3.1722 \(\int \frac{x}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=72 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{5/2}}-\frac{3 b x \sqrt{a+\frac{b}{x}}}{4 a^2}+\frac{x^2 \sqrt{a+\frac{b}{x}}}{2 a} \]

[Out]

(-3*b*Sqrt[a + b/x]*x)/(4*a^2) + (Sqrt[a + b/x]*x^2)/(2*a) + (3*b^2*ArcTanh[Sqrt
[a + b/x]/Sqrt[a]])/(4*a^(5/2))

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Rubi [A]  time = 0.0933906, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{5/2}}-\frac{3 b x \sqrt{a+\frac{b}{x}}}{4 a^2}+\frac{x^2 \sqrt{a+\frac{b}{x}}}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a + b/x],x]

[Out]

(-3*b*Sqrt[a + b/x]*x)/(4*a^2) + (Sqrt[a + b/x]*x^2)/(2*a) + (3*b^2*ArcTanh[Sqrt
[a + b/x]/Sqrt[a]])/(4*a^(5/2))

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Rubi in Sympy [A]  time = 9.01738, size = 60, normalized size = 0.83 \[ \frac{x^{2} \sqrt{a + \frac{b}{x}}}{2 a} - \frac{3 b x \sqrt{a + \frac{b}{x}}}{4 a^{2}} + \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b/x)**(1/2),x)

[Out]

x**2*sqrt(a + b/x)/(2*a) - 3*b*x*sqrt(a + b/x)/(4*a**2) + 3*b**2*atanh(sqrt(a +
b/x)/sqrt(a))/(4*a**(5/2))

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Mathematica [A]  time = 0.099932, size = 66, normalized size = 0.92 \[ \frac{3 b^2 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{8 a^{5/2}}+\frac{x \sqrt{a+\frac{b}{x}} (2 a x-3 b)}{4 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a + b/x],x]

[Out]

(Sqrt[a + b/x]*x*(-3*b + 2*a*x))/(4*a^2) + (3*b^2*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt
[a + b/x]*x])/(8*a^(5/2))

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Maple [B]  time = 0.013, size = 146, normalized size = 2. \[ -{\frac{x}{8}\sqrt{{\frac{ax+b}{x}}} \left ( -4\,\sqrt{a{x}^{2}+bx}x{a}^{7/2}-2\,\sqrt{a{x}^{2}+bx}b{a}^{5/2}+8\,b\sqrt{x \left ( ax+b \right ) }{a}^{5/2}+{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){a}^{2}-4\,{b}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{2} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x)^(1/2),x)

[Out]

-1/8*((a*x+b)/x)^(1/2)*x*(-4*(a*x^2+b*x)^(1/2)*x*a^(7/2)-2*(a*x^2+b*x)^(1/2)*b*a
^(5/2)+8*b*(x*(a*x+b))^(1/2)*a^(5/2)+b^2*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a
*x+b)/a^(1/2))*a^2-4*b^2*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a
^2)/(x*(a*x+b))^(1/2)/a^(9/2)

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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(x/sqrt(a + b/x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240279, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (2 \, a x^{2} - 3 \, b x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{8 \, a^{\frac{5}{2}}}, -\frac{3 \, b^{2} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) -{\left (2 \, a x^{2} - 3 \, b x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{4 \, \sqrt{-a} a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(a + b/x),x, algorithm="fricas")

[Out]

[1/8*(3*b^2*log(2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) + 2*(2*a*x^2 - 3*
b*x)*sqrt(a)*sqrt((a*x + b)/x))/a^(5/2), -1/4*(3*b^2*arctan(a/(sqrt(-a)*sqrt((a*
x + b)/x))) - (2*a*x^2 - 3*b*x)*sqrt(-a)*sqrt((a*x + b)/x))/(sqrt(-a)*a^2)]

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Sympy [A]  time = 13.2237, size = 100, normalized size = 1.39 \[ \frac{x^{\frac{5}{2}}}{2 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{\sqrt{b} x^{\frac{3}{2}}}{4 a \sqrt{\frac{a x}{b} + 1}} - \frac{3 b^{\frac{3}{2}} \sqrt{x}}{4 a^{2} \sqrt{\frac{a x}{b} + 1}} + \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b/x)**(1/2),x)

[Out]

x**(5/2)/(2*sqrt(b)*sqrt(a*x/b + 1)) - sqrt(b)*x**(3/2)/(4*a*sqrt(a*x/b + 1)) -
3*b**(3/2)*sqrt(x)/(4*a**2*sqrt(a*x/b + 1)) + 3*b**2*asinh(sqrt(a)*sqrt(x)/sqrt(
b))/(4*a**(5/2))

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GIAC/XCAS [A]  time = 0.25098, size = 120, normalized size = 1.67 \[ -\frac{1}{4} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} - \frac{5 \, a \sqrt{\frac{a x + b}{x}} - \frac{3 \,{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}}{{\left (a - \frac{a x + b}{x}\right )}^{2} a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(a + b/x),x, algorithm="giac")

[Out]

-1/4*b^2*(3*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^2) - (5*a*sqrt((a*x +
b)/x) - 3*(a*x + b)*sqrt((a*x + b)/x)/x)/((a - (a*x + b)/x)^2*a^2))