3.1684 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^{5/2}} \, dx\)

Optimal. Leaf size=73 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2}}+\frac{\sqrt{x}}{4 a b (a x+b)}-\frac{\sqrt{x}}{2 a (a x+b)^2} \]

[Out]

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

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Rubi [A]  time = 0.0694629, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2}}+\frac{\sqrt{x}}{4 a b (a x+b)}-\frac{\sqrt{x}}{2 a (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.2391, size = 58, normalized size = 0.79 \[ - \frac{\sqrt{x}}{2 a \left (a x + b\right )^{2}} + \frac{\sqrt{x}}{4 a b \left (a x + b\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 a^{\frac{3}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**3/x**(5/2),x)

[Out]

-sqrt(x)/(2*a*(a*x + b)**2) + sqrt(x)/(4*a*b*(a*x + b)) + atan(sqrt(a)*sqrt(x)/s
qrt(b))/(4*a**(3/2)*b**(3/2))

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Mathematica [A]  time = 0.059393, size = 62, normalized size = 0.85 \[ \frac{\frac{\sqrt{a} \sqrt{b} \sqrt{x} (a x-b)}{(a x+b)^2}+\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^3*x^(5/2)),x]

[Out]

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

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Maple [A]  time = 0.016, size = 52, normalized size = 0.7 \[ 2\,{\frac{1}{ \left ( ax+b \right ) ^{2}} \left ( 1/8\,{\frac{{x}^{3/2}}{b}}-1/8\,{\frac{\sqrt{x}}{a}} \right ) }+{\frac{1}{4\,ab}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^(5/2),x)

[Out]

2*(1/8*x^(3/2)/b-1/8*x^(1/2)/a)/(a*x+b)^2+1/4/b/a/(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)^3*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244173, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{-a b}{\left (a x - b\right )} \sqrt{x} +{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (a x - b\right )}}{a x + b}\right )}{8 \,{\left (a^{3} b x^{2} + 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt{-a b}}, \frac{\sqrt{a b}{\left (a x - b\right )} \sqrt{x} -{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \arctan \left (\frac{b}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (a^{3} b x^{2} + 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(5/2)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**3/x**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217877, size = 70, normalized size = 0.96 \[ \frac{\arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a b} + \frac{a x^{\frac{3}{2}} - b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(5/2)),x, algorithm="giac")

[Out]

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