3.1714 \(\int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{x} \, dx\)

Optimal. Leaf size=73 \[ 2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 a^2 \sqrt{a+\frac{b}{x}}-\frac{2}{3} a \left (a+\frac{b}{x}\right )^{3/2}-\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2} \]

[Out]

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

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Rubi [A]  time = 0.110465, antiderivative size = 73, 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 \[ 2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 a^2 \sqrt{a+\frac{b}{x}}-\frac{2}{3} a \left (a+\frac{b}{x}\right )^{3/2}-\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.6851, size = 60, normalized size = 0.82 \[ 2 a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - 2 a^{2} \sqrt{a + \frac{b}{x}} - \frac{2 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.108304, size = 70, normalized size = 0.96 \[ a^{5/2} \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )-\frac{2 \sqrt{a+\frac{b}{x}} \left (23 a^2 x^2+11 a b x+3 b^2\right )}{15 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(5/2)/x,x]

[Out]

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

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Maple [B]  time = 0.015, size = 137, normalized size = 1.9 \[{\frac{1}{15\,b{x}^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 15\,{a}^{5/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) b{x}^{4}+30\,{a}^{3}\sqrt{a{x}^{2}+bx}{x}^{4}-30\,{a}^{2} \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{2}-16\,a \left ( a{x}^{2}+bx \right ) ^{3/2}bx-6\, \left ( a{x}^{2}+bx \right ) ^{3/2}{b}^{2} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)/x,x)

[Out]

1/15*((a*x+b)/x)^(1/2)/x^3/b*(15*a^(5/2)*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a
*x+b)/a^(1/2))*b*x^4+30*a^3*(a*x^2+b*x)^(1/2)*x^4-30*a^2*(a*x^2+b*x)^(3/2)*x^2-1
6*a*(a*x^2+b*x)^(3/2)*b*x-6*(a*x^2+b*x)^(3/2)*b^2)/(x*(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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239777, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{\frac{5}{2}} x^{2} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \,{\left (23 \, a^{2} x^{2} + 11 \, a b x + 3 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{15 \, x^{2}}, \frac{2 \,{\left (15 \, \sqrt{-a} a^{2} x^{2} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) -{\left (23 \, a^{2} x^{2} + 11 \, a b x + 3 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}\right )}}{15 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15*(15*a^(5/2)*x^2*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(23*a^2
*x^2 + 11*a*b*x + 3*b^2)*sqrt((a*x + b)/x))/x^2, 2/15*(15*sqrt(-a)*a^2*x^2*arcta
n(sqrt((a*x + b)/x)/sqrt(-a)) - (23*a^2*x^2 + 11*a*b*x + 3*b^2)*sqrt((a*x + b)/x
))/x^2]

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Sympy [A]  time = 13.8969, size = 97, normalized size = 1.33 \[ - \frac{46 a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x}}}{15} - a^{\frac{5}{2}} \log{\left (\frac{b}{a x} \right )} + 2 a^{\frac{5}{2}} \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )} - \frac{22 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x}}}{15 x} - \frac{2 \sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x}}}{5 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**(5/2)/x,x)

[Out]

-46*a**(5/2)*sqrt(1 + b/(a*x))/15 - a**(5/2)*log(b/(a*x)) + 2*a**(5/2)*log(sqrt(
1 + b/(a*x)) + 1) - 22*a**(3/2)*b*sqrt(1 + b/(a*x))/(15*x) - 2*sqrt(a)*b**2*sqrt
(1 + b/(a*x))/(5*x**2)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError