3.1764 \(\int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{x^{3/2}} \, dx\)

Optimal. Leaf size=77 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x}}}{4 \sqrt{x}}-\frac{\left (a+\frac{b}{x}\right )^{3/2}}{2 \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0864968, antiderivative size = 77, 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 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x}}}{4 \sqrt{x}}-\frac{\left (a+\frac{b}{x}\right )^{3/2}}{2 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2)/x^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.99027, size = 66, normalized size = 0.86 \[ - \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{4 \sqrt{b}} - \frac{3 a \sqrt{a + \frac{b}{x}}}{4 \sqrt{x}} - \frac{\left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{2 \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**2*atanh(sqrt(b)/(sqrt(x)*sqrt(a + b/x)))/(4*sqrt(b)) - 3*a*sqrt(a + b/x)/(
4*sqrt(x)) - (a + b/x)**(3/2)/(2*sqrt(x))

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Mathematica [A]  time = 0.198958, size = 80, normalized size = 1.04 \[ -\frac{3 a^2 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )}{4 \sqrt{b}}+\frac{3 a^2 \log (x)}{8 \sqrt{b}}-\frac{\sqrt{a+\frac{b}{x}} (5 a x+2 b)}{4 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b/x]*(2*b + 5*a*x))/(4*x^(3/2)) - (3*a^2*Log[b + Sqrt[b]*Sqrt[a + b/x
]*Sqrt[x]])/(4*Sqrt[b]) + (3*a^2*Log[x])/(8*Sqrt[b])

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Maple [A]  time = 0.023, size = 74, normalized size = 1. \[ -{\frac{1}{4}\sqrt{{\frac{ax+b}{x}}} \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{2}{x}^{2}+5\,xa\sqrt{ax+b}\sqrt{b}+2\,{b}^{3/2}\sqrt{ax+b} \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ax+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)/x^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253894, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} x^{2} \log \left (-\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} -{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) - 2 \,{\left (5 \, a x + 2 \, b\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{8 \, \sqrt{b} x^{2}}, \frac{3 \, a^{2} x^{2} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) -{\left (5 \, a x + 2 \, b\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{4 \, \sqrt{-b} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(3*a^2*x^2*log(-(2*b*sqrt(x)*sqrt((a*x + b)/x) - (a*x + 2*b)*sqrt(b))/x) -
2*(5*a*x + 2*b)*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x))/(sqrt(b)*x^2), 1/4*(3*a^2*x^2
*arctan(b/(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x))) - (5*a*x + 2*b)*sqrt(-b)*sqrt(x)
*sqrt((a*x + b)/x))/(sqrt(-b)*x^2)]

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Sympy [A]  time = 50.0428, size = 76, normalized size = 0.99 \[ - \frac{5 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}}{4 \sqrt{x}} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x}}}{2 x^{\frac{3}{2}}} - \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{4 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a**(3/2)*sqrt(1 + b/(a*x))/(4*sqrt(x)) - sqrt(a)*b*sqrt(1 + b/(a*x))/(2*x**(3
/2)) - 3*a**2*asinh(sqrt(b)/(sqrt(a)*sqrt(x)))/(4*sqrt(b))

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GIAC/XCAS [A]  time = 0.269391, size = 74, normalized size = 0.96 \[ \frac{1}{4} \, a^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{5 \,{\left (a x + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{a x + b} b}{a^{2} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a^2*(3*arctan(sqrt(a*x + b)/sqrt(-b))/sqrt(-b) - (5*(a*x + b)^(3/2) - 3*sqrt
(a*x + b)*b)/(a^2*x^2))