Optimal. Leaf size=39 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a c-b c x}}\right )}{b \sqrt{c}} \]
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Rubi [A] time = 0.0430537, 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 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a c-b c x}}\right )}{b \sqrt{c}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
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Rubi in Sympy [A] time = 7.54457, size = 36, normalized size = 0.92 \[ - \frac{2 \operatorname{atan}{\left (\frac{\sqrt{a c - b c x}}{\sqrt{c} \sqrt{a + b x}} \right )}}{b \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0345239, size = 49, normalized size = 1.26 \[ \frac{\sqrt{a-b x} \tan ^{-1}\left (\frac{b x}{\sqrt{a-b x} \sqrt{a+b x}}\right )}{b \sqrt{c (a-b x)}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]
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Maple [B] time = 0.007, size = 71, normalized size = 1.8 \[{1\sqrt{ \left ( bx+a \right ) \left ( -bcx+ac \right ) }\arctan \left ({x\sqrt{{b}^{2}c}{\frac{1}{\sqrt{-{b}^{2}c{x}^{2}+{a}^{2}c}}}} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{-bcx+ac}}}{\frac{1}{\sqrt{{b}^{2}c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(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, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221725, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b x +{\left (2 \, b^{2} x^{2} - a^{2}\right )} \sqrt{-c}\right )}{2 \, b \sqrt{-c}}, \frac{\arctan \left (\frac{b \sqrt{c} x}{\sqrt{-b c x + a c} \sqrt{b x + a}}\right )}{b \sqrt{c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b*c*x + a*c)*sqrt(b*x + a)),x, algorithm="fricas")
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Sympy [A] time = 10.8277, size = 90, normalized size = 2.31 \[ - \frac{i{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b \sqrt{c}} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(1/2)/(-b*c*x+a*c)**(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*c*x + a*c)*sqrt(b*x + a)),x, algorithm="giac")
[Out]