Optimal. Leaf size=51 \[ \frac{c x \sqrt{a+\frac{b}{x}}}{a}-\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.109843, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{c x \sqrt{a+\frac{b}{x}}}{a}-\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(c + d/x)/Sqrt[a + b/x],x]
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Rubi in Sympy [A] time = 9.44179, size = 42, normalized size = 0.82 \[ \frac{c x \sqrt{a + \frac{b}{x}}}{a} + \frac{2 \left (a d - \frac{b c}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c+d/x)/(a+b/x)**(1/2),x)
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Mathematica [A] time = 0.0753147, size = 62, normalized size = 1.22 \[ \frac{(2 a d-b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{3/2}}+\frac{c x \sqrt{a+\frac{b}{x}}}{a} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d/x)/Sqrt[a + b/x],x]
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Maple [B] time = 0.017, size = 179, normalized size = 3.5 \[ -{\frac{x}{2\,b}\sqrt{{\frac{ax+b}{x}}} \left ( 2\,\sqrt{x \left ( ax+b \right ) }d{a}^{5/2}-2\,\sqrt{x \left ( ax+b \right ) }cb{a}^{3/2}-2\,d\sqrt{a{x}^{2}+bx}{a}^{5/2}-d\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) b{a}^{2}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) db{a}^{2}+{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) ca \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c+d/x)/(a+b/x)^(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((c + d/x)/sqrt(a + b/x),x, algorithm="maxima")
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Fricas [A] time = 0.249634, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{a} c x \sqrt{\frac{a x + b}{x}} -{\left (b c - 2 \, a d\right )} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right )}{2 \, a^{\frac{3}{2}}}, \frac{\sqrt{-a} c x \sqrt{\frac{a x + b}{x}} +{\left (b c - 2 \, a d\right )} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right )}{\sqrt{-a} a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c + d/x)/sqrt(a + b/x),x, algorithm="fricas")
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Sympy [A] time = 25.0676, size = 73, normalized size = 1.43 \[ \frac{\sqrt{b} c \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{a} + \frac{2 d \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{\sqrt{a}} - \frac{b c \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c+d/x)/(a+b/x)**(1/2),x)
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GIAC/XCAS [A] time = 0.250741, size = 99, normalized size = 1.94 \[ -b{\left (\frac{c \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a} - \frac{{\left (b c - 2 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c + d/x)/sqrt(a + b/x),x, algorithm="giac")
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