Optimal. Leaf size=42 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{a c-b c x}}{a \sqrt{c}}\right )}{a \sqrt{c}} \]
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Rubi [A] time = 0.0937899, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{a c-b c x}}{a \sqrt{c}}\right )}{a \sqrt{c}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
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Rubi in Sympy [A] time = 9.53729, size = 36, normalized size = 0.86 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x} \sqrt{a c - b c x}}{a \sqrt{c}} \right )}}{a \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0664701, size = 54, normalized size = 1.29 \[ \frac{\sqrt{a-b x} \left (\log (x)-\log \left (\sqrt{a-b x} \sqrt{a+b x}+a\right )\right )}{a \sqrt{c (a-b x)}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
[Out]
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Maple [B] time = 0.042, size = 85, normalized size = 2. \[ -{1\sqrt{bx+a}\sqrt{-c \left ( bx-a \right ) }\ln \left ( 2\,{\frac{{a}^{2}c+\sqrt{{a}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}{x}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}{\frac{1}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(b*x+a)^(1/2)/(-b*c*x+a*c)^(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*c*x + a*c)*sqrt(b*x + a)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231843, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{2 \, \sqrt{-b c x + a c} \sqrt{b x + a} a +{\left (b^{2} x^{2} - 2 \, a^{2}\right )} \sqrt{c}}{x^{2}}\right )}{2 \, a \sqrt{c}}, -\frac{\arctan \left (\frac{a \sqrt{-c}}{\sqrt{-b c x + a c} \sqrt{b x + a}}\right )}{a \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*x),x, algorithm="fricas")
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Sympy [A] time = 12.7479, size = 83, normalized size = 1.98 \[ \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} a \sqrt{c}} - \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} a \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)
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GIAC/XCAS [A] time = 0.226588, size = 88, normalized size = 2.1 \[ -\frac{2 \, \sqrt{-c} \arctan \left (\frac{{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2}}{2 \, a c^{2}}\right )}{a{\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*x),x, algorithm="giac")
[Out]