3.1781 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^{5/2}} \, dx\)

Optimal. Leaf size=52 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{b \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0790806, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{b \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x]*x^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.20154, size = 41, normalized size = 0.79 \[ \frac{a \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{b^{\frac{3}{2}}} - \frac{\sqrt{a + \frac{b}{x}}}{b \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(1/2)/x**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.117005, size = 64, normalized size = 1.23 \[ \frac{-\frac{\sqrt{b} \sqrt{a+\frac{b}{x}}}{\sqrt{x}}+a \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )-\frac{1}{2} a \log (x)}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b/x]*x^(5/2)),x]

[Out]

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

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Maple [A]  time = 0.017, size = 55, normalized size = 1.1 \[ -{1\sqrt{{\frac{ax+b}{x}}} \left ( -{\it Artanh} \left ({1\sqrt{ax+b}{\frac{1}{\sqrt{b}}}} \right ) ax+\sqrt{ax+b}\sqrt{b} \right ){\frac{1}{\sqrt{x}}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ax+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(1/2)/x^(5/2),x)

[Out]

-((a*x+b)/x)^(1/2)*(-arctanh((a*x+b)^(1/2)/b^(1/2))*a*x+(a*x+b)^(1/2)*b^(1/2))/x
^(1/2)/b^(3/2)/(a*x+b)^(1/2)

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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(1/(sqrt(a + b/x)*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251408, size = 1, normalized size = 0.02 \[ \left [\frac{a x \log \left (\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} +{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{2 \, b^{\frac{3}{2}} x}, -\frac{a x \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) + \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{\sqrt{-b} b x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^(5/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 147.587, size = 44, normalized size = 0.85 \[ - \frac{\sqrt{a} \sqrt{1 + \frac{b}{a x}}}{b \sqrt{x}} + \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(1/2)/x**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.250556, size = 59, normalized size = 1.13 \[ -a{\left (\frac{\arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{\sqrt{a x + b}}{a b x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^(5/2)),x, algorithm="giac")

[Out]

-a*(arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b) + sqrt(a*x + b)/(a*b*x))