relax (x) + a n \log \relax (x) - {\left (b x^{n} + a\right )} \log \left (b x^{n} + a\right ) + a}{a^{2} b n x^{n} + a^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 45, normalized size = 1.15 \[ \frac {1}{\left (b \,x^{n}+a \right ) a n}+\frac {\ln \left (x^{n}\right )}{a^{2} n}-\frac {\ln \left (b \,x^{n}+a \right )}{a^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 43, normalized size = 1.10 \[ \frac {1}{a b n x^{n} + a^{2} n} - \frac {\log \left (b x^{n} + a\right )}{a^{2} n} + \frac {\log \left (x^{n}\right )}{a^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 39, normalized size = 1.00 \[ \frac {\ln \relax (x)}{a^2}+\frac {1}{a\,n\,\left (a+b\,x^n\right )}-\frac {\ln \left (a+b\,x^n\right )}{a^2\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.55, size = 160, normalized size = 4.10 \[ \begin {cases} \tilde {\infty } \log {\relax (x )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac {x^{- 2 n}}{2 b^{2} n} & \text {for}\: a = 0 \\\tilde {\infty } \log {\relax (x )} & \text {for}\: b = - a x^{- n} \\\frac {\log {\relax (x )}}{\left (a + b\right )^{2}} & \text {for}\: n = 0 \\\frac {\log {\relax (x )}}{a^{2}} & \text {for}\: b = 0 \\\frac {a n \log {\relax (x )}}{a^{3} n + a^{2} b n x^{n}} - \frac {a \log {\left (\frac {a}{b} + x^{n} \right )}}{a^{3} n + a^{2} b n x^{n}} + \frac {a}{a^{3} n + a^{2} b n x^{n}} + \frac {b n x^{n} \log {\relax (x )}}{a^{3} n + a^{2} b n x^{n}} - \frac {b x^{n} \log {\left (\frac {a}{b} + x^{n} \right )}}{a^{3} n + a^{2} b n x^{n}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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