3.1733 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x} \, dx\)

Optimal. Leaf size=42 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{a+\frac{b}{x}}} \]

[Out]

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

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Rubi [A]  time = 0.0714656, antiderivative size = 42, 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 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(3/2)*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.9679, size = 32, normalized size = 0.76 \[ - \frac{2}{a \sqrt{a + \frac{b}{x}}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0763326, size = 57, normalized size = 1.36 \[ \frac{\log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{a^{3/2}}-\frac{2 x \sqrt{a+\frac{b}{x}}}{a (a x+b)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 198, normalized size = 4.7 \[{\frac{x}{b \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( \ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}{a}^{2}b-2\,{a}^{5/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}+2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) xa{b}^{2}+2\,{a}^{3/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}-4\,{a}^{3/2}\sqrt{x \left ( ax+b \right ) }xb+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){b}^{3}-2\,\sqrt{a}\sqrt{x \left ( ax+b \right ) }{b}^{2} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(3/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248218, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\frac{a x + b}{x}} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) - 2 \, \sqrt{a}}{a^{\frac{3}{2}} \sqrt{\frac{a x + b}{x}}}, -\frac{2 \,{\left (\sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) + \sqrt{-a}\right )}}{\sqrt{-a} a \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.63779, size = 148, normalized size = 3.52 \[ - \frac{2 a^{3} x \sqrt{1 + \frac{b}{a x}}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} - \frac{a^{3} x \log{\left (\frac{b}{a x} \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} + \frac{2 a^{3} x \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x} \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(3/2)/x,x)

[Out]

-2*a**3*x*sqrt(1 + b/(a*x))/(a**(9/2)*x + a**(7/2)*b) - a**3*x*log(b/(a*x))/(a**
(9/2)*x + a**(7/2)*b) + 2*a**3*x*log(sqrt(1 + b/(a*x)) + 1)/(a**(9/2)*x + a**(7/
2)*b) - a**2*b*log(b/(a*x))/(a**(9/2)*x + a**(7/2)*b) + 2*a**2*b*log(sqrt(1 + b/
(a*x)) + 1)/(a**(9/2)*x + a**(7/2)*b)

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GIAC/XCAS [A]  time = 0.263957, size = 70, normalized size = 1.67 \[ -2 \, b{\left (\frac{\arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b} + \frac{1}{a b \sqrt{\frac{a x + b}{x}}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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