Optimal. Leaf size=159 \[ \frac{1}{a^2 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}+\frac{1}{2 a n \left (a+b x^n\right ) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}+\frac{\log (x) \left (a+b x^n\right )}{a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a^3 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A] time = 0.0830614, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {1355, 266, 44} \[ \frac{1}{a^2 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}+\frac{1}{2 a n \left (a+b x^n\right ) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}+\frac{\log (x) \left (a+b x^n\right )}{a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a^3 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^n\right )\right ) \int \frac{1}{x \left (a b+b^2 x^n\right )^3} \, dx}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac{\left (b^2 \left (a b+b^2 x^n\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (a b+b^2 x\right )^3} \, dx,x,x^n\right )}{n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac{\left (b^2 \left (a b+b^2 x^n\right )\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^3 b^3 x}-\frac{1}{a b^2 (a+b x)^3}-\frac{1}{a^2 b^2 (a+b x)^2}-\frac{1}{a^3 b^2 (a+b x)}\right ) \, dx,x,x^n\right )}{n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac{1}{a^2 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}+\frac{1}{2 a n \left (a+b x^n\right ) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}+\frac{\left (a+b x^n\right ) \log (x)}{a^3 \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^3 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end{align*}
Mathematica [A] time = 0.0673037, size = 76, normalized size = 0.48 \[ \frac{\left (a+b x^n\right )^3 \left (\frac{1}{a^2 \left (a+b x^n\right )}-\frac{\log \left (a+b x^n\right )}{a^3}+\frac{n \log (x)}{a^3}+\frac{1}{2 a \left (a+b x^n\right )^2}\right )}{n \left (\left (a+b x^n\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 104, normalized size = 0.7 \begin{align*}{\frac{\ln \left ( x \right ) }{ \left ( a+b{x}^{n} \right ){a}^{3}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{2\,b{x}^{n}+3\,a}{2\, \left ( a+b{x}^{n} \right ) ^{3}{a}^{2}n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}-{\frac{1}{ \left ( a+b{x}^{n} \right ){a}^{3}n}\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 = 0.962489, size = 95, normalized size = 0.6 \begin{align*} \frac{2 \, b x^{n} + 3 \, a}{2 \,{\left (a^{2} b^{2} n x^{2 \, n} + 2 \, a^{3} b n x^{n} + a^{4} n\right )}} + \frac{\log \left (x\right )}{a^{3}} - \frac{\log \left (\frac{b x^{n} + a}{b}\right )}{a^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86974, size = 244, normalized size = 1.53 \begin{align*} \frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, a^{2} n \log \left (x\right ) + 3 \, a^{2} + 2 \,{\left (2 \, a b n \log \left (x\right ) + a b\right )} x^{n} - 2 \,{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (a^{3} b^{2} n x^{2 \, n} + 2 \, a^{4} b n x^{n} + a^{5} n\right )}} \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 \left (\left (a + b x^{n}\right )^{2}\right )^{\frac{3}{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}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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