Optimal. Leaf size=196 \[ \frac{3 a^2 b^2 x^n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac{3 a b^3 x^{2 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]
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Rubi [A] time = 0.0516225, antiderivative size = 196, 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, 43} \[ \frac{3 a^2 b^2 x^n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac{3 a b^3 x^{2 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{x} \, dx &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int \frac{\left (a b+b^2 x^n\right )^3}{x} \, dx}{b^2 \left (a b+b^2 x^n\right )}\\ &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^3}{x} \, dx,x,x^n\right )}{b^2 n \left (a b+b^2 x^n\right )}\\ &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \operatorname{Subst}\left (\int \left (3 a^2 b^4+\frac{a^3 b^3}{x}+3 a b^5 x+b^6 x^2\right ) \, dx,x,x^n\right )}{b^2 n \left (a b+b^2 x^n\right )}\\ &=\frac{3 a^2 b^2 x^n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac{3 a b^3 x^{2 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )}+\frac{a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \log (x)}{a+b x^n}\\ \end{align*}
Mathematica [A] time = 0.0385719, size = 68, normalized size = 0.35 \[ \frac{\left (\left (a+b x^n\right )^2\right )^{3/2} \left (3 a^2 b x^n+a^3 n \log (x)+\frac{3}{2} a b^2 x^{2 n}+\frac{1}{3} b^3 x^{3 n}\right )}{n \left (a+b x^n\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 127, normalized size = 0.7 \begin{align*}{\frac{{a}^{3}\ln \left ( x \right ) }{a+b{x}^{n}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{{b}^{3} \left ({x}^{n} \right ) ^{3}}{ \left ( 3\,a+3\,b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{3\,a{b}^{2} \left ({x}^{n} \right ) ^{2}}{ \left ( 2\,a+2\,b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+3\,{\frac{\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}{a}^{2}b{x}^{n}}{ \left ( a+b{x}^{n} \right ) n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983491, size = 58, normalized size = 0.3 \begin{align*} a^{3} \log \left (x\right ) + \frac{2 \, b^{3} x^{3 \, n} + 9 \, a b^{2} x^{2 \, n} + 18 \, a^{2} b x^{n}}{6 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56861, size = 99, normalized size = 0.51 \begin{align*} \frac{6 \, a^{3} n \log \left (x\right ) + 2 \, b^{3} x^{3 \, n} + 9 \, a b^{2} x^{2 \, n} + 18 \, a^{2} b x^{n}}{6 \, 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{\left (\left (a + b x^{n}\right )^{2}\right )^{\frac{3}{2}}}{x}\, 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{{\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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