3.1763 \(\int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=69 \[ 2 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}-\frac{3 b \sqrt{a+\frac{b}{x}}}{\sqrt{x}}-3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0947284, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ 2 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}-\frac{3 b \sqrt{a+\frac{b}{x}}}{\sqrt{x}}-3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2)/Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 8.35803, size = 60, normalized size = 0.87 \[ - 3 a \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )} - \frac{3 b \sqrt{a + \frac{b}{x}}}{\sqrt{x}} + 2 \sqrt{x} \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.180888, size = 71, normalized size = 1.03 \[ \frac{\sqrt{a+\frac{b}{x}} (2 a x-b)}{\sqrt{x}}-3 a \sqrt{b} \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+\frac{3}{2} a \sqrt{b} \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 70, normalized size = 1. \[ -{1\sqrt{{\frac{ax+b}{x}}} \left ({b}^{{\frac{3}{2}}}\sqrt{ax+b}-2\,xa\sqrt{ax+b}\sqrt{b}+3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) xab \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{ax+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)/x^(1/2),x)

[Out]

-((a*x+b)/x)^(1/2)*(b^(3/2)*(a*x+b)^(1/2)-2*x*a*(a*x+b)^(1/2)*b^(1/2)+3*arctanh(
(a*x+b)^(1/2)/b^(1/2))*x*a*b)/x^(1/2)/(a*x+b)^(1/2)/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 = 0.244022, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a \sqrt{b} x \log \left (\frac{a x - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) + 2 \,{\left (2 \, a x - b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{2 \, x}, -\frac{3 \, a \sqrt{-b} x \arctan \left (\frac{\sqrt{x} \sqrt{\frac{a x + b}{x}}}{\sqrt{-b}}\right ) -{\left (2 \, a x - b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 48.0271, size = 92, normalized size = 1.33 \[ \frac{2 a^{\frac{3}{2}} \sqrt{x}}{\sqrt{1 + \frac{b}{a x}}} + \frac{\sqrt{a} b}{\sqrt{x} \sqrt{1 + \frac{b}{a x}}} - 3 a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )} - \frac{b^{2}}{\sqrt{a} x^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.26841, size = 68, normalized size = 0.99 \[{\left (\frac{3 \, b \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \, \sqrt{a x + b} - \frac{\sqrt{a x + b} b}{a x}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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