Optimal. Leaf size=93 \[ -\frac{c (3 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 d^{5/2}}+\frac{\sqrt{c+\frac{d}{x^2}} (3 b c-4 a d)}{8 d^2 x}-\frac{b \sqrt{c+\frac{d}{x^2}}}{4 d x^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.173637, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{c (3 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 d^{5/2}}+\frac{\sqrt{c+\frac{d}{x^2}} (3 b c-4 a d)}{8 d^2 x}-\frac{b \sqrt{c+\frac{d}{x^2}}}{4 d x^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)/(Sqrt[c + d/x^2]*x^4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.415, size = 82, normalized size = 0.88 \[ - \frac{b \sqrt{c + \frac{d}{x^{2}}}}{4 d x^{3}} + \frac{c \left (4 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{8 d^{\frac{5}{2}}} - \frac{\sqrt{c + \frac{d}{x^{2}}} \left (4 a d - 3 b c\right )}{8 d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)/x**4/(c+d/x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.181506, size = 129, normalized size = 1.39 \[ \frac{-\sqrt{d} \left (c x^2+d\right ) \left (4 a d x^2-3 b c x^2+2 b d\right )+c x^4 \log (x) \sqrt{c x^2+d} (3 b c-4 a d)+c x^4 \sqrt{c x^2+d} (4 a d-3 b c) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )}{8 d^{5/2} x^5 \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)/(Sqrt[c + d/x^2]*x^4),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.032, size = 148, normalized size = 1.6 \[ -{\frac{1}{8\,{x}^{5}}\sqrt{c{x}^{2}+d} \left ( -4\,ac\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{4}{d}^{3}+4\,a\sqrt{c{x}^{2}+d}{x}^{2}{d}^{7/2}-3\,bc\sqrt{c{x}^{2}+d}{x}^{2}{d}^{5/2}+3\,b{c}^{2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{4}{d}^{2}+2\,b\sqrt{c{x}^{2}+d}{d}^{7/2} \right ){\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}{d}^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)/x^4/(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)/(sqrt(c + d/x^2)*x^4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.235741, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, b c^{2} - 4 \, a c d\right )} \sqrt{d} x^{3} \log \left (-\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) + 2 \,{\left (2 \, b d^{2} -{\left (3 \, b c d - 4 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \, d^{3} x^{3}}, \frac{{\left (3 \, b c^{2} - 4 \, a c d\right )} \sqrt{-d} x^{3} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left (2 \, b d^{2} -{\left (3 \, b c d - 4 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \, d^{3} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/(sqrt(c + d/x^2)*x^4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 12.8797, size = 150, normalized size = 1.61 \[ - \frac{a \sqrt{c} \sqrt{1 + \frac{d}{c x^{2}}}}{2 d x} + \frac{a c \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{2 d^{\frac{3}{2}}} + \frac{3 b c^{\frac{3}{2}}}{8 d^{2} x \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{b \sqrt{c}}{8 d x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{8 d^{\frac{5}{2}}} - \frac{b}{4 \sqrt{c} x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)/x**4/(c+d/x**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{\sqrt{c + \frac{d}{x^{2}}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/(sqrt(c + d/x^2)*x^4),x, algorithm="giac")
[Out]