3.1702 \(\int \left (a+\frac{b}{x}\right )^{3/2} \, dx\)

Optimal. Leaf size=54 \[ x \left (a+\frac{b}{x}\right )^{3/2}-3 b \sqrt{a+\frac{b}{x}}+3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0839546, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ x \left (a+\frac{b}{x}\right )^{3/2}-3 b \sqrt{a+\frac{b}{x}}+3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.35797, size = 44, normalized size = 0.81 \[ 3 \sqrt{a} b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - 3 b \sqrt{a + \frac{b}{x}} + x \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0381999, size = 56, normalized size = 1.04 \[ \sqrt{a+\frac{b}{x}} (a x-2 b)+\frac{3}{2} \sqrt{a} b \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(3/2),x]

[Out]

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

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Maple [B]  time = 0.011, size = 94, normalized size = 1.7 \[{\frac{1}{2\,x}\sqrt{{\frac{ax+b}{x}}} \left ( 3\,\sqrt{a}b\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}+6\,a\sqrt{a{x}^{2}+bx}{x}^{2}-4\, \left ( a{x}^{2}+bx \right ) ^{3/2} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2),x)

[Out]

1/2*((a*x+b)/x)^(1/2)/x*(3*a^(1/2)*b*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b
)/a^(1/2))*x^2+6*a*(a*x^2+b*x)^(1/2)*x^2-4*(a*x^2+b*x)^(3/2))/(x*(a*x+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((a + b/x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235524, size = 1, normalized size = 0.02 \[ \left [\frac{3}{2} \, \sqrt{a} b \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) +{\left (a x - 2 \, b\right )} \sqrt{\frac{a x + b}{x}}, 3 \, \sqrt{-a} b \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) +{\left (a x - 2 \, b\right )} \sqrt{\frac{a x + b}{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.99481, size = 92, normalized size = 1.7 \[ 3 \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} + \frac{a^{2} x^{\frac{3}{2}}}{\sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{a \sqrt{b} \sqrt{x}}{\sqrt{\frac{a x}{b} + 1}} - \frac{2 b^{\frac{3}{2}}}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**(3/2),x)

[Out]

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

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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, algorithm="giac")

[Out]

Exception raised: TypeError