Optimal. Leaf size=36 \[ \frac{2}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 a}{3 b^2 \left (a+\frac{b}{x}\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0575557, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 a}{3 b^2 \left (a+\frac{b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^(5/2)*x^3),x]
[Out]
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Rubi in Sympy [A] time = 6.80363, size = 29, normalized size = 0.81 \[ - \frac{2 a}{3 b^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{2}{b^{2} \sqrt{a + \frac{b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(5/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.0378262, size = 34, normalized size = 0.94 \[ \frac{2 x \sqrt{a+\frac{b}{x}} (2 a x+3 b)}{3 b^2 (a x+b)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^(5/2)*x^3),x]
[Out]
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Maple [A] time = 0.007, size = 33, normalized size = 0.9 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax+3\,b \right ) }{3\,{b}^{2}{x}^{2}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(5/2)/x^3,x)
[Out]
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Maxima [A] time = 1.43444, size = 41, normalized size = 1.14 \[ \frac{2}{\sqrt{a + \frac{b}{x}} b^{2}} - \frac{2 \, a}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(5/2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233373, size = 45, normalized size = 1.25 \[ \frac{2 \,{\left (2 \, a x + 3 \, b\right )}}{3 \,{\left (a b^{2} x + b^{3}\right )} \sqrt{\frac{a x + b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(5/2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 11.2442, size = 82, normalized size = 2.28 \[ \begin{cases} \frac{4 a x}{3 a b^{2} x \sqrt{a + \frac{b}{x}} + 3 b^{3} \sqrt{a + \frac{b}{x}}} + \frac{6 b}{3 a b^{2} x \sqrt{a + \frac{b}{x}} + 3 b^{3} \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{2 a^{\frac{5}{2}} x^{2}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)**(5/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.269592, size = 49, normalized size = 1.36 \[ -\frac{2 \,{\left (a - \frac{3 \,{\left (a x + b\right )}}{x}\right )} x}{3 \,{\left (a x + b\right )} b^{2} \sqrt{\frac{a x + b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(5/2)*x^3),x, algorithm="giac")
[Out]