Optimal. Leaf size=70 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{5/2} \sqrt{b}}-\frac{3 \sqrt{x}}{4 a^2 (a x+b)}-\frac{x^{3/2}}{2 a (a x+b)^2} \]
[Out]
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Rubi [A] time = 0.0677862, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{5/2} \sqrt{b}}-\frac{3 \sqrt{x}}{4 a^2 (a x+b)}-\frac{x^{3/2}}{2 a (a x+b)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^3*x^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 11.8574, size = 61, normalized size = 0.87 \[ - \frac{x^{\frac{3}{2}}}{2 a \left (a x + b\right )^{2}} - \frac{3 \sqrt{x}}{4 a^{2} \left (a x + b\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 a^{\frac{5}{2}} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**3/x**(3/2),x)
[Out]
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Mathematica [A] time = 0.0626015, size = 59, normalized size = 0.84 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{5/2} \sqrt{b}}-\frac{\sqrt{x} (5 a x+3 b)}{4 a^2 (a x+b)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^3*x^(3/2)),x]
[Out]
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Maple [A] time = 0.016, size = 50, normalized size = 0.7 \[ 2\,{\frac{1}{ \left ( ax+b \right ) ^{2}} \left ( -5/8\,{\frac{{x}^{3/2}}{a}}-3/8\,{\frac{b\sqrt{x}}{{a}^{2}}} \right ) }+{\frac{3}{4\,{a}^{2}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^3/x^(3/2),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^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.245491, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{-a b}{\left (5 \, a x + 3 \, b\right )} \sqrt{x} - 3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (a x - b\right )}}{a x + b}\right )}{8 \,{\left (a^{4} x^{2} + 2 \, a^{3} b x + a^{2} b^{2}\right )} \sqrt{-a b}}, -\frac{\sqrt{a b}{\left (5 \, a x + 3 \, b\right )} \sqrt{x} + 3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \arctan \left (\frac{b}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (a^{4} x^{2} + 2 \, a^{3} b x + a^{2} b^{2}\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^3*x^(3/2)),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**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221305, size = 63, normalized size = 0.9 \[ \frac{3 \, \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{2}} - \frac{5 \, a x^{\frac{3}{2}} + 3 \, b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^3*x^(3/2)),x, algorithm="giac")
[Out]