3.1712 \(\int \left (a+\frac{b}{x}\right )^{5/2} x \, dx\)

Optimal. Leaf size=84 \[ -\frac{15}{4} b^2 \sqrt{a+\frac{b}{x}}+\frac{15}{4} \sqrt{a} b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+\frac{1}{2} x^2 \left (a+\frac{b}{x}\right )^{5/2}+\frac{5}{4} b x \left (a+\frac{b}{x}\right )^{3/2} \]

[Out]

(-15*b^2*Sqrt[a + b/x])/4 + (5*b*(a + b/x)^(3/2)*x)/4 + ((a + b/x)^(5/2)*x^2)/2
+ (15*Sqrt[a]*b^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/4

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Rubi [A]  time = 0.111506, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{15}{4} b^2 \sqrt{a+\frac{b}{x}}+\frac{15}{4} \sqrt{a} b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+\frac{1}{2} x^2 \left (a+\frac{b}{x}\right )^{5/2}+\frac{5}{4} b x \left (a+\frac{b}{x}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)*x,x]

[Out]

(-15*b^2*Sqrt[a + b/x])/4 + (5*b*(a + b/x)^(3/2)*x)/4 + ((a + b/x)^(5/2)*x^2)/2
+ (15*Sqrt[a]*b^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/4

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Rubi in Sympy [A]  time = 11.061, size = 70, normalized size = 0.83 \[ \frac{15 \sqrt{a} b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{4} - \frac{15 b^{2} \sqrt{a + \frac{b}{x}}}{4} + \frac{5 b x \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{4} + \frac{x^{2} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*sqrt(a)*b**2*atanh(sqrt(a + b/x)/sqrt(a))/4 - 15*b**2*sqrt(a + b/x)/4 + 5*b*x
*(a + b/x)**(3/2)/4 + x**2*(a + b/x)**(5/2)/2

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Mathematica [A]  time = 0.0674233, size = 73, normalized size = 0.87 \[ \frac{1}{4} \sqrt{a+\frac{b}{x}} \left (2 a^2 x^2+9 a b x-8 b^2\right )+\frac{15}{8} \sqrt{a} b^2 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(5/2)*x,x]

[Out]

(Sqrt[a + b/x]*(-8*b^2 + 9*a*b*x + 2*a^2*x^2))/4 + (15*Sqrt[a]*b^2*Log[b + 2*a*x
+ 2*Sqrt[a]*Sqrt[a + b/x]*x])/8

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Maple [A]  time = 0.017, size = 117, normalized size = 1.4 \[{\frac{1}{8\,x}\sqrt{{\frac{ax+b}{x}}} \left ( 15\,\sqrt{a}{b}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}+4\,{a}^{2}\sqrt{a{x}^{2}+bx}{x}^{3}+34\,a\sqrt{a{x}^{2}+bx}b{x}^{2}-16\, \left ( a{x}^{2}+bx \right ) ^{3/2}b \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)*x,x)

[Out]

1/8*((a*x+b)/x)^(1/2)/x*(15*a^(1/2)*b^2*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*
x+b)/a^(1/2))*x^2+4*a^2*(a*x^2+b*x)^(1/2)*x^3+34*a*(a*x^2+b*x)^(1/2)*b*x^2-16*(a
*x^2+b*x)^(3/2)*b)/(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 = 0.235525, size = 1, normalized size = 0.01 \[ \left [\frac{15}{8} \, \sqrt{a} b^{2} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + \frac{1}{4} \,{\left (2 \, a^{2} x^{2} + 9 \, a b x - 8 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}, \frac{15}{4} \, \sqrt{-a} b^{2} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) + \frac{1}{4} \,{\left (2 \, a^{2} x^{2} + 9 \, a b x - 8 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[15/8*sqrt(a)*b^2*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 1/4*(2*a^2*x^
2 + 9*a*b*x - 8*b^2)*sqrt((a*x + b)/x), 15/4*sqrt(-a)*b^2*arctan(sqrt((a*x + b)/
x)/sqrt(-a)) + 1/4*(2*a^2*x^2 + 9*a*b*x - 8*b^2)*sqrt((a*x + b)/x)]

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Sympy [A]  time = 15.9782, size = 126, normalized size = 1.5 \[ \frac{15 \sqrt{a} b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4} + \frac{a^{3} x^{\frac{5}{2}}}{2 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{11 a^{2} \sqrt{b} x^{\frac{3}{2}}}{4 \sqrt{\frac{a x}{b} + 1}} + \frac{a b^{\frac{3}{2}} \sqrt{x}}{4 \sqrt{\frac{a x}{b} + 1}} - \frac{2 b^{\frac{5}{2}}}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**(5/2)*x,x)

[Out]

15*sqrt(a)*b**2*asinh(sqrt(a)*sqrt(x)/sqrt(b))/4 + a**3*x**(5/2)/(2*sqrt(b)*sqrt
(a*x/b + 1)) + 11*a**2*sqrt(b)*x**(3/2)/(4*sqrt(a*x/b + 1)) + a*b**(3/2)*sqrt(x)
/(4*sqrt(a*x/b + 1)) - 2*b**(5/2)/(sqrt(x)*sqrt(a*x/b + 1))

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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