Optimal. Leaf size=17 \[ \frac{\left (\sqrt{a+x^2}+x\right )^b}{b} \]
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Rubi [A] time = 0.0535661, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2122, 30} \[ \frac{\left (\sqrt{a+x^2}+x\right )^b}{b} \]
Antiderivative was successfully verified.
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Rule 2122
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (x+\sqrt{a+x^2}\right )^b}{\sqrt{a+x^2}} \, dx &=\operatorname{Subst}\left (\int x^{-1+b} \, dx,x,x+\sqrt{a+x^2}\right )\\ &=\frac{\left (x+\sqrt{a+x^2}\right )^b}{b}\\ \end{align*}
Mathematica [A] time = 0.0067221, size = 17, normalized size = 1. \[ \frac{\left (\sqrt{a+x^2}+x\right )^b}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{ \left ( x+\sqrt{{x}^{2}+a} \right ) ^{b}{\frac{1}{\sqrt{{x}^{2}+a}}}}\, 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{{\left (x + \sqrt{x^{2} + a}\right )}^{b}}{\sqrt{x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99166, size = 34, normalized size = 2. \begin{align*} \frac{{\left (x + \sqrt{x^{2} + a}\right )}^{b}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.67894, size = 313, normalized size = 18.41 \begin{align*} \begin{cases} - \frac{\sqrt{a} a^{\frac{b}{2}} \sinh{\left (- b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{b x \sqrt{\frac{a}{x^{2}} + 1}} - \frac{2 a^{\frac{b}{2}} \cosh{\left (b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )} \Gamma \left (1 - \frac{b}{2}\right )}{b^{2} \Gamma \left (- \frac{b}{2}\right )} + \frac{a^{\frac{b}{2}} x \cosh{\left (- b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{\sqrt{a} b} - \frac{a^{\frac{b}{2}} x \sinh{\left (- b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{\sqrt{a} b \sqrt{\frac{a}{x^{2}} + 1}} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{a}\right |} > 1 \\- \frac{a^{\frac{b}{2}} \sinh{\left (- b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{b \sqrt{1 + \frac{x^{2}}{a}}} - \frac{2 a^{\frac{b}{2}} \cosh{\left (b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )} \Gamma \left (1 - \frac{b}{2}\right )}{b^{2} \Gamma \left (- \frac{b}{2}\right )} - \frac{a^{\frac{b}{2}} x^{2} \sinh{\left (- b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{a b \sqrt{1 + \frac{x^{2}}{a}}} + \frac{a^{\frac{b}{2}} x \cosh{\left (- b \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{a}} \right )} \right )}}{\sqrt{a} b} & \text{otherwise} \end{cases} \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 (x + \sqrt{x^{2} + a}\right )}^{b}}{\sqrt{x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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