3.156 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2}} \, dx\)

Optimal. Leaf size=61 \[ -\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{3 x \sqrt{a+\frac{b}{x}}}{a^2}-\frac{2 x}{a \sqrt{a+\frac{b}{x}}} \]

[Out]

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

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Rubi [A]  time = 0.0834458, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{3 x \sqrt{a+\frac{b}{x}}}{a^2}-\frac{2 x}{a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(-3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.025, size = 51, normalized size = 0.84 \[ - \frac{2 x}{a \sqrt{a + \frac{b}{x}}} + \frac{3 x \sqrt{a + \frac{b}{x}}}{a^{2}} - \frac{3 b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.100055, size = 67, normalized size = 1.1 \[ \frac{x \sqrt{a+\frac{b}{x}} (a x+3 b)}{a^2 (a x+b)}-\frac{3 b \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(-3/2),x]

[Out]

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

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Maple [B]  time = 0.005, size = 203, normalized size = 3.3 \[ -{\frac{x}{2\, \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( -6\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}+4\,{a}^{7/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}-12\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }xb+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{4}b-6\,{a}^{5/2}\sqrt{x \left ( ax+b \right ) }{b}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{3}{b}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{2}{b}^{3} \right ){a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(3/2),x)

[Out]

-1/2*((a*x+b)/x)^(1/2)*x/a^(9/2)*(-6*a^(9/2)*(x*(a*x+b))^(1/2)*x^2+4*a^(7/2)*(x*
(a*x+b))^(3/2)-12*a^(7/2)*(x*(a*x+b))^(1/2)*x*b+3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^
(1/2)+2*a*x+b)/a^(1/2))*x^2*a^4*b-6*a^(5/2)*(x*(a*x+b))^(1/2)*b^2+6*ln(1/2*(2*(x
*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a^3*b^2+3*ln(1/2*(2*(x*(a*x+b))^(1/2
)*a^(1/2)+2*a*x+b)/a^(1/2))*a^2*b^3)/(x*(a*x+b))^(1/2)/(a*x+b)^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((a + b/x)^(-3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251474, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b \sqrt{\frac{a x + b}{x}} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (a x + 3 \, b\right )} \sqrt{a}}{2 \, a^{\frac{5}{2}} \sqrt{\frac{a x + b}{x}}}, \frac{3 \, b \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (a x + 3 \, b\right )} \sqrt{-a}}{\sqrt{-a} a^{2} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(-3/2),x, algorithm="fricas")

[Out]

[1/2*(3*b*sqrt((a*x + b)/x)*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a))
+ 2*(a*x + 3*b)*sqrt(a))/(a^(5/2)*sqrt((a*x + b)/x)), (3*b*sqrt((a*x + b)/x)*arc
tan(a/(sqrt(-a)*sqrt((a*x + b)/x))) + (a*x + 3*b)*sqrt(-a))/(sqrt(-a)*a^2*sqrt((
a*x + b)/x))]

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Sympy [A]  time = 11.4526, size = 71, normalized size = 1.16 \[ \frac{x^{\frac{3}{2}}}{a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{3 \sqrt{b} \sqrt{x}}{a^{2} \sqrt{\frac{a x}{b} + 1}} - \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.250133, size = 116, normalized size = 1.9 \[ b{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{2 \, a - \frac{3 \,{\left (a x + b\right )}}{x}}{{\left (a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )} a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(-3/2),x, algorithm="giac")

[Out]

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