Optimal. Leaf size=74 \[ -\frac{16 b^2}{3 a^3 \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{8 b \sqrt{x}}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{3/2}}{3 a \sqrt{a+\frac{b}{x}}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0818078, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{16 b^2}{3 a^3 \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{8 b \sqrt{x}}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{3/2}}{3 a \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/(a + b/x)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 6.73129, size = 63, normalized size = 0.85 \[ \frac{2 x^{\frac{3}{2}}}{3 a \sqrt{a + \frac{b}{x}}} - \frac{8 b \sqrt{x}}{3 a^{2} \sqrt{a + \frac{b}{x}}} - \frac{16 b^{2}}{3 a^{3} \sqrt{x} \sqrt{a + \frac{b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(a+b/x)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0510546, size = 48, normalized size = 0.65 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (a^2 x^2-4 a b x-8 b^2\right )}{3 a^3 (a x+b)} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/(a + b/x)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 43, normalized size = 0.6 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ({a}^{2}{x}^{2}-4\,abx-8\,{b}^{2} \right ) }{3\,{a}^{3}}{x}^{-{\frac{3}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(a+b/x)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.44394, size = 74, normalized size = 1. \[ \frac{2 \,{\left ({\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} x^{\frac{3}{2}} - 6 \, \sqrt{a + \frac{b}{x}} b \sqrt{x}\right )}}{3 \, a^{3}} - \frac{2 \, b^{2}}{\sqrt{a + \frac{b}{x}} a^{3} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(a + b/x)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.227952, size = 50, normalized size = 0.68 \[ \frac{2 \,{\left (a^{2} x^{2} - 4 \, a b x - 8 \, b^{2}\right )}}{3 \, a^{3} \sqrt{x} \sqrt{\frac{a x + b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(a + b/x)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 16.802, size = 206, normalized size = 2.78 \[ \frac{2 a^{3} b^{\frac{9}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac{6 a^{2} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac{24 a b^{\frac{13}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac{16 b^{\frac{15}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(a+b/x)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.234486, size = 59, normalized size = 0.8 \[ \frac{16 \, b^{\frac{3}{2}}}{3 \, a^{3}} + \frac{2 \,{\left ({\left (a x + b\right )}^{\frac{3}{2}} - 6 \, \sqrt{a x + b} b - \frac{3 \, b^{2}}{\sqrt{a x + b}}\right )}}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(a + b/x)^(3/2),x, algorithm="giac")
[Out]