Optimal. Leaf size=59 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 c^{3/2}}+\frac{a x^2 \sqrt{c+\frac{d}{x^2}}}{2 c} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.139945, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 c^{3/2}}+\frac{a x^2 \sqrt{c+\frac{d}{x^2}}}{2 c} \]
Antiderivative was successfully verified.
[In] Int[((a + b/x^2)*x)/Sqrt[c + d/x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 11.3052, size = 48, normalized size = 0.81 \[ \frac{a x^{2} \sqrt{c + \frac{d}{x^{2}}}}{2 c} - \frac{\left (\frac{a d}{2} - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)*x/(c+d/x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0749944, size = 82, normalized size = 1.39 \[ \frac{\sqrt{c x^2+d} (2 b c-a d) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )+a \sqrt{c} x \left (c x^2+d\right )}{2 c^{3/2} x \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b/x^2)*x)/Sqrt[c + d/x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 90, normalized size = 1.5 \[{\frac{1}{2\,x}\sqrt{c{x}^{2}+d} \left ( ax\sqrt{c{x}^{2}+d}{c}^{{\frac{3}{2}}}+2\,b\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{2}-ad\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) c \right ){\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}{c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)*x/(c+d/x^2)^(1/2),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^2)*x/sqrt(c + d/x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.233514, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, a c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, b c - a d\right )} \sqrt{c} \log \left (2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right )}{4 \, c^{2}}, \frac{a c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, b c - a d\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right )}{2 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x/sqrt(c + d/x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 12.4467, size = 66, normalized size = 1.12 \[ \frac{a \sqrt{d} x \sqrt{\frac{c x^{2}}{d} + 1}}{2 c} - \frac{a d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 c^{\frac{3}{2}}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{\sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)*x/(c+d/x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.246198, size = 107, normalized size = 1.81 \[ -\frac{1}{2} \, d{\left (\frac{a \sqrt{\frac{c x^{2} + d}{x^{2}}}}{{\left (c - \frac{c x^{2} + d}{x^{2}}\right )} c} + \frac{{\left (2 \, b c - a d\right )} \arctan \left (\frac{\sqrt{\frac{c x^{2} + d}{x^{2}}}}{\sqrt{-c}}\right )}{\sqrt{-c} c d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x/sqrt(c + d/x^2),x, algorithm="giac")
[Out]