Optimal. Leaf size=70 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{a} b^{5/2}}+\frac{3 \sqrt{x}}{4 b^2 (a x+b)}+\frac{\sqrt{x}}{2 b (a x+b)^2} \]
[Out]
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Rubi [A] time = 0.0681785, 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 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{a} b^{5/2}}+\frac{3 \sqrt{x}}{4 b^2 (a x+b)}+\frac{\sqrt{x}}{2 b (a x+b)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^3*x^(7/2)),x]
[Out]
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Rubi in Sympy [A] time = 11.5646, size = 61, normalized size = 0.87 \[ \frac{\sqrt{x}}{2 b \left (a x + b\right )^{2}} + \frac{3 \sqrt{x}}{4 b^{2} \left (a x + b\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 \sqrt{a} b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**3/x**(7/2),x)
[Out]
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Mathematica [A] time = 0.051301, size = 59, normalized size = 0.84 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{a} b^{5/2}}+\frac{\sqrt{x} (3 a x+5 b)}{4 b^2 (a x+b)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^3*x^(7/2)),x]
[Out]
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Maple [A] time = 0.01, size = 53, normalized size = 0.8 \[{\frac{1}{2\,b \left ( ax+b \right ) ^{2}}\sqrt{x}}+{\frac{3}{4\,{b}^{2} \left ( ax+b \right ) }\sqrt{x}}+{\frac{3}{4\,{b}^{2}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^3/x^(7/2),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^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24224, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{-a b}{\left (3 \, a x + 5 \, b\right )} \sqrt{x} + 3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (a x - b\right )}}{a x + b}\right )}{8 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt{-a b}}, \frac{\sqrt{a b}{\left (3 \, a x + 5 \, b\right )} \sqrt{x} - 3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \arctan \left (\frac{b}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^3*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)**3/x**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.222764, size = 63, normalized size = 0.9 \[ \frac{3 \, \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{2}} + \frac{3 \, a x^{\frac{3}{2}} + 5 \, b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^3*x^(7/2)),x, algorithm="giac")
[Out]