3.1690 \(\int \sqrt{a+\frac{b}{x}} x^2 \, dx\)

Optimal. Leaf size=93 \[ \frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{5/2}}-\frac{b^2 x \sqrt{a+\frac{b}{x}}}{8 a^2}+\frac{1}{3} x^3 \sqrt{a+\frac{b}{x}}+\frac{b x^2 \sqrt{a+\frac{b}{x}}}{12 a} \]

[Out]

-(b^2*Sqrt[a + b/x]*x)/(8*a^2) + (b*Sqrt[a + b/x]*x^2)/(12*a) + (Sqrt[a + b/x]*x
^3)/3 + (b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(5/2))

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Rubi [A]  time = 0.12909, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{5/2}}-\frac{b^2 x \sqrt{a+\frac{b}{x}}}{8 a^2}+\frac{1}{3} x^3 \sqrt{a+\frac{b}{x}}+\frac{b x^2 \sqrt{a+\frac{b}{x}}}{12 a} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]*x^2,x]

[Out]

-(b^2*Sqrt[a + b/x]*x)/(8*a^2) + (b*Sqrt[a + b/x]*x^2)/(12*a) + (Sqrt[a + b/x]*x
^3)/3 + (b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(5/2))

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Rubi in Sympy [A]  time = 12.2253, size = 73, normalized size = 0.78 \[ \frac{x^{3} \sqrt{a + \frac{b}{x}}}{3} + \frac{b x^{2} \sqrt{a + \frac{b}{x}}}{12 a} - \frac{b^{2} x \sqrt{a + \frac{b}{x}}}{8 a^{2}} + \frac{b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*sqrt(a + b/x)/3 + b*x**2*sqrt(a + b/x)/(12*a) - b**2*x*sqrt(a + b/x)/(8*a**
2) + b**3*atanh(sqrt(a + b/x)/sqrt(a))/(8*a**(5/2))

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Mathematica [A]  time = 0.0946769, size = 79, normalized size = 0.85 \[ \frac{2 \sqrt{a} x \sqrt{a+\frac{b}{x}} \left (8 a^2 x^2+2 a b x-3 b^2\right )+3 b^3 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{48 a^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x]*x^2,x]

[Out]

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

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Maple [A]  time = 0.012, size = 115, normalized size = 1.2 \[{\frac{x}{48}\sqrt{{\frac{ax+b}{x}}} \left ( 16\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{5/2}-12\,\sqrt{a{x}^{2}+bx}{a}^{5/2}xb-6\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{b}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{3} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b/x)^(1/2),x)

[Out]

1/48*((a*x+b)/x)^(1/2)*x*(16*(a*x^2+b*x)^(3/2)*a^(5/2)-12*(a*x^2+b*x)^(1/2)*a^(5
/2)*x*b-6*(a*x^2+b*x)^(1/2)*a^(3/2)*b^2+3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*
a*x+b)/a^(1/2))*a*b^3)/(x*(a*x+b))^(1/2)/a^(7/2)

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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(sqrt(a + b/x)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240159, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{3} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (8 \, a^{2} x^{3} + 2 \, a b x^{2} - 3 \, b^{2} x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{48 \, a^{\frac{5}{2}}}, -\frac{3 \, b^{3} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) -{\left (8 \, a^{2} x^{3} + 2 \, a b x^{2} - 3 \, b^{2} x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{24 \, \sqrt{-a} a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(3*b^3*log(2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) + 2*(8*a^2*x^3 +
2*a*b*x^2 - 3*b^2*x)*sqrt(a)*sqrt((a*x + b)/x))/a^(5/2), -1/24*(3*b^3*arctan(a/
(sqrt(-a)*sqrt((a*x + b)/x))) - (8*a^2*x^3 + 2*a*b*x^2 - 3*b^2*x)*sqrt(-a)*sqrt(
(a*x + b)/x))/(sqrt(-a)*a^2)]

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Sympy [A]  time = 19.3046, size = 122, normalized size = 1.31 \[ \frac{a x^{\frac{7}{2}}}{3 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{5 \sqrt{b} x^{\frac{5}{2}}}{12 \sqrt{\frac{a x}{b} + 1}} - \frac{b^{\frac{3}{2}} x^{\frac{3}{2}}}{24 a \sqrt{\frac{a x}{b} + 1}} - \frac{b^{\frac{5}{2}} \sqrt{x}}{8 a^{2} \sqrt{\frac{a x}{b} + 1}} + \frac{b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.241532, size = 127, normalized size = 1.37 \[ -\frac{b^{3}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right ){\rm sign}\left (x\right )}{16 \, a^{\frac{5}{2}}} + \frac{b^{3}{\rm ln}\left ({\left | b \right |}\right ){\rm sign}\left (x\right )}{16 \, a^{\frac{5}{2}}} + \frac{1}{24} \, \sqrt{a x^{2} + b x}{\left (2 \,{\left (4 \, x{\rm sign}\left (x\right ) + \frac{b{\rm sign}\left (x\right )}{a}\right )} x - \frac{3 \, b^{2}{\rm sign}\left (x\right )}{a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*b^3*ln(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))*sign(x)/a^(5/2
) + 1/16*b^3*ln(abs(b))*sign(x)/a^(5/2) + 1/24*sqrt(a*x^2 + b*x)*(2*(4*x*sign(x)
+ b*sign(x)/a)*x - 3*b^2*sign(x)/a^2)