Optimal. Leaf size=97 \[ \frac{2 \sqrt{a+b x}}{b-c}-\frac{2 \sqrt{a+c x}}{b-c}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{b-c}+\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{b-c} \]
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Rubi [A] time = 0.150429, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{2 \sqrt{a+b x}}{b-c}-\frac{2 \sqrt{a+c x}}{b-c}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{b-c}+\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{b-c} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x] + Sqrt[a + c*x])^(-1),x]
[Out]
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Rubi in Sympy [A] time = 13.1754, size = 76, normalized size = 0.78 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{b - c} + \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + c x}}{\sqrt{a}} \right )}}{b - c} + \frac{2 \sqrt{a + b x}}{b - c} - \frac{2 \sqrt{a + c x}}{b - c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/((b*x+a)**(1/2)+(c*x+a)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0620588, size = 75, normalized size = 0.77 \[ \frac{2 \left (\sqrt{a+b x}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\sqrt{a+c x}+\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )\right )}{b-c} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x] + Sqrt[a + c*x])^(-1),x]
[Out]
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Maple [A] time = 0.005, size = 73, normalized size = 0.8 \[{\frac{1}{b-c} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }-{\frac{1}{b-c} \left ( 2\,\sqrt{cx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/((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. \[ \int \frac{1}{\sqrt{b x + a} + \sqrt{c x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a) + sqrt(c*x + a)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279846, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{a} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \sqrt{a} \log \left (\frac{c x - 2 \, \sqrt{c x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \, \sqrt{b x + a} + 2 \, \sqrt{c x + a}}{b - c}, -\frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) - \sqrt{-a} \arctan \left (\frac{\sqrt{c x + a}}{\sqrt{-a}}\right ) - \sqrt{b x + a} + \sqrt{c x + a}\right )}}{b - c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(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. \[ \int \frac{1}{\sqrt{a + b x} + \sqrt{a + c x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((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/(sqrt(b*x + a) + sqrt(c*x + a)),x, algorithm="giac")
[Out]