Optimal. Leaf size=71 \[ 5 a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+x \left (a+\frac{b}{x}\right )^{5/2}-\frac{5}{3} b \left (a+\frac{b}{x}\right )^{3/2}-5 a b \sqrt{a+\frac{b}{x}} \]
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Rubi [A] time = 0.11446, 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]
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Rubi in Sympy [A] time = 9.51426, 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)
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Mathematica [A] time = 0.112019, 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]
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Maple [A] time = 0.013, 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)
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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")
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Fricas [A] time = 0.2386, 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")
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Sympy [A] time = 12.9866, 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)
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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")
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