3.1711 \(\int \left (a+\frac{b}{x}\right )^{5/2} x^2 \, dx\)

Optimal. Leaf size=87 \[ \frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 \sqrt{a}}+\frac{5}{8} b^2 x \sqrt{a+\frac{b}{x}}+\frac{1}{3} x^3 \left (a+\frac{b}{x}\right )^{5/2}+\frac{5}{12} b x^2 \left (a+\frac{b}{x}\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.119454, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 \sqrt{a}}+\frac{5}{8} b^2 x \sqrt{a+\frac{b}{x}}+\frac{1}{3} x^3 \left (a+\frac{b}{x}\right )^{5/2}+\frac{5}{12} b x^2 \left (a+\frac{b}{x}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)*x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.717, size = 73, normalized size = 0.84 \[ \frac{5 b^{2} x \sqrt{a + \frac{b}{x}}}{8} + \frac{5 b x^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{12} + \frac{x^{3} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{3} + \frac{5 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{8 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.117392, size = 74, normalized size = 0.85 \[ \frac{1}{24} x \sqrt{a+\frac{b}{x}} \left (8 a^2 x^2+26 a b x+33 b^2\right )+\frac{5 b^3 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{16 \sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*x*(33*b^2 + 26*a*b*x + 8*a^2*x^2))/24 + (5*b^3*Log[b + 2*a*x + 2*
Sqrt[a]*Sqrt[a + b/x]*x])/(16*Sqrt[a])

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Maple [A]  time = 0.012, size = 114, normalized size = 1.3 \[{\frac{x}{48}\sqrt{{\frac{ax+b}{x}}} \left ( 16\,{a}^{3/2} \left ( a{x}^{2}+bx \right ) ^{3/2}+36\,{a}^{3/2}b\sqrt{a{x}^{2}+bx}x+15\,{b}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) +66\,{b}^{2}\sqrt{a{x}^{2}+bx}\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)*x^2,x)

[Out]

1/48*((a*x+b)/x)^(1/2)*x*(16*a^(3/2)*(a*x^2+b*x)^(3/2)+36*a^(3/2)*b*(a*x^2+b*x)^
(1/2)*x+15*b^3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))+66*b^2*(a*x
^2+b*x)^(1/2)*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)^(5/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238937, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{3} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (8 \, a^{2} x^{3} + 26 \, a b x^{2} + 33 \, b^{2} x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{48 \, \sqrt{a}}, -\frac{15 \, b^{3} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) -{\left (8 \, a^{2} x^{3} + 26 \, a b x^{2} + 33 \, b^{2} x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{24 \, \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*b^3*log(2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) + 2*(8*a^2*x^3
+ 26*a*b*x^2 + 33*b^2*x)*sqrt(a)*sqrt((a*x + b)/x))/sqrt(a), -1/24*(15*b^3*arcta
n(a/(sqrt(-a)*sqrt((a*x + b)/x))) - (8*a^2*x^3 + 26*a*b*x^2 + 33*b^2*x)*sqrt(-a)
*sqrt((a*x + b)/x))/sqrt(-a)]

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Sympy [A]  time = 19.3715, size = 102, normalized size = 1.17 \[ \frac{a^{2} \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a x}{b} + 1}}{3} + \frac{13 a b^{\frac{3}{2}} x^{\frac{3}{2}} \sqrt{\frac{a x}{b} + 1}}{12} + \frac{11 b^{\frac{5}{2}} \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{8} + \frac{5 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{8 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*sqrt(b)*x**(5/2)*sqrt(a*x/b + 1)/3 + 13*a*b**(3/2)*x**(3/2)*sqrt(a*x/b + 1)
/12 + 11*b**(5/2)*sqrt(x)*sqrt(a*x/b + 1)/8 + 5*b**3*asinh(sqrt(a)*sqrt(x)/sqrt(
b))/(8*sqrt(a))

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GIAC/XCAS [A]  time = 0.24466, size = 126, normalized size = 1.45 \[ -\frac{5 \, b^{3}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right ){\rm sign}\left (x\right )}{16 \, \sqrt{a}} + \frac{5 \, b^{3}{\rm ln}\left ({\left | b \right |}\right ){\rm sign}\left (x\right )}{16 \, \sqrt{a}} + \frac{1}{24} \, \sqrt{a x^{2} + b x}{\left (33 \, b^{2}{\rm sign}\left (x\right ) + 2 \,{\left (4 \, a^{2} x{\rm sign}\left (x\right ) + 13 \, a b{\rm sign}\left (x\right )\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/16*b^3*ln(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))*sign(x)/sqrt(a
) + 5/16*b^3*ln(abs(b))*sign(x)/sqrt(a) + 1/24*sqrt(a*x^2 + b*x)*(33*b^2*sign(x)
+ 2*(4*a^2*x*sign(x) + 13*a*b*sign(x))*x)