3.1762 \(\int \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x} \, dx\)

Optimal. Leaf size=70 \[ -2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )+\frac{2}{3} x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}+2 b \sqrt{x} \sqrt{a+\frac{b}{x}} \]

[Out]

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

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Rubi [A]  time = 0.10432, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )+\frac{2}{3} x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}+2 b \sqrt{x} \sqrt{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2)*Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 10.0489, size = 60, normalized size = 0.86 \[ - 2 b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )} + 2 b \sqrt{x} \sqrt{a + \frac{b}{x}} + \frac{2 x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.11073, size = 68, normalized size = 0.97 \[ -2 b^{3/2} \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+\frac{2}{3} \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+4 b)+b^{3/2} \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 63, normalized size = 0.9 \[ -{\frac{2}{3}\sqrt{{\frac{ax+b}{x}}}\sqrt{x} \left ( 3\,{b}^{3/2}{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) -xa\sqrt{ax+b}-4\,\sqrt{ax+b}b \right ){\frac{1}{\sqrt{ax+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)*x^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254045, size = 1, normalized size = 0.01 \[ \left [b^{\frac{3}{2}} \log \left (\frac{a x - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) + \frac{2}{3} \,{\left (a x + 4 \, b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}, -2 \, \sqrt{-b} b \arctan \left (\frac{\sqrt{x} \sqrt{\frac{a x + b}{x}}}{\sqrt{-b}}\right ) + \frac{2}{3} \,{\left (a x + 4 \, b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 94.2167, size = 71, normalized size = 1.01 \[ \frac{2 a \sqrt{b} x \sqrt{\frac{a x}{b} + 1}}{3} + \frac{8 b^{\frac{3}{2}} \sqrt{\frac{a x}{b} + 1}}{3} + b^{\frac{3}{2}} \log{\left (\frac{a x}{b} \right )} - 2 b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a x}{b} + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.238729, size = 59, normalized size = 0.84 \[ \frac{2 \, b^{2} \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + \frac{2}{3} \,{\left (a x + b\right )}^{\frac{3}{2}} + 2 \, \sqrt{a x + b} b \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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