Optimal. Leaf size=93 \[ \frac{b^2 x^{n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+2) \left (a b+b^2 x^n\right )}+\frac{a x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left (a+b x^n\right )} \]
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Rubi [A] time = 0.0247172, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 14} \[ \frac{b^2 x^{n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+2) \left (a b+b^2 x^n\right )}+\frac{a x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 14
Rubi steps
\begin{align*} \int x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \, dx &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int x \left (a b+b^2 x^n\right ) \, dx}{a b+b^2 x^n}\\ &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int \left (a b x+b^2 x^{1+n}\right ) \, dx}{a b+b^2 x^n}\\ &=\frac{a x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left (a+b x^n\right )}+\frac{b^2 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 )}\\ \end{align*}
Mathematica [A] time = 0.0222121, size = 46, normalized size = 0.49 \[ \frac{x^2 \sqrt{\left (a+b x^n\right )^2} \left (a (n+2)+2 b x^n\right )}{2 (n+2) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 61, normalized size = 0.7 \begin{align*}{\frac{a{x}^{2}}{2\,a+2\,b{x}^{n}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{b{x}^{2}{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( 2+n \right ) }\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997503, size = 34, normalized size = 0.37 \begin{align*} \frac{2 \, b x^{2} x^{n} + a{\left (n + 2\right )} x^{2}}{2 \,{\left (n + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57154, size = 61, normalized size = 0.66 \begin{align*} \frac{2 \, b x^{2} x^{n} +{\left (a n + 2 \, a\right )} x^{2}}{2 \,{\left (n + 2\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 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 [A] time = 1.11286, size = 72, normalized size = 0.77 \begin{align*} \frac{2 \, b x^{2} x^{n} \mathrm{sgn}\left (b x^{n} + a\right ) + a n x^{2} \mathrm{sgn}\left (b x^{n} + a\right ) + 2 \, a x^{2} \mathrm{sgn}\left (b x^{n} + a\right )}{2 \,{\left (n + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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