3.1675 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^{3/2}} \, dx\)

Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2} \sqrt{b}}-\frac{\sqrt{x}}{a (a x+b)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0513336, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2} \sqrt{b}}-\frac{\sqrt{x}}{a (a x+b)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x^(3/2)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.833, size = 37, normalized size = 0.8 \[ - \frac{\sqrt{x}}{a \left (a x + b\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0360141, size = 46, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2} \sqrt{b}}-\frac{\sqrt{x}}{a (a x+b)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 37, normalized size = 0.8 \[ -{\frac{1}{a \left ( ax+b \right ) }\sqrt{x}}+{\frac{1}{a}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^(3/2),x)

[Out]

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

_______________________________________________________________________________________

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/((a + b/x)^2*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.238939, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (a x + b\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (a x - b\right )}}{a x + b}\right ) - 2 \, \sqrt{-a b} \sqrt{x}}{2 \,{\left (a^{2} x + a b\right )} \sqrt{-a b}}, -\frac{{\left (a x + b\right )} \arctan \left (\frac{b}{\sqrt{a b} \sqrt{x}}\right ) + \sqrt{a b} \sqrt{x}}{{\left (a^{2} x + a b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^(3/2)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 84.2155, size = 337, normalized size = 7.33 \[ \begin{cases} \tilde{\infty } x^{\frac{3}{2}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 x^{\frac{3}{2}}}{3 b^{2}} & \text{for}\: a = 0 \\- \frac{2}{a^{2} \sqrt{x}} & \text{for}\: b = 0 \\- \frac{2 i a \sqrt{b} \sqrt{x} \sqrt{\frac{1}{a}}}{2 i a^{3} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{2} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{a x \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{2 i a^{3} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{2} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{a x \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{2 i a^{3} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{2} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{b \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{2 i a^{3} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{2} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{b \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{2 i a^{3} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{2} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo*x**(3/2), Eq(a, 0) & Eq(b, 0)), (2*x**(3/2)/(3*b**2), Eq(a, 0)),
(-2/(a**2*sqrt(x)), Eq(b, 0)), (-2*I*a*sqrt(b)*sqrt(x)*sqrt(1/a)/(2*I*a**3*sqrt(
b)*x*sqrt(1/a) + 2*I*a**2*b**(3/2)*sqrt(1/a)) + a*x*log(-I*sqrt(b)*sqrt(1/a) + s
qrt(x))/(2*I*a**3*sqrt(b)*x*sqrt(1/a) + 2*I*a**2*b**(3/2)*sqrt(1/a)) - a*x*log(I
*sqrt(b)*sqrt(1/a) + sqrt(x))/(2*I*a**3*sqrt(b)*x*sqrt(1/a) + 2*I*a**2*b**(3/2)*
sqrt(1/a)) + b*log(-I*sqrt(b)*sqrt(1/a) + sqrt(x))/(2*I*a**3*sqrt(b)*x*sqrt(1/a)
+ 2*I*a**2*b**(3/2)*sqrt(1/a)) - b*log(I*sqrt(b)*sqrt(1/a) + sqrt(x))/(2*I*a**3
*sqrt(b)*x*sqrt(1/a) + 2*I*a**2*b**(3/2)*sqrt(1/a)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.232223, size = 49, normalized size = 1.07 \[ \frac{\arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{\sqrt{x}}{{\left (a x + b\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^(3/2)),x, algorithm="giac")

[Out]

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