Optimal. Leaf size=54 \[ \frac{\tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 \sqrt{a} \sqrt{b c-a d}} \]
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Rubi [A] time = 0.118259, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 \sqrt{a} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
[In] Int[x/((a + b*x^4)*Sqrt[c + d*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 16.1979, size = 46, normalized size = 0.85 \[ \frac{\operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{2 \sqrt{a} \sqrt{a d - b c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**4+a)/(d*x**4+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0389525, size = 54, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 \sqrt{a} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
[In] Integrate[x/((a + b*x^4)*Sqrt[c + d*x^4]),x]
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Maple [B] time = 0.008, size = 322, normalized size = 6. \[ -{\frac{1}{4}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{1}{4}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^4+a)/(d*x^4+c)^(1/2),x)
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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/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.290364, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{6} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2}\right )} \sqrt{d x^{4} + c} +{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{8 \, \sqrt{-a b c + a^{2} d}}, \frac{\arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} x^{2}}\right )}{4 \, \sqrt{a b c - a^{2} d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x**4+a)/(d*x**4+c)**(1/2),x)
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GIAC/XCAS [A] time = 0.223819, size = 97, normalized size = 1.8 \[ -\frac{\sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x^{2} - \sqrt{d x^{4} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt{a b c d - a^{2} d^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="giac")
[Out]