3.1667 \(\int \frac{1}{\left (a+\frac{b}{x}\right ) x^{3/2}} \, dx\)

Optimal. Leaf size=29 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b}} \]

[Out]

(2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(Sqrt[a]*Sqrt[b])

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Rubi [A]  time = 0.0357949, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)*x^(3/2)),x]

[Out]

(2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(Sqrt[a]*Sqrt[b])

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Rubi in Sympy [A]  time = 6.27091, size = 27, normalized size = 0.93 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{\sqrt{a} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atan(sqrt(a)*sqrt(x)/sqrt(b))/(sqrt(a)*sqrt(b))

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Mathematica [A]  time = 0.00949198, size = 29, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(Sqrt[a]*Sqrt[b])

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Maple [A]  time = 0.007, size = 19, normalized size = 0.7 \[ 2\,{\frac{1}{\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)/x^(3/2),x)

[Out]

2/(a*b)^(1/2)*arctan(a*x^(1/2)/(a*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/((a + b/x)*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244768, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (a x - b\right )}}{a x + b}\right )}{\sqrt{-a b}}, -\frac{2 \, \arctan \left (\frac{b}{\sqrt{a b} \sqrt{x}}\right )}{\sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.6707, size = 94, normalized size = 3.24 \[ \begin{cases} \tilde{\infty } \sqrt{x} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{a \sqrt{x}} & \text{for}\: b = 0 \\\frac{2 \sqrt{x}}{b} & \text{for}\: a = 0 \\- \frac{i \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a \sqrt{b} \sqrt{\frac{1}{a}}} + \frac{i \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a \sqrt{b} \sqrt{\frac{1}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)/x**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.222967, size = 24, normalized size = 0.83 \[ \frac{2 \, \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(a*sqrt(x)/sqrt(a*b))/sqrt(a*b)