3.1700 \(\int \left (a+\frac{b}{x}\right )^{3/2} x^2 \, dx\)

Optimal. Leaf size=90 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{b^2 x \sqrt{a+\frac{b}{x}}}{8 a}+\frac{1}{3} x^3 \left (a+\frac{b}{x}\right )^{3/2}+\frac{1}{4} b x^2 \sqrt{a+\frac{b}{x}} \]

[Out]

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

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Rubi [A]  time = 0.123743, antiderivative size = 90, 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 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{b^2 x \sqrt{a+\frac{b}{x}}}{8 a}+\frac{1}{3} x^3 \left (a+\frac{b}{x}\right )^{3/2}+\frac{1}{4} b x^2 \sqrt{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2)*x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.9163, size = 70, normalized size = 0.78 \[ \frac{b x^{2} \sqrt{a + \frac{b}{x}}}{4} + \frac{x^{3} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3} + \frac{b^{2} x \sqrt{a + \frac{b}{x}}}{8 a} - \frac{b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.109118, size = 79, normalized size = 0.88 \[ \frac{2 \sqrt{a} x \sqrt{a+\frac{b}{x}} \left (8 a^2 x^2+14 a b x+3 b^2\right )-3 b^3 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{48 a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a]*Sqrt[a + b/x]*x*(3*b^2 + 14*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^(3/2))

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Maple [A]  time = 0.012, size = 115, normalized size = 1.3 \[ -{\frac{x}{48}\sqrt{{\frac{ax+b}{x}}} \left ( -16\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{5/2}-12\,\sqrt{a{x}^{2}+bx}{a}^{5/2}xb-6\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{b}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{3} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)*x^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^(5/2)

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24039, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{3} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (8 \, a^{2} x^{3} + 14 \, a b x^{2} + 3 \, b^{2} x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{48 \, a^{\frac{3}{2}}}, \frac{3 \, b^{3} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (8 \, a^{2} x^{3} + 14 \, a b x^{2} + 3 \, b^{2} x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{24 \, \sqrt{-a} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(3/2)*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
+ 14*a*b*x^2 + 3*b^2*x)*sqrt(a)*sqrt((a*x + b)/x))/a^(3/2), 1/24*(3*b^3*arctan(a
/(sqrt(-a)*sqrt((a*x + b)/x))) + (8*a^2*x^3 + 14*a*b*x^2 + 3*b^2*x)*sqrt(-a)*sqr
t((a*x + b)/x))/(sqrt(-a)*a)]

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Sympy [A]  time = 18.7739, size = 124, normalized size = 1.38 \[ \frac{a^{2} x^{\frac{7}{2}}}{3 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{11 a \sqrt{b} x^{\frac{5}{2}}}{12 \sqrt{\frac{a x}{b} + 1}} + \frac{17 b^{\frac{3}{2}} x^{\frac{3}{2}}}{24 \sqrt{\frac{a x}{b} + 1}} + \frac{b^{\frac{5}{2}} \sqrt{x}}{8 a \sqrt{\frac{a x}{b} + 1}} - \frac{b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.24759, size = 126, normalized size = 1.4 \[ \frac{b^{3}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right ){\rm sign}\left (x\right )}{16 \, a^{\frac{3}{2}}} - \frac{b^{3}{\rm ln}\left ({\left | b \right |}\right ){\rm sign}\left (x\right )}{16 \, a^{\frac{3}{2}}} + \frac{1}{24} \, \sqrt{a x^{2} + b x}{\left (2 \,{\left (4 \, a x{\rm sign}\left (x\right ) + 7 \, b{\rm sign}\left (x\right )\right )} x + \frac{3 \, b^{2}{\rm sign}\left (x\right )}{a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(3/2)*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^(3/2)
- 1/16*b^3*ln(abs(b))*sign(x)/a^(3/2) + 1/24*sqrt(a*x^2 + b*x)*(2*(4*a*x*sign(x
) + 7*b*sign(x))*x + 3*b^2*sign(x)/a)