Optimal. Leaf size=39 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b d x-a d}}\right )}{b \sqrt{d}} \]
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Rubi [A] time = 0.048064, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b d x-a d}}\right )}{b \sqrt{d}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b*x]*Sqrt[-(a*d) + b*d*x]),x]
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Rubi in Sympy [A] time = 7.33575, size = 34, normalized size = 0.87 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{- a d + b d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b \sqrt{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(1/2)/(b*d*x-a*d)**(1/2),x)
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Mathematica [A] time = 0.0631371, size = 43, normalized size = 1.1 \[ \frac{\log \left (\sqrt{d} \sqrt{a+b x} \sqrt{-d (a-b x)}+b d x\right )}{b \sqrt{d}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b*x]*Sqrt[-(a*d) + b*d*x]),x]
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Maple [B] time = 0.013, size = 76, normalized size = 2. \[{1\sqrt{ \left ( bx+a \right ) \left ( bdx-ad \right ) }\ln \left ({{b}^{2}dx{\frac{1}{\sqrt{{b}^{2}d}}}}+\sqrt{{b}^{2}d{x}^{2}-{a}^{2}d} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{bdx-ad}}}{\frac{1}{\sqrt{{b}^{2}d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(1/2)/(b*d*x-a*d)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*d*x - a*d)*sqrt(b*x + a)),x, algorithm="maxima")
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Fricas [A] time = 0.215054, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (2 \, \sqrt{b d x - a d} \sqrt{b x + a} b x +{\left (2 \, b^{2} x^{2} - a^{2}\right )} \sqrt{d}\right )}{2 \, b \sqrt{d}}, \frac{\arctan \left (\frac{b \sqrt{-d} x}{\sqrt{b d x - a d} \sqrt{b x + a}}\right )}{b \sqrt{-d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*d*x - a*d)*sqrt(b*x + a)),x, algorithm="fricas")
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Sympy [A] time = 10.76, size = 88, normalized size = 2.26 \[ \frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b \sqrt{d}} - \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{a^{2} e^{2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b \sqrt{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(1/2)/(b*d*x-a*d)**(1/2),x)
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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*d*x - a*d)*sqrt(b*x + a)),x, algorithm="giac")
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