Optimal. Leaf size=85 \[ \frac{\log (x) \left (a+b x^n\right )}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A] time = 0.0361088, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {1355, 266, 36, 29, 31} \[ \frac{\log (x) \left (a+b x^n\right )}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \, dx &=\frac{\left (a b+b^2 x^n\right ) \int \frac{1}{x \left (a b+b^2 x^n\right )} \, dx}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac{\left (a b+b^2 x^n\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (a b+b^2 x\right )} \, dx,x,x^n\right )}{n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac{\left (a b+b^2 x^n\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^n\right )}{a b n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left (b \left (a b+b^2 x^n\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+b^2 x} \, dx,x,x^n\right )}{a n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac{\left (a+b x^n\right ) \log (x)}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end{align*}
Mathematica [A] time = 0.0148707, size = 42, normalized size = 0.49 \[ \frac{\left (a+b x^n\right ) \left (n \log (x)-\log \left (a+b x^n\right )\right )}{a n \sqrt{\left (a+b x^n\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 66, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x \right ) }{ \left ( a+b{x}^{n} \right ) a}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}-{\frac{1}{ \left ( a+b{x}^{n} \right ) an}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}\ln \left ({x}^{n}+{\frac{a}{b}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993845, size = 36, normalized size = 0.42 \begin{align*} \frac{\log \left (x\right )}{a} - \frac{\log \left (\frac{b x^{n} + a}{b}\right )}{a n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59713, size = 47, normalized size = 0.55 \begin{align*} \frac{n \log \left (x\right ) - \log \left (b x^{n} + a\right )}{a n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\left (a + b x^{n}\right )^{2}}}\, dx \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}{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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