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

Optimal. Leaf size=71 \[ 5 a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+x \left (a+\frac{b}{x}\right )^{5/2}-\frac{5}{3} b \left (a+\frac{b}{x}\right )^{3/2}-5 a b \sqrt{a+\frac{b}{x}} \]

[Out]

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

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Rubi [A]  time = 0.101272, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ 5 a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+x \left (a+\frac{b}{x}\right )^{5/2}-\frac{5}{3} b \left (a+\frac{b}{x}\right )^{3/2}-5 a b \sqrt{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.5882, size = 60, normalized size = 0.85 \[ 5 a^{\frac{3}{2}} b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )} - 5 a b \sqrt{a + \frac{b}{x}} - \frac{5 b \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3} + x \left (a + \frac{b}{x}\right )^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.102443, size = 71, normalized size = 1. \[ \frac{5}{2} a^{3/2} b \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )+\sqrt{a+\frac{b}{x}} \left (a^2 x-\frac{14 a b}{3}-\frac{2 b^2}{3 x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 112, normalized size = 1.6 \[{\frac{1}{6\,{x}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 15\,{a}^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}b+30\,{a}^{2}\sqrt{a{x}^{2}+bx}{x}^{3}-24\,a \left ( a{x}^{2}+bx \right ) ^{3/2}x-4\, \left ( a{x}^{2}+bx \right ) ^{3/2}b \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2),x)

[Out]

1/6*((a*x+b)/x)^(1/2)/x^2*(15*a^(3/2)*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+
b)/a^(1/2))*x^3*b+30*a^2*(a*x^2+b*x)^(1/2)*x^3-24*a*(a*x^2+b*x)^(3/2)*x-4*(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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256117, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{\frac{3}{2}} b x \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (3 \, a^{2} x^{2} - 14 \, a b x - 2 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{6 \, x}, \frac{15 \, \sqrt{-a} a b x \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right ) +{\left (3 \, a^{2} x^{2} - 14 \, a b x - 2 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{3 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.3606, size = 99, normalized size = 1.39 \[ a^{\frac{5}{2}} x \sqrt{1 + \frac{b}{a x}} - \frac{14 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x}}}{3} - \frac{5 a^{\frac{3}{2}} b \log{\left (\frac{b}{a x} \right )}}{2} + 5 a^{\frac{3}{2}} b \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )} - \frac{2 \sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x}}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**(5/2),x)

[Out]

a**(5/2)*x*sqrt(1 + b/(a*x)) - 14*a**(3/2)*b*sqrt(1 + b/(a*x))/3 - 5*a**(3/2)*b*
log(b/(a*x))/2 + 5*a**(3/2)*b*log(sqrt(1 + b/(a*x)) + 1) - 2*sqrt(a)*b**2*sqrt(1
+ b/(a*x))/(3*x)

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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, algorithm="giac")

[Out]

Exception raised: TypeError