Optimal. Leaf size=55 \[ \frac{x \left (a+b x^n\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A] time = 0.0173405, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1343, 245} \[ \frac{x \left (a+b x^n\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 1343
Rule 245
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \, dx &=\frac{\left (2 a b+2 b^2 x^n\right ) \int \frac{1}{2 a b+2 b^2 x^n} \, dx}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac{x \left (a+b x^n\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end{align*}
Mathematica [A] time = 0.0103412, size = 44, normalized size = 0.8 \[ \frac{x \left (a+b x^n\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a \sqrt{\left (a+b x^n\right )^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{{a}^{2}+2\,ab{x}^{n}+{b}^{2}{x}^{2\,n}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}, x\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}{\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 [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}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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