Optimal. Leaf size=73 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2}}+\frac{\sqrt{x}}{4 a b (a x+b)}-\frac{\sqrt{x}}{2 a (a x+b)^2} \]
[Out]
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Rubi [A] time = 0.0694629, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2}}+\frac{\sqrt{x}}{4 a b (a x+b)}-\frac{\sqrt{x}}{2 a (a x+b)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^3*x^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 12.2391, size = 58, normalized size = 0.79 \[ - \frac{\sqrt{x}}{2 a \left (a x + b\right )^{2}} + \frac{\sqrt{x}}{4 a b \left (a x + b\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 a^{\frac{3}{2}} b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**3/x**(5/2),x)
[Out]
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Mathematica [A] time = 0.059393, size = 62, normalized size = 0.85 \[ \frac{\frac{\sqrt{a} \sqrt{b} \sqrt{x} (a x-b)}{(a x+b)^2}+\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^3*x^(5/2)),x]
[Out]
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Maple [A] time = 0.016, size = 52, normalized size = 0.7 \[ 2\,{\frac{1}{ \left ( ax+b \right ) ^{2}} \left ( 1/8\,{\frac{{x}^{3/2}}{b}}-1/8\,{\frac{\sqrt{x}}{a}} \right ) }+{\frac{1}{4\,ab}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^3/x^(5/2),x)
[Out]
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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)^3*x^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.244173, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{-a b}{\left (a x - b\right )} \sqrt{x} +{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (a x - b\right )}}{a x + b}\right )}{8 \,{\left (a^{3} b x^{2} + 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt{-a b}}, \frac{\sqrt{a b}{\left (a x - b\right )} \sqrt{x} -{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \arctan \left (\frac{b}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (a^{3} b x^{2} + 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^3*x^(5/2)),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)**3/x**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.217877, size = 70, normalized size = 0.96 \[ \frac{\arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a b} + \frac{a x^{\frac{3}{2}} - b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^3*x^(5/2)),x, algorithm="giac")
[Out]