Optimal. Leaf size=50 \[ \frac{x^4 \sqrt{a+\frac{b}{x^4}}}{4 a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0938088, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{x^4 \sqrt{a+\frac{b}{x^4}}}{4 a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^3/Sqrt[a + b/x^4],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 6.99084, size = 41, normalized size = 0.82 \[ \frac{x^{4} \sqrt{a + \frac{b}{x^{4}}}}{4 a} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(a+b/x**4)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0616594, size = 78, normalized size = 1.56 \[ \frac{\sqrt{a} x^2 \left (a x^4+b\right )-b \sqrt{a x^4+b} \log \left (\sqrt{a} \sqrt{a x^4+b}+a x^2\right )}{4 a^{3/2} x^2 \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/Sqrt[a + b/x^4],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.02, size = 70, normalized size = 1.4 \[ -{\frac{1}{4\,{x}^{2}}\sqrt{a{x}^{4}+b} \left ( -{x}^{2}\sqrt{a{x}^{4}+b}{a}^{{\frac{3}{2}}}+b\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ) a \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{a}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(a+b/x^4)^(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(x^3/sqrt(a + b/x^4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.253831, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} + \sqrt{a} b \log \left (2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (2 \, a x^{4} + b\right )} \sqrt{a}\right )}{8 \, a^{2}}, \frac{a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} + \sqrt{-a} b \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}\right )}{4 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(a + b/x^4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 9.0067, size = 46, normalized size = 0.92 \[ \frac{\sqrt{b} x^{2} \sqrt{\frac{a x^{4}}{b} + 1}}{4 a} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{4 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(a+b/x**4)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.256846, size = 90, normalized size = 1.8 \[ \frac{1}{4} \, b{\left (\frac{\arctan \left (\frac{\sqrt{\frac{a x^{4} + b}{x^{4}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{\sqrt{\frac{a x^{4} + b}{x^{4}}}}{{\left (a - \frac{a x^{4} + b}{x^{4}}\right )} a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(a + b/x^4),x, algorithm="giac")
[Out]