3.1771 \(\int \left (a+\frac{b}{x}\right )^{5/2} \sqrt{x} \, dx\)

Optimal. Leaf size=94 \[ -5 a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )-\frac{5 b^2 \sqrt{a+\frac{b}{x}}}{\sqrt{x}}+\frac{2}{3} x^{3/2} \left (a+\frac{b}{x}\right )^{5/2}+\frac{10}{3} b \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2} \]

[Out]

(-5*b^2*Sqrt[a + b/x])/Sqrt[x] + (10*b*(a + b/x)^(3/2)*Sqrt[x])/3 + (2*(a + b/x)
^(5/2)*x^(3/2))/3 - 5*a*b^(3/2)*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])]

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Rubi [A]  time = 0.130284, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -5 a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )-\frac{5 b^2 \sqrt{a+\frac{b}{x}}}{\sqrt{x}}+\frac{2}{3} x^{3/2} \left (a+\frac{b}{x}\right )^{5/2}+\frac{10}{3} b \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)*Sqrt[x],x]

[Out]

(-5*b^2*Sqrt[a + b/x])/Sqrt[x] + (10*b*(a + b/x)^(3/2)*Sqrt[x])/3 + (2*(a + b/x)
^(5/2)*x^(3/2))/3 - 5*a*b^(3/2)*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])]

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Rubi in Sympy [A]  time = 11.6508, size = 82, normalized size = 0.87 \[ - 5 a b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )} - \frac{5 b^{2} \sqrt{a + \frac{b}{x}}}{\sqrt{x}} + \frac{10 b \sqrt{x} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3} + \frac{2 x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(5/2)*x**(1/2),x)

[Out]

-5*a*b**(3/2)*atanh(sqrt(b)/(sqrt(x)*sqrt(a + b/x))) - 5*b**2*sqrt(a + b/x)/sqrt
(x) + 10*b*sqrt(x)*(a + b/x)**(3/2)/3 + 2*x**(3/2)*(a + b/x)**(5/2)/3

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Mathematica [A]  time = 0.231975, size = 85, normalized size = 0.9 \[ \frac{\sqrt{a+\frac{b}{x}} \left (2 a^2 x^2+14 a b x-3 b^2\right )}{3 \sqrt{x}}-5 a b^{3/2} \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+\frac{5}{2} a b^{3/2} \log (x) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*(-3*b^2 + 14*a*b*x + 2*a^2*x^2))/(3*Sqrt[x]) - 5*a*b^(3/2)*Log[b
+ Sqrt[b]*Sqrt[a + b/x]*Sqrt[x]] + (5*a*b^(3/2)*Log[x])/2

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Maple [A]  time = 0.022, size = 91, normalized size = 1. \[ -{\frac{1}{3}\sqrt{{\frac{ax+b}{x}}} \left ( -2\,{x}^{2}{a}^{2}\sqrt{b}\sqrt{ax+b}+15\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) xa{b}^{2}-14\,xa{b}^{3/2}\sqrt{ax+b}+3\,{b}^{5/2}\sqrt{ax+b} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{ax+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)*x^(1/2),x)

[Out]

-1/3*((a*x+b)/x)^(1/2)/x^(1/2)*(-2*x^2*a^2*b^(1/2)*(a*x+b)^(1/2)+15*arctanh((a*x
+b)^(1/2)/b^(1/2))*x*a*b^2-14*x*a*b^(3/2)*(a*x+b)^(1/2)+3*b^(5/2)*(a*x+b)^(1/2))
/(a*x+b)^(1/2)/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)*sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25096, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a b^{\frac{3}{2}} x \log \left (\frac{a x - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) + 2 \,{\left (2 \, a^{2} x^{2} + 14 \, a b x - 3 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{6 \, x}, -\frac{15 \, a \sqrt{-b} b x \arctan \left (\frac{\sqrt{x} \sqrt{\frac{a x + b}{x}}}{\sqrt{-b}}\right ) -{\left (2 \, a^{2} x^{2} + 14 \, a b x - 3 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(15*a*b^(3/2)*x*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) +
2*(2*a^2*x^2 + 14*a*b*x - 3*b^2)*sqrt(x)*sqrt((a*x + b)/x))/x, -1/3*(15*a*sqrt(-
b)*b*x*arctan(sqrt(x)*sqrt((a*x + b)/x)/sqrt(-b)) - (2*a^2*x^2 + 14*a*b*x - 3*b^
2)*sqrt(x)*sqrt((a*x + b)/x))/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**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.294612, size = 88, normalized size = 0.94 \[ \frac{1}{3} \,{\left (\frac{15 \, b^{2} \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \,{\left (a x + b\right )}^{\frac{3}{2}} + 12 \, \sqrt{a x + b} b - \frac{3 \, \sqrt{a x + b} b^{2}}{a x}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(15*b^2*arctan(sqrt(a*x + b)/sqrt(-b))/sqrt(-b) + 2*(a*x + b)^(3/2) + 12*sqr
t(a*x + b)*b - 3*sqrt(a*x + b)*b^2/(a*x))*a