Optimal. Leaf size=75 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{5/2}}+\frac{2}{b^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{2}{3 b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.112363, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{5/2}}+\frac{2}{b^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{2}{3 b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^(5/2)*x^(7/2)),x]
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Rubi in Sympy [A] time = 11.6955, size = 63, normalized size = 0.84 \[ \frac{2}{3 b x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{2}{b^{2} \sqrt{x} \sqrt{a + \frac{b}{x}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(5/2)/x**(7/2),x)
[Out]
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Mathematica [A] time = 0.247573, size = 79, normalized size = 1.05 \[ -\frac{2 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )}{b^{5/2}}+\frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (3 a x+4 b)}{3 b^2 (a x+b)^2}+\frac{\log (x)}{b^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^(5/2)*x^(7/2)),x]
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Maple [A] time = 0.023, size = 85, normalized size = 1.1 \[{\frac{2}{3\, \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}}\sqrt{x} \left ( -3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}xa+4\,{b}^{3/2}+3\,ax\sqrt{b}-3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) b\sqrt{ax+b} \right ){b}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(5/2)/x^(7/2),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(1/((a + b/x)^(5/2)*x^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.251399, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a x + b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}} \log \left (-\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} -{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) + 2 \,{\left (3 \, a x + 4 \, b\right )} \sqrt{b}}{3 \,{\left (a b^{2} x + b^{3}\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}, \frac{2 \,{\left (3 \,{\left (a x + b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a x + 4 \, b\right )} \sqrt{-b}\right )}}{3 \,{\left (a b^{2} x + b^{3}\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(5/2)*x^(7/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)**(5/2)/x**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.236917, size = 105, normalized size = 1.4 \[ \frac{2 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} - \frac{2 \,{\left (3 \, \sqrt{b} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 4 \, \sqrt{-b}\right )}}{3 \, \sqrt{-b} b^{\frac{5}{2}}} + \frac{2 \,{\left (3 \, a x + 4 \, b\right )}}{3 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(5/2)*x^(7/2)),x, algorithm="giac")
[Out]