Optimal. Leaf size=103 \[ -\frac{\sqrt{a+b x}}{x (b-c)}+\frac{\sqrt{a+c x}}{x (b-c)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)} \]
[Out]
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Rubi [A] time = 0.193023, 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 \[ -\frac{\sqrt{a+b x}}{x (b-c)}+\frac{\sqrt{a+c x}}{x (b-c)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(Sqrt[a + b*x] + Sqrt[a + c*x])),x]
[Out]
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Rubi in Sympy [A] time = 17.4099, size = 76, normalized size = 0.74 \[ - \frac{\sqrt{a + b x}}{x \left (b - c\right )} + \frac{\sqrt{a + c x}}{x \left (b - c\right )} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a} \left (b - c\right )} + \frac{c \operatorname{atanh}{\left (\frac{\sqrt{a + c x}}{\sqrt{a}} \right )}}{\sqrt{a} \left (b - c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/((b*x+a)**(1/2)+(c*x+a)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.120457, size = 81, normalized size = 0.79 \[ \frac{-\sqrt{a+b x}-\frac{b x \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}+\sqrt{a+c x}+\frac{c x \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a}}}{b x-c x} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(Sqrt[a + b*x] + Sqrt[a + c*x])),x]
[Out]
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Maple [A] time = 0.006, size = 88, normalized size = 0.9 \[ 2\,{\frac{b}{b-c} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-2\,{\frac{c}{b-c} \left ( -1/2\,{\frac{\sqrt{cx+a}}{cx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/((b*x+a)^(1/2)+(c*x+a)^(1/2)),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x*(sqrt(b*x + a) + sqrt(c*x + a))),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.290659, size = 1, normalized size = 0.01 \[ \left [-\frac{b x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + c x \log \left (\frac{{\left (c x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x + a} a}{x}\right ) + 2 \, \sqrt{b x + a} \sqrt{a} - 2 \, \sqrt{c x + a} \sqrt{a}}{2 \, \sqrt{a}{\left (b - c\right )} x}, \frac{b x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) - c x \arctan \left (\frac{a}{\sqrt{c x + a} \sqrt{-a}}\right ) - \sqrt{b x + a} \sqrt{-a} + \sqrt{c x + a} \sqrt{-a}}{\sqrt{-a}{\left (b - c\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x*(sqrt(b*x + a) + sqrt(c*x + a))),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (\sqrt{a + b x} + \sqrt{a + c x}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/((b*x+a)**(1/2)+(c*x+a)**(1/2)),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x*(sqrt(b*x + a) + sqrt(c*x + a))),x, algorithm="giac")
[Out]