Optimal. Leaf size=98 \[ \frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{11/2}}-\frac{9 b^3 \sqrt{x}}{a^5}+\frac{3 b^2 x^{3/2}}{a^4}-\frac{9 b x^{5/2}}{5 a^3}+\frac{9 x^{7/2}}{7 a^2}-\frac{x^{9/2}}{a (a x+b)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.121125, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{11/2}}-\frac{9 b^3 \sqrt{x}}{a^5}+\frac{3 b^2 x^{3/2}}{a^4}-\frac{9 b x^{5/2}}{5 a^3}+\frac{9 x^{7/2}}{7 a^2}-\frac{x^{9/2}}{a (a x+b)} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)/(a + b/x)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 22.2268, size = 92, normalized size = 0.94 \[ - \frac{x^{\frac{9}{2}}}{a \left (a x + b\right )} + \frac{9 x^{\frac{7}{2}}}{7 a^{2}} - \frac{9 b x^{\frac{5}{2}}}{5 a^{3}} + \frac{3 b^{2} x^{\frac{3}{2}}}{a^{4}} - \frac{9 b^{3} \sqrt{x}}{a^{5}} + \frac{9 b^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)/(a+b/x)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0989349, size = 90, normalized size = 0.92 \[ \frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{11/2}}+\frac{\sqrt{x} \left (10 a^4 x^4-18 a^3 b x^3+42 a^2 b^2 x^2-210 a b^3 x-315 b^4\right )}{35 a^5 (a x+b)} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)/(a + b/x)^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.015, size = 83, normalized size = 0.9 \[{\frac{2}{7\,{a}^{2}}{x}^{{\frac{7}{2}}}}-{\frac{4\,b}{5\,{a}^{3}}{x}^{{\frac{5}{2}}}}+2\,{\frac{{b}^{2}{x}^{3/2}}{{a}^{4}}}-8\,{\frac{{b}^{3}\sqrt{x}}{{a}^{5}}}-{\frac{{b}^{4}}{{a}^{5} \left ( ax+b \right ) }\sqrt{x}}+9\,{\frac{{b}^{4}}{{a}^{5}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)/(a+b/x)^2,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^(5/2)/(a + b/x)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.240039, size = 1, normalized size = 0.01 \[ \left [\frac{315 \,{\left (a b^{3} x + b^{4}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (10 \, a^{4} x^{4} - 18 \, a^{3} b x^{3} + 42 \, a^{2} b^{2} x^{2} - 210 \, a b^{3} x - 315 \, b^{4}\right )} \sqrt{x}}{70 \,{\left (a^{6} x + a^{5} b\right )}}, \frac{315 \,{\left (a b^{3} x + b^{4}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{a}}}\right ) +{\left (10 \, a^{4} x^{4} - 18 \, a^{3} b x^{3} + 42 \, a^{2} b^{2} x^{2} - 210 \, a b^{3} x - 315 \, b^{4}\right )} \sqrt{x}}{35 \,{\left (a^{6} x + a^{5} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(a + b/x)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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(x**(5/2)/(a+b/x)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.218882, size = 119, normalized size = 1.21 \[ \frac{9 \, b^{4} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{5}} - \frac{b^{4} \sqrt{x}}{{\left (a x + b\right )} a^{5}} + \frac{2 \,{\left (5 \, a^{12} x^{\frac{7}{2}} - 14 \, a^{11} b x^{\frac{5}{2}} + 35 \, a^{10} b^{2} x^{\frac{3}{2}} - 140 \, a^{9} b^{3} \sqrt{x}\right )}}{35 \, a^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(a + b/x)^2,x, algorithm="giac")
[Out]