3.1668 \(\int \frac{1}{\left (a+\frac{b}{x}\right ) x^{5/2}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{2}{b \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0493267, antiderivative size = 40, 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 \[ -\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{2}{b \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)*x^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.46567, size = 37, normalized size = 0.92 \[ - \frac{2 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{3}{2}}} - \frac{2}{b \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0233428, size = 40, normalized size = 1. \[ -\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{3/2}}-\frac{2}{b \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 32, normalized size = 0.8 \[ -2\,{\frac{a}{b\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) }-2\,{\frac{1}{b\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)/x^(5/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24552, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{x} \sqrt{-\frac{a}{b}} \log \left (\frac{a x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right ) - 2}{b \sqrt{x}}, \frac{2 \,{\left (\sqrt{x} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) - 1\right )}}{b \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 30.4701, size = 102, normalized size = 2.55 \[ \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{3 a x^{\frac{3}{2}}} & \text{for}\: b = 0 \\- \frac{2}{b \sqrt{x}} & \text{for}\: a = 0 \\- \frac{2}{b \sqrt{x}} + \frac{i \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{i \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)/x**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220667, size = 42, normalized size = 1.05 \[ -\frac{2 \, a \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b} - \frac{2}{b \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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