Optimal. Leaf size=95 \[ \frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{16 b^{3/2}}-\frac{a^2 \sqrt{a+\frac{b}{x^2}}}{16 b x}-\frac{a \sqrt{a+\frac{b}{x^2}}}{8 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{6 x^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.143646, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{16 b^{3/2}}-\frac{a^2 \sqrt{a+\frac{b}{x^2}}}{16 b x}-\frac{a \sqrt{a+\frac{b}{x^2}}}{8 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{6 x^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)^(3/2)/x^4,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.1463, size = 78, normalized size = 0.82 \[ \frac{a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{16 b^{\frac{3}{2}}} - \frac{a^{2} \sqrt{a + \frac{b}{x^{2}}}}{16 b x} - \frac{a \sqrt{a + \frac{b}{x^{2}}}}{8 x^{3}} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)**(3/2)/x**4,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.1085, size = 111, normalized size = 1.17 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (-3 a^3 x^6 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+3 a^3 x^6 \log (x)+\sqrt{b} \sqrt{a x^2+b} \left (3 a^2 x^4+14 a b x^2+8 b^2\right )\right )}{48 b^{3/2} x^5 \sqrt{a x^2+b}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)^(3/2)/x^4,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.016, size = 145, normalized size = 1.5 \[{\frac{1}{48\,{b}^{3}{x}^{3}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{6}{a}^{3}- \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{6}{a}^{3}+ \left ( a{x}^{2}+b \right ) ^{{\frac{5}{2}}}{x}^{4}{a}^{2}-3\,\sqrt{a{x}^{2}+b}{x}^{6}{a}^{3}b+2\, \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{2}ab-8\, \left ( a{x}^{2}+b \right ) ^{5/2}{b}^{2} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)^(3/2)/x^4,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)^(3/2)/x^4,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.258503, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{3} \sqrt{b} x^{5} \log \left (-\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (3 \, a^{2} b x^{4} + 14 \, a b^{2} x^{2} + 8 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{96 \, b^{2} x^{5}}, -\frac{3 \, a^{3} \sqrt{-b} x^{5} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (3 \, a^{2} b x^{4} + 14 \, a b^{2} x^{2} + 8 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{48 \, b^{2} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(3/2)/x^4,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 18.8757, size = 119, normalized size = 1.25 \[ - \frac{a^{\frac{5}{2}}}{16 b x \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{17 a^{\frac{3}{2}}}{48 x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{11 \sqrt{a} b}{24 x^{5} \sqrt{1 + \frac{b}{a x^{2}}}} + \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{16 b^{\frac{3}{2}}} - \frac{b^{2}}{6 \sqrt{a} x^{7} \sqrt{1 + \frac{b}{a x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)**(3/2)/x**4,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.264781, size = 111, normalized size = 1.17 \[ -\frac{1}{48} \, a^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{3 \,{\left (a x^{2} + b\right )}^{\frac{5}{2}} + 8 \,{\left (a x^{2} + b\right )}^{\frac{3}{2}} b - 3 \, \sqrt{a x^{2} + b} b^{2}}{a^{3} b x^{6}}\right )}{\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(3/2)/x^4,x, algorithm="giac")
[Out]