Optimal. Leaf size=103 \[ -\frac{\sqrt{a+b x}}{x (a-c)}+\frac{\sqrt{b x+c}}{x (a-c)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (a-c)}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{\sqrt{c} (a-c)} \]
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Rubi [A] time = 0.102121, 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 = {2104, 47, 63, 208} \[ -\frac{\sqrt{a+b x}}{x (a-c)}+\frac{\sqrt{b x+c}}{x (a-c)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (a-c)}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{\sqrt{c} (a-c)} \]
Antiderivative was successfully verified.
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Rule 2104
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (\sqrt{a+b x}+\sqrt{c+b x}\right )} \, dx &=-\frac{b \int \frac{\sqrt{a+b x}}{x^2} \, dx}{-a b+b c}+\frac{b \int \frac{\sqrt{c+b x}}{x^2} \, dx}{-a b+b c}\\ &=-\frac{\sqrt{a+b x}}{(a-c) x}+\frac{\sqrt{c+b x}}{(a-c) x}+\frac{b \int \frac{1}{x \sqrt{a+b x}} \, dx}{2 (a-c)}-\frac{b \int \frac{1}{x \sqrt{c+b x}} \, dx}{2 (a-c)}\\ &=-\frac{\sqrt{a+b x}}{(a-c) x}+\frac{\sqrt{c+b x}}{(a-c) x}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{a-c}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{c}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{c+b x}\right )}{a-c}\\ &=-\frac{\sqrt{a+b x}}{(a-c) x}+\frac{\sqrt{c+b x}}{(a-c) x}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (a-c)}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{c+b x}}{\sqrt{c}}\right )}{(a-c) \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.278175, size = 99, normalized size = 0.96 \[ \frac{\frac{b x \sqrt{\frac{b x}{c}+1} \tanh ^{-1}\left (\sqrt{\frac{b x}{c}+1}\right )+b x+c}{\sqrt{b x+c}}-\frac{b x \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x}{a}+1}\right )+a+b x}{\sqrt{a+b x}}}{x (a-c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 88, normalized size = 0.9 \begin{align*} 2\,{\frac{b}{a-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{b}{a-c} \left ( -1/2\,{\frac{\sqrt{bx+c}}{bx}}-1/2\,{\frac{1}{\sqrt{c}}{\it Artanh} \left ({\frac{\sqrt{bx+c}}{\sqrt{c}}} \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^{2}{\left (\sqrt{b x + a} + \sqrt{b x + c}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06672, size = 980, normalized size = 9.51 \begin{align*} \left [-\frac{\sqrt{a} b c x \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + a b \sqrt{c} x \log \left (\frac{b x - 2 \, \sqrt{b x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \, \sqrt{b x + a} a c - 2 \, \sqrt{b x + c} a c}{2 \,{\left (a^{2} c - a c^{2}\right )} x}, -\frac{2 \, a b \sqrt{-c} x \arctan \left (\frac{\sqrt{b x + c} \sqrt{-c}}{c}\right ) + \sqrt{a} b c x \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \, \sqrt{b x + a} a c - 2 \, \sqrt{b x + c} a c}{2 \,{\left (a^{2} c - a c^{2}\right )} x}, \frac{2 \, \sqrt{-a} b c x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) - a b \sqrt{c} x \log \left (\frac{b x - 2 \, \sqrt{b x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \, \sqrt{b x + a} a c + 2 \, \sqrt{b x + c} a c}{2 \,{\left (a^{2} c - a c^{2}\right )} x}, \frac{\sqrt{-a} b c x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) - a b \sqrt{-c} x \arctan \left (\frac{\sqrt{b x + c} \sqrt{-c}}{c}\right ) - \sqrt{b x + a} a c + \sqrt{b x + c} a c}{{\left (a^{2} c - a c^{2}\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^{2} \left (\sqrt{a + b x} + \sqrt{b x + c}\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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