Optimal. Leaf size=69 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{3/2}}+\frac{1}{2} x^2 \sqrt{a+\frac{b}{x}}+\frac{b x \sqrt{a+\frac{b}{x}}}{4 a} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0903853, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{3/2}}+\frac{1}{2} x^2 \sqrt{a+\frac{b}{x}}+\frac{b x \sqrt{a+\frac{b}{x}}}{4 a} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b/x]*x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.8529, size = 53, normalized size = 0.77 \[ \frac{x^{2} \sqrt{a + \frac{b}{x}}}{2} + \frac{b x \sqrt{a + \frac{b}{x}}}{4 a} - \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(a+b/x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0963773, size = 64, normalized size = 0.93 \[ \frac{x \sqrt{a+\frac{b}{x}} (2 a x+b)}{4 a}-\frac{b^2 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{8 a^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x]*x,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 96, normalized size = 1.4 \[{\frac{x}{8}\sqrt{{\frac{ax+b}{x}}} \left ( 4\,\sqrt{a{x}^{2}+bx}{a}^{5/2}x+2\,\sqrt{a{x}^{2}+bx}{a}^{3/2}b-{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) a \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(a+b/x)^(1/2),x)
[Out]
_______________________________________________________________________________________
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(sqrt(a + b/x)*x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.241606, size = 1, normalized size = 0.01 \[ \left [\frac{b^{2} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (2 \, a x^{2} + b x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{8 \, a^{\frac{3}{2}}}, \frac{b^{2} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (2 \, a x^{2} + b x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{4 \, \sqrt{-a} a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x)*x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 12.1334, size = 97, normalized size = 1.41 \[ \frac{a x^{\frac{5}{2}}}{2 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{3 \sqrt{b} x^{\frac{3}{2}}}{4 \sqrt{\frac{a x}{b} + 1}} + \frac{b^{\frac{3}{2}} \sqrt{x}}{4 a \sqrt{\frac{a x}{b} + 1}} - \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(a+b/x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.244618, size = 105, normalized size = 1.52 \[ -\frac{b^{2}{\rm ln}\left ({\left | b \right |}\right ){\rm sign}\left (x\right )}{8 \, a^{\frac{3}{2}}} + \frac{1}{8} \,{\left (2 \, \sqrt{a x^{2} + b x}{\left (2 \, x + \frac{b}{a}\right )} + \frac{b^{2}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right )}{a^{\frac{3}{2}}}\right )}{\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x)*x,x, algorithm="giac")
[Out]