Optimal. Leaf size=52 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 b^{3/2}}-\frac{x^2}{2 b \sqrt{a+b x^4}} \]
[Out]
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Rubi [A] time = 0.0744296, antiderivative size = 52, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 b^{3/2}}-\frac{x^2}{2 b \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[x^5/(a + b*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 7.89916, size = 42, normalized size = 0.81 \[ - \frac{x^{2}}{2 b \sqrt{a + b x^{4}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{2 b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(b*x**4+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0421878, size = 55, normalized size = 1.06 \[ \frac{\log \left (\sqrt{b} \sqrt{a+b x^4}+b x^2\right )}{2 b^{3/2}}-\frac{x^2}{2 b \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/(a + b*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.016, size = 42, normalized size = 0.8 \[ -{\frac{{x}^{2}}{2\,b}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{1}{2}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(b*x^4+a)^(3/2),x)
[Out]
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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^5/(b*x^4 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.327374, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, \sqrt{b x^{4} + a} \sqrt{b} x^{2} -{\left (b x^{4} + a\right )} \log \left (-2 \, \sqrt{b x^{4} + a} b x^{2} -{\left (2 \, b x^{4} + a\right )} \sqrt{b}\right )}{4 \,{\left (b^{2} x^{4} + a b\right )} \sqrt{b}}, -\frac{\sqrt{b x^{4} + a} \sqrt{-b} x^{2} -{\left (b x^{4} + a\right )} \arctan \left (\frac{\sqrt{-b} x^{2}}{\sqrt{b x^{4} + a}}\right )}{2 \,{\left (b^{2} x^{4} + a b\right )} \sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b*x^4 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.97639, size = 44, normalized size = 0.85 \[ \frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} - \frac{x^{2}}{2 \sqrt{a} b \sqrt{1 + \frac{b x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(b*x**4+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.239285, size = 58, normalized size = 1.12 \[ -\frac{x^{2}}{2 \, \sqrt{b x^{4} + a} b} - \frac{{\rm ln}\left ({\left | -\sqrt{b} x^{2} + \sqrt{b x^{4} + a} \right |}\right )}{2 \, b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b*x^4 + a)^(3/2),x, algorithm="giac")
[Out]