Optimal. Leaf size=34 \[ \sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0087351, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {50, 54, 215} \[ \sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \frac{\sqrt{1+a x}}{\sqrt{x}} \, dx &=\sqrt{x} \sqrt{1+a x}+\frac{1}{2} \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx\\ &=\sqrt{x} \sqrt{1+a x}+\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )\\ &=\sqrt{x} \sqrt{1+a x}+\frac{\sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.013254, size = 34, normalized size = 1. \[ \sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.167, size = 57, normalized size = 1.7 \begin{align*} \sqrt{x}\sqrt{ax+1}+{\frac{1}{2}\sqrt{ \left ( ax+1 \right ) x}\ln \left ({ \left ({\frac{1}{2}}+ax \right ){\frac{1}{\sqrt{a}}}}+\sqrt{a{x}^{2}+x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{ax+1}}}{\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59336, size = 244, normalized size = 7.18 \begin{align*} \left [\frac{2 \, \sqrt{a x + 1} a \sqrt{x} + \sqrt{a} \log \left (2 \, a x + 2 \, \sqrt{a x + 1} \sqrt{a} \sqrt{x} + 1\right )}{2 \, a}, \frac{\sqrt{a x + 1} a \sqrt{x} - \sqrt{-a} \arctan \left (\frac{\sqrt{a x + 1} \sqrt{-a}}{a \sqrt{x}}\right )}{a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.92732, size = 29, normalized size = 0.85 \begin{align*} \sqrt{x} \sqrt{a x + 1} + \frac{\operatorname{asinh}{\left (\sqrt{a} \sqrt{x} \right )}}{\sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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