3.1773 \(\int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{x^{3/2}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 \sqrt{b}}-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 \sqrt{x}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{12 \sqrt{x}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{3 \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.110722, antiderivative size = 100, 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 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 \sqrt{b}}-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 \sqrt{x}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{12 \sqrt{x}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)/x^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.61431, size = 87, normalized size = 0.87 \[ - \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{8 \sqrt{b}} - \frac{5 a^{2} \sqrt{a + \frac{b}{x}}}{8 \sqrt{x}} - \frac{5 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{12 \sqrt{x}} - \frac{\left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{3 \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.268146, size = 91, normalized size = 0.91 \[ -\frac{5 a^3 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )}{8 \sqrt{b}}+\frac{5 a^3 \log (x)}{16 \sqrt{b}}-\frac{\sqrt{a+\frac{b}{x}} \left (33 a^2 x^2+26 a b x+8 b^2\right )}{24 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.024, size = 92, normalized size = 0.9 \[ -{\frac{1}{24}\sqrt{{\frac{ax+b}{x}}} \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{3}{x}^{3}+33\,{x}^{2}{a}^{2}\sqrt{b}\sqrt{ax+b}+26\,xa{b}^{3/2}\sqrt{ax+b}+8\,{b}^{5/2}\sqrt{ax+b} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ax+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)/x^(3/2),x)

[Out]

-1/24*((a*x+b)/x)^(1/2)/x^(5/2)*(15*arctanh((a*x+b)^(1/2)/b^(1/2))*a^3*x^3+33*x^
2*a^2*b^(1/2)*(a*x+b)^(1/2)+26*x*a*b^(3/2)*(a*x+b)^(1/2)+8*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)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248528, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} x^{3} \log \left (-\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} -{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) - 2 \,{\left (33 \, a^{2} x^{2} + 26 \, a b x + 8 \, b^{2}\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{48 \, \sqrt{b} x^{3}}, \frac{15 \, a^{3} x^{3} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) -{\left (33 \, a^{2} x^{2} + 26 \, a b x + 8 \, b^{2}\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{24 \, \sqrt{-b} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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 = 0.312344, size = 90, normalized size = 0.9 \[ \frac{1}{24} \, a^{3}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{33 \,{\left (a x + b\right )}^{\frac{5}{2}} - 40 \,{\left (a x + b\right )}^{\frac{3}{2}} b + 15 \, \sqrt{a x + b} b^{2}}{a^{3} x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*a^3*(15*arctan(sqrt(a*x + b)/sqrt(-b))/sqrt(-b) - (33*(a*x + b)^(5/2) - 40*
(a*x + b)^(3/2)*b + 15*sqrt(a*x + b)*b^2)/(a^3*x^3))