Optimal. Leaf size=103 \[ -\frac{\sqrt{a+b x}}{x (b-c)}+\frac{\sqrt{a+c x}}{x (b-c)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)} \]
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Rubi [A] time = 0.0945461, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2103, 47, 63, 208} \[ -\frac{\sqrt{a+b x}}{x (b-c)}+\frac{\sqrt{a+c x}}{x (b-c)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)} \]
Antiderivative was successfully verified.
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Rule 2103
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x \left (\sqrt{a+b x}+\sqrt{a+c x}\right )} \, dx &=\frac{\int \frac{\sqrt{a+b x}}{x^2} \, dx}{b-c}-\frac{\int \frac{\sqrt{a+c x}}{x^2} \, dx}{b-c}\\ &=-\frac{\sqrt{a+b x}}{(b-c) x}+\frac{\sqrt{a+c x}}{(b-c) x}+\frac{b \int \frac{1}{x \sqrt{a+b x}} \, dx}{2 (b-c)}-\frac{c \int \frac{1}{x \sqrt{a+c x}} \, dx}{2 (b-c)}\\ &=-\frac{\sqrt{a+b x}}{(b-c) x}+\frac{\sqrt{a+c x}}{(b-c) x}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{b-c}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x}\right )}{b-c}\\ &=-\frac{\sqrt{a+b x}}{(b-c) x}+\frac{\sqrt{a+c x}}{(b-c) x}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)}\\ \end{align*}
Mathematica [A] time = 0.198739, size = 135, normalized size = 1.31 \[ \frac{-\frac{a}{\sqrt{a+b x}}-\frac{b x}{\sqrt{a+b x}}-\frac{b x \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x}{a}+1}\right )}{\sqrt{a+b x}}+\frac{a}{\sqrt{a+c x}}+\frac{c x}{\sqrt{a+c x}}+\frac{c x \sqrt{\frac{c x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{c x}{a}+1}\right )}{\sqrt{a+c x}}}{b x-c x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 88, normalized size = 0.9 \begin{align*} 2\,{\frac{b}{b-c} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-2\,{\frac{c}{b-c} \left ( -1/2\,{\frac{\sqrt{cx+a}}{cx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \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 \frac{1}{x{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20451, size = 450, normalized size = 4.37 \begin{align*} \left [-\frac{\sqrt{a} b x \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \sqrt{a} c x \log \left (\frac{c x - 2 \, \sqrt{c x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \, \sqrt{b x + a} a - 2 \, \sqrt{c x + a} a}{2 \,{\left (a b - a c\right )} x}, \frac{\sqrt{-a} b x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) - \sqrt{-a} c x \arctan \left (\frac{\sqrt{c x + a} \sqrt{-a}}{a}\right ) - \sqrt{b x + a} a + \sqrt{c x + a} a}{{\left (a b - a c\right )} 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}{x \left (\sqrt{a + b x} + \sqrt{a + c x}\right )}\, dx \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: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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