Optimal. Leaf size=68 \[ -\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}+\frac{2 b^2 \sqrt{x}}{a^3}-\frac{2 b x^{3/2}}{3 a^2}+\frac{2 x^{5/2}}{5 a} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0766494, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}+\frac{2 b^2 \sqrt{x}}{a^3}-\frac{2 b x^{3/2}}{3 a^2}+\frac{2 x^{5/2}}{5 a} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)/(a + b/x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.8625, size = 65, normalized size = 0.96 \[ \frac{2 x^{\frac{5}{2}}}{5 a} - \frac{2 b x^{\frac{3}{2}}}{3 a^{2}} + \frac{2 b^{2} \sqrt{x}}{a^{3}} - \frac{2 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)/(a+b/x),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0456197, size = 61, normalized size = 0.9 \[ \frac{2 \sqrt{x} \left (3 a^2 x^2-5 a b x+15 b^2\right )}{15 a^3}-\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)/(a + b/x),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 54, normalized size = 0.8 \[{\frac{2}{5\,a}{x}^{{\frac{5}{2}}}}-{\frac{2\,b}{3\,{a}^{2}}{x}^{{\frac{3}{2}}}}+2\,{\frac{{b}^{2}\sqrt{x}}{{a}^{3}}}-2\,{\frac{{b}^{3}}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)/(a+b/x),x)
[Out]
_______________________________________________________________________________________
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(x^(3/2)/(a + b/x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.239522, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{2} \sqrt{-\frac{b}{a}} \log \left (\frac{a x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (3 \, a^{2} x^{2} - 5 \, a b x + 15 \, b^{2}\right )} \sqrt{x}}{15 \, a^{3}}, -\frac{2 \,{\left (15 \, b^{2} \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) -{\left (3 \, a^{2} x^{2} - 5 \, a b x + 15 \, b^{2}\right )} \sqrt{x}\right )}}{15 \, a^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(a + b/x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 23.197, size = 121, normalized size = 1.78 \[ \begin{cases} \frac{2 x^{\frac{5}{2}}}{5 a} - \frac{2 b x^{\frac{3}{2}}}{3 a^{2}} + \frac{2 b^{2} \sqrt{x}}{a^{3}} + \frac{i b^{\frac{5}{2}} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{4} \sqrt{\frac{1}{a}}} - \frac{i b^{\frac{5}{2}} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{4} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\\frac{2 x^{\frac{7}{2}}}{7 b} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)/(a+b/x),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.217959, size = 80, normalized size = 1.18 \[ -\frac{2 \, b^{3} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} + \frac{2 \,{\left (3 \, a^{4} x^{\frac{5}{2}} - 5 \, a^{3} b x^{\frac{3}{2}} + 15 \, a^{2} b^{2} \sqrt{x}\right )}}{15 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(a + b/x),x, algorithm="giac")
[Out]