3.1703 \(\int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{x} \, dx\)

Optimal. Leaf size=55 \[ 2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 a \sqrt{a+\frac{b}{x}}-\frac{2}{3} \left (a+\frac{b}{x}\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0865256, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ 2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 a \sqrt{a+\frac{b}{x}}-\frac{2}{3} \left (a+\frac{b}{x}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.54465, size = 44, normalized size = 0.8 \[ 2 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - 2 a \sqrt{a + \frac{b}{x}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0855327, size = 57, normalized size = 1.04 \[ a^{3/2} \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )-\frac{2 \sqrt{a+\frac{b}{x}} (4 a x+b)}{3 x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(3/2)/x,x]

[Out]

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

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Maple [B]  time = 0.016, size = 115, normalized size = 2.1 \[{\frac{1}{3\,b{x}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 3\,{a}^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}b+6\,{a}^{2}\sqrt{a{x}^{2}+bx}{x}^{3}-6\,a \left ( a{x}^{2}+bx \right ) ^{3/2}x-2\, \left ( a{x}^{2}+bx \right ) ^{3/2}b \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)/x,x)

[Out]

1/3*((a*x+b)/x)^(1/2)/x^2*(3*a^(3/2)*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b
)/a^(1/2))*x^3*b+6*a^2*(a*x^2+b*x)^(1/2)*x^3-6*a*(a*x^2+b*x)^(3/2)*x-2*(a*x^2+b*
x)^(3/2)*b)/(x*(a*x+b))^(1/2)/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((a + b/x)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237224, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a^{\frac{3}{2}} x \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \,{\left (4 \, a x + b\right )} \sqrt{\frac{a x + b}{x}}}{3 \, x}, \frac{2 \,{\left (3 \, \sqrt{-a} a x \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) -{\left (4 \, a x + b\right )} \sqrt{\frac{a x + b}{x}}\right )}}{3 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.53769, size = 71, normalized size = 1.29 \[ - \frac{8 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}}{3} - a^{\frac{3}{2}} \log{\left (\frac{b}{a x} \right )} + 2 a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )} - \frac{2 \sqrt{a} b \sqrt{1 + \frac{b}{a x}}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**(3/2)/x,x)

[Out]

-8*a**(3/2)*sqrt(1 + b/(a*x))/3 - a**(3/2)*log(b/(a*x)) + 2*a**(3/2)*log(sqrt(1
+ b/(a*x)) + 1) - 2*sqrt(a)*b*sqrt(1 + b/(a*x))/(3*x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError