Optimal. Leaf size=51 \[ 2 x \sqrt{\frac{a}{x^2}+\frac{b}{x}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+\frac{b}{x}}}\right ) \]
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Rubi [A] time = 0.0765409, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {1979, 2007, 2013, 620, 206} \[ 2 x \sqrt{\frac{a}{x^2}+\frac{b}{x}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+\frac{b}{x}}}\right ) \]
Antiderivative was successfully verified.
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Rule 1979
Rule 2007
Rule 2013
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \sqrt{\frac{a+b x}{x^2}} \, dx &=\int \sqrt{\frac{a}{x^2}+\frac{b}{x}} \, dx\\ &=2 \sqrt{\frac{a}{x^2}+\frac{b}{x}} x+a \int \frac{1}{\sqrt{\frac{a}{x^2}+\frac{b}{x}} x^2} \, dx\\ &=2 \sqrt{\frac{a}{x^2}+\frac{b}{x}} x-a \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^2}} \, dx,x,\frac{1}{x}\right )\\ &=2 \sqrt{\frac{a}{x^2}+\frac{b}{x}} x-(2 a) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{1}{\sqrt{\frac{a}{x^2}+\frac{b}{x}} x}\right )\\ &=2 \sqrt{\frac{a}{x^2}+\frac{b}{x}} x-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{\frac{a}{x^2}+\frac{b}{x}} x}\right )\\ \end{align*}
Mathematica [A] time = 0.0256985, size = 58, normalized size = 1.14 \[ \frac{2 x \sqrt{\frac{a+b x}{x^2}} \left (\sqrt{a+b x}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\right )}{\sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 47, normalized size = 0.9 \begin{align*} 2\,{\frac{x}{\sqrt{bx+a}}\sqrt{{\frac{bx+a}{{x}^{2}}}} \left ( -\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) +\sqrt{bx+a} \right ) } \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 \sqrt{\frac{b x + a}{x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.916763, size = 231, normalized size = 4.53 \begin{align*} \left [2 \, x \sqrt{\frac{b x + a}{x^{2}}} + \sqrt{a} \log \left (\frac{b x - 2 \, \sqrt{a} x \sqrt{\frac{b x + a}{x^{2}}} + 2 \, a}{x}\right ), 2 \, x \sqrt{\frac{b x + a}{x^{2}}} + 2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x \sqrt{\frac{b x + a}{x^{2}}}}{a}\right )\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 \sqrt{\frac{a + b x}{x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12506, size = 88, normalized size = 1.73 \begin{align*} 2 \,{\left (\frac{a \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{b x + a}\right )} \mathrm{sgn}\left (x\right ) - \frac{2 \,{\left (a \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) + \sqrt{-a} \sqrt{a}\right )} \mathrm{sgn}\left (x\right )}{\sqrt{-a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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