3.1770 \(\int \left (a+\frac{b}{x}\right )^{5/2} x^{3/2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.135878, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )+2 b^2 \sqrt{x} \sqrt{a+\frac{b}{x}}+\frac{2}{3} b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}+\frac{2}{5} x^{5/2} \left (a+\frac{b}{x}\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)*x^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.2526, size = 80, normalized size = 0.86 \[ - 2 b^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )} + 2 b^{2} \sqrt{x} \sqrt{a + \frac{b}{x}} + \frac{2 b x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3} + \frac{2 x^{\frac{5}{2}} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.137847, size = 80, normalized size = 0.86 \[ \frac{2}{15} \sqrt{x} \sqrt{a+\frac{b}{x}} \left (3 a^2 x^2+11 a b x+23 b^2\right )-2 b^{5/2} \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+b^{5/2} \log (x) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(23*b^2 + 11*a*b*x + 3*a^2*x^2))/15 - 2*b^(5/2)*Log[b +
Sqrt[b]*Sqrt[a + b/x]*Sqrt[x]] + b^(5/2)*Log[x]

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Maple [A]  time = 0.015, size = 81, normalized size = 0.9 \[ -{\frac{2}{15}\sqrt{{\frac{ax+b}{x}}}\sqrt{x} \left ( -3\,{x}^{2}{a}^{2}\sqrt{ax+b}+15\,{b}^{5/2}{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) -11\,xab\sqrt{ax+b}-23\,\sqrt{ax+b}{b}^{2} \right ){\frac{1}{\sqrt{ax+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)*x^(3/2),x)

[Out]

-2/15*((a*x+b)/x)^(1/2)*x^(1/2)*(-3*x^2*a^2*(a*x+b)^(1/2)+15*b^(5/2)*arctanh((a*
x+b)^(1/2)/b^(1/2))-11*x*a*b*(a*x+b)^(1/2)-23*(a*x+b)^(1/2)*b^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)^(5/2)*x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24861, size = 1, normalized size = 0.01 \[ \left [b^{\frac{5}{2}} \log \left (\frac{a x - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) + \frac{2}{15} \,{\left (3 \, a^{2} x^{2} + 11 \, a b x + 23 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}, -2 \, \sqrt{-b} b^{2} \arctan \left (\frac{\sqrt{x} \sqrt{\frac{a x + b}{x}}}{\sqrt{-b}}\right ) + \frac{2}{15} \,{\left (3 \, a^{2} x^{2} + 11 \, a b x + 23 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[b^(5/2)*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) + 2/15*(3*a^2*
x^2 + 11*a*b*x + 23*b^2)*sqrt(x)*sqrt((a*x + b)/x), -2*sqrt(-b)*b^2*arctan(sqrt(
x)*sqrt((a*x + b)/x)/sqrt(-b)) + 2/15*(3*a^2*x^2 + 11*a*b*x + 23*b^2)*sqrt(x)*sq
rt((a*x + b)/x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.237444, size = 76, normalized size = 0.82 \[ \frac{2 \, b^{3} \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + \frac{2}{5} \,{\left (a x + b\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (a x + b\right )}^{\frac{3}{2}} b + 2 \, \sqrt{a x + b} b^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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