3.1701 \(\int \left (a+\frac{b}{x}\right )^{3/2} x \, dx\)

Optimal. Leaf size=66 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 \sqrt{a}}+\frac{1}{2} x^2 \left (a+\frac{b}{x}\right )^{3/2}+\frac{3}{4} b x \sqrt{a+\frac{b}{x}} \]

[Out]

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

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Rubi [A]  time = 0.0876603, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 \sqrt{a}}+\frac{1}{2} x^2 \left (a+\frac{b}{x}\right )^{3/2}+\frac{3}{4} b x \sqrt{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2)*x,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.8261, size = 54, normalized size = 0.82 \[ \frac{3 b x \sqrt{a + \frac{b}{x}}}{4} + \frac{x^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{2} + \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.089129, size = 63, normalized size = 0.95 \[ \frac{3 b^2 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{8 \sqrt{a}}+\frac{1}{4} x \sqrt{a+\frac{b}{x}} (2 a x+5 b) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(3/2)*x,x]

[Out]

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

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Maple [A]  time = 0.013, size = 95, normalized size = 1.4 \[{\frac{x}{8}\sqrt{{\frac{ax+b}{x}}} \left ( 4\,{a}^{3/2}\sqrt{a{x}^{2}+bx}x+3\,{b}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) +10\,\sqrt{a{x}^{2}+bx}b\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)*x,x)

[Out]

1/8*((a*x+b)/x)^(1/2)*x*(4*a^(3/2)*(a*x^2+b*x)^(1/2)*x+3*b^2*ln(1/2*(2*(a*x^2+b*
x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))+10*(a*x^2+b*x)^(1/2)*b*a^(1/2))/(x*(a*x+b))^(
1/2)/a^(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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236976, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b^{2} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (2 \, a x^{2} + 5 \, b x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{8 \, \sqrt{a}}, -\frac{3 \, b^{2} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) -{\left (2 \, a x^{2} + 5 \, b x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{4 \, \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.2653, size = 75, normalized size = 1.14 \[ \frac{a \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a x}{b} + 1}}{2} + \frac{5 b^{\frac{3}{2}} \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{4} + \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**(3/2)*x,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.238864, size = 107, normalized size = 1.62 \[ -\frac{3 \, b^{2}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right ){\rm sign}\left (x\right )}{8 \, \sqrt{a}} + \frac{3 \, b^{2}{\rm ln}\left ({\left | b \right |}\right ){\rm sign}\left (x\right )}{8 \, \sqrt{a}} + \frac{1}{4} \, \sqrt{a x^{2} + b x}{\left (2 \, a x{\rm sign}\left (x\right ) + 5 \, b{\rm sign}\left (x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/8*b^2*ln(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))*sign(x)/sqrt(a)
+ 3/8*b^2*ln(abs(b))*sign(x)/sqrt(a) + 1/4*sqrt(a*x^2 + b*x)*(2*a*x*sign(x) + 5
*b*sign(x))