Optimal. Leaf size=82 \[ -\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 x \sqrt{a+\frac{b}{x}}}{a^3}-\frac{10 x}{3 a^2 \sqrt{a+\frac{b}{x}}}-\frac{2 x}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.106736, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 x \sqrt{a+\frac{b}{x}}}{a^3}-\frac{10 x}{3 a^2 \sqrt{a+\frac{b}{x}}}-\frac{2 x}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x)^(-5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 11.1873, size = 70, normalized size = 0.85 \[ - \frac{2 x}{3 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{10 x}{3 a^{2} \sqrt{a + \frac{b}{x}}} + \frac{5 x \sqrt{a + \frac{b}{x}}}{a^{3}} - \frac{5 b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.126751, size = 82, normalized size = 1. \[ \frac{x \sqrt{a+\frac{b}{x}} \left (3 a^2 x^2+20 a b x+15 b^2\right )}{3 a^3 (a x+b)^2}-\frac{5 b \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x)^(-5/2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.014, size = 276, normalized size = 3.4 \[ -{\frac{x}{6\, \left ( ax+b \right ) ^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( -30\,{a}^{13/2}\sqrt{x \left ( ax+b \right ) }{x}^{3}+24\,{a}^{11/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}x-90\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}b+20\,b{a}^{9/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}-90\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }x{b}^{2}+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{6}b-30\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }{b}^{3}+45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{5}{b}^{2}+45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{4}{b}^{3}+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3}{b}^{4} \right ){a}^{-{\frac{13}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(5/2),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((a + b/x)^(-5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.248177, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a b x + b^{2}\right )} \sqrt{\frac{a x + b}{x}} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (3 \, a^{2} x^{2} + 20 \, a b x + 15 \, b^{2}\right )} \sqrt{a}}{6 \,{\left (a^{4} x + a^{3} b\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}, \frac{15 \,{\left (a b x + b^{2}\right )} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a^{2} x^{2} + 20 \, a b x + 15 \, b^{2}\right )} \sqrt{-a}}{3 \,{\left (a^{4} x + a^{3} b\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(-5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 17.3799, size = 774, normalized size = 9.44 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.270145, size = 132, normalized size = 1.61 \[ \frac{1}{3} \, b{\left (\frac{2 \,{\left (a + \frac{6 \,{\left (a x + b\right )}}{x}\right )} x}{{\left (a x + b\right )} a^{3} \sqrt{\frac{a x + b}{x}}} + \frac{15 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} - \frac{3 \, \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(-5/2),x, algorithm="giac")
[Out]