3.1765 \(\int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{x^{5/2}} \, dx\)

Optimal. Leaf size=103 \[ \frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 b^{3/2}}-\frac{a^2 \sqrt{a+\frac{b}{x}}}{8 b \sqrt{x}}-\frac{a \sqrt{a+\frac{b}{x}}}{4 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{3/2}}{3 x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.154021, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 b^{3/2}}-\frac{a^2 \sqrt{a+\frac{b}{x}}}{8 b \sqrt{x}}-\frac{a \sqrt{a+\frac{b}{x}}}{4 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{3/2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2)/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.4583, size = 82, normalized size = 0.8 \[ \frac{a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{8 b^{\frac{3}{2}}} - \frac{a^{2} \sqrt{a + \frac{b}{x}}}{8 b \sqrt{x}} - \frac{a \sqrt{a + \frac{b}{x}}}{4 x^{\frac{3}{2}}} - \frac{\left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(3/2)/x**(5/2),x)

[Out]

a**3*atanh(sqrt(b)/(sqrt(x)*sqrt(a + b/x)))/(8*b**(3/2)) - a**2*sqrt(a + b/x)/(8
*b*sqrt(x)) - a*sqrt(a + b/x)/(4*x**(3/2)) - (a + b/x)**(3/2)/(3*x**(3/2))

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Mathematica [A]  time = 0.299942, size = 89, normalized size = 0.86 \[ \frac{6 a^3 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )-3 a^3 \log (x)-\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} \left (3 a^2 x^2+14 a b x+8 b^2\right )}{x^{5/2}}}{48 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(3/2)/x^(5/2),x]

[Out]

((-2*Sqrt[b]*Sqrt[a + b/x]*(8*b^2 + 14*a*b*x + 3*a^2*x^2))/x^(5/2) + 6*a^3*Log[b
+ Sqrt[b]*Sqrt[a + b/x]*Sqrt[x]] - 3*a^3*Log[x])/(48*b^(3/2))

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Maple [A]  time = 0.024, size = 92, normalized size = 0.9 \[ -{\frac{1}{24}\sqrt{{\frac{ax+b}{x}}} \left ( -3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{3}{x}^{3}+8\,{b}^{5/2}\sqrt{ax+b}+14\,xa{b}^{3/2}\sqrt{ax+b}+3\,{x}^{2}{a}^{2}\sqrt{b}\sqrt{ax+b} \right ){x}^{-{\frac{5}{2}}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ax+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)/x^(5/2),x)

[Out]

-1/24*((a*x+b)/x)^(1/2)*(-3*arctanh((a*x+b)^(1/2)/b^(1/2))*a^3*x^3+8*b^(5/2)*(a*
x+b)^(1/2)+14*x*a*b^(3/2)*(a*x+b)^(1/2)+3*x^2*a^2*b^(1/2)*(a*x+b)^(1/2))/x^(5/2)
/b^(3/2)/(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^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247099, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{3} x^{3} \log \left (\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} +{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) - 2 \,{\left (3 \, a^{2} x^{2} + 14 \, a b x + 8 \, b^{2}\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{48 \, b^{\frac{3}{2}} x^{3}}, -\frac{3 \, a^{3} x^{3} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a^{2} x^{2} + 14 \, a b x + 8 \, b^{2}\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{24 \, \sqrt{-b} b x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(3/2)/x^(5/2),x, algorithm="fricas")

[Out]

[1/48*(3*a^3*x^3*log((2*b*sqrt(x)*sqrt((a*x + b)/x) + (a*x + 2*b)*sqrt(b))/x) -
2*(3*a^2*x^2 + 14*a*b*x + 8*b^2)*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x))/(b^(3/2)*x^3
), -1/24*(3*a^3*x^3*arctan(b/(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x))) + (3*a^2*x^2
+ 14*a*b*x + 8*b^2)*sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x))/(sqrt(-b)*b*x^3)]

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Sympy [A]  time = 156.394, size = 124, normalized size = 1.2 \[ - \frac{a^{\frac{5}{2}}}{8 b \sqrt{x} \sqrt{1 + \frac{b}{a x}}} - \frac{17 a^{\frac{3}{2}}}{24 x^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}} - \frac{11 \sqrt{a} b}{12 x^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x}}} + \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{8 b^{\frac{3}{2}}} - \frac{b^{2}}{3 \sqrt{a} x^{\frac{7}{2}} \sqrt{1 + \frac{b}{a x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**(3/2)/x**(5/2),x)

[Out]

-a**(5/2)/(8*b*sqrt(x)*sqrt(1 + b/(a*x))) - 17*a**(3/2)/(24*x**(3/2)*sqrt(1 + b/
(a*x))) - 11*sqrt(a)*b/(12*x**(5/2)*sqrt(1 + b/(a*x))) + a**3*asinh(sqrt(b)/(sqr
t(a)*sqrt(x)))/(8*b**(3/2)) - b**2/(3*sqrt(a)*x**(7/2)*sqrt(1 + b/(a*x)))

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GIAC/XCAS [A]  time = 0.29852, size = 97, normalized size = 0.94 \[ -\frac{1}{24} \, a^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{3 \,{\left (a x + b\right )}^{\frac{5}{2}} + 8 \,{\left (a x + b\right )}^{\frac{3}{2}} b - 3 \, \sqrt{a x + b} b^{2}}{a^{3} b x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(3/2)/x^(5/2),x, algorithm="giac")

[Out]

-1/24*a^3*(3*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b) + (3*(a*x + b)^(5/2) +
8*(a*x + b)^(3/2)*b - 3*sqrt(a*x + b)*b^2)/(a^3*b*x^3))