3.1789 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^{5/2}} \, dx\)

Optimal. Leaf size=52 \[ \frac{2}{b \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0778538, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{2}{b \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.1495, size = 42, normalized size = 0.81 \[ \frac{2}{b \sqrt{x} \sqrt{a + \frac{b}{x}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.146486, size = 66, normalized size = 1.27 \[ \frac{\frac{2 \sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}}{a x+b}-2 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+\log (x)}{b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 52, normalized size = 1. \[ 2\,{\frac{\sqrt{x}}{{b}^{3/2} \left ( ax+b \right ) }\sqrt{{\frac{ax+b}{x}}} \left ( -{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}+\sqrt{b} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(3/2)/x^(5/2),x)

[Out]

2*((a*x+b)/x)^(1/2)*x^(1/2)*(-arctanh((a*x+b)^(1/2)/b^(1/2))*(a*x+b)^(1/2)+b^(1/
2))/b^(3/2)/(a*x+b)

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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)^(3/2)*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250448, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{x} \sqrt{\frac{a x + b}{x}} \log \left (-\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} -{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) + 2 \, \sqrt{b}}{b^{\frac{3}{2}} \sqrt{x} \sqrt{\frac{a x + b}{x}}}, \frac{2 \,{\left (\sqrt{x} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) + \sqrt{-b}\right )}}{\sqrt{-b} b \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(sqrt(x)*sqrt((a*x + b)/x)*log(-(2*b*sqrt(x)*sqrt((a*x + b)/x) - (a*x + 2*b)*sq
rt(b))/x) + 2*sqrt(b))/(b^(3/2)*sqrt(x)*sqrt((a*x + b)/x)), 2*(sqrt(x)*sqrt((a*x
+ b)/x)*arctan(b/(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x))) + sqrt(-b))/(sqrt(-b)*b*
sqrt(x)*sqrt((a*x + b)/x))]

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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)**(3/2)/x**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228967, size = 90, normalized size = 1.73 \[ \frac{2 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{2 \,{\left (\sqrt{b} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b}\right )}}{\sqrt{-b} b^{\frac{3}{2}}} + \frac{2}{\sqrt{a x + b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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