3.1756 \(\int \frac{\sqrt{a+\frac{b}{x}}}{x^{5/2}} \, dx\)

Optimal. Leaf size=80 \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{2 x^{3/2}}-\frac{a \sqrt{a+\frac{b}{x}}}{4 b \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.116938, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{2 x^{3/2}}-\frac{a \sqrt{a+\frac{b}{x}}}{4 b \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.6502, size = 63, normalized size = 0.79 \[ \frac{a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{4 b^{\frac{3}{2}}} - \frac{a \sqrt{a + \frac{b}{x}}}{4 b \sqrt{x}} - \frac{\sqrt{a + \frac{b}{x}}}{2 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.222108, size = 77, normalized size = 0.96 \[ \frac{2 a^2 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )-a^2 \log (x)-\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} (a x+2 b)}{x^{3/2}}}{8 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.023, size = 73, normalized size = 0.9 \[ -{\frac{1}{4}\sqrt{{\frac{ax+b}{x}}} \left ( -{\it Artanh} \left ({1\sqrt{ax+b}{\frac{1}{\sqrt{b}}}} \right ){a}^{2}{x}^{2}+2\,{b}^{3/2}\sqrt{ax+b}+xa\sqrt{ax+b}\sqrt{b} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ax+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)/x^(5/2),x)

[Out]

-1/4*((a*x+b)/x)^(1/2)*(-arctanh((a*x+b)^(1/2)/b^(1/2))*a^2*x^2+2*b^(3/2)*(a*x+b
)^(1/2)+x*a*(a*x+b)^(1/2)*b^(1/2))/x^(3/2)/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(sqrt(a + b/x)/x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24984, size = 1, normalized size = 0.01 \[ \left [\frac{a^{2} x^{2} \log \left (\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} +{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) - 2 \,{\left (a x + 2 \, b\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{8 \, b^{\frac{3}{2}} x^{2}}, -\frac{a^{2} x^{2} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (a x + 2 \, b\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{4 \, \sqrt{-b} b x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 101.326, size = 97, normalized size = 1.21 \[ - \frac{a^{\frac{3}{2}}}{4 b \sqrt{x} \sqrt{1 + \frac{b}{a x}}} - \frac{3 \sqrt{a}}{4 x^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}} + \frac{a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{4 b^{\frac{3}{2}}} - \frac{b}{2 \sqrt{a} x^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272548, size = 78, normalized size = 0.98 \[ -\frac{1}{4} \, a^{2}{\left (\frac{\arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{{\left (a x + b\right )}^{\frac{3}{2}} + \sqrt{a x + b} b}{a^{2} b x^{2}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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