Optimal. Leaf size=88 \[ \frac{b^2 x^{n+1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+1) \left (a b+b^2 x^n\right )}+\frac{a x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]
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Rubi [A] time = 0.0185866, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1343} \[ \frac{b^2 x^{n+1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+1) \left (a b+b^2 x^n\right )}+\frac{a x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]
Antiderivative was successfully verified.
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Rule 1343
Rubi steps
\begin{align*} \int \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 \left (2 a b+2 b^2 x^n\right ) \, dx}{2 a b+2 b^2 x^n}\\ &=\frac{a x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n}+\frac{b^2 x^{1+n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1+n) \left (a b+b^2 x^n\right )}\\ \end{align*}
Mathematica [A] time = 0.01403, size = 39, normalized size = 0.44 \[ \frac{x \sqrt{\left (a+b x^n\right )^2} \left (a n+a+b x^n\right )}{(n+1) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 56, normalized size = 0.6 \begin{align*}{\frac{ax}{a+b{x}^{n}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{bx{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( 1+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 = 0.960536, size = 26, normalized size = 0.3 \begin{align*} \frac{a{\left (n + 1\right )} x + b x x^{n}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59111, size = 45, normalized size = 0.51 \begin{align*} \frac{b x x^{n} +{\left (a n + a\right )} x}{n + 1} \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 \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1015, size = 34, normalized size = 0.39 \begin{align*}{\left (a x + \frac{b x^{n + 1}}{n + 1}\right )} \mathrm{sgn}\left (b x^{n} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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