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