Optimal. Leaf size=103 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a+b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} c}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a+b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} c} \]
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Rubi [A] time = 0.141018, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a+b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} c}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a+b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} c} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x^4]/(a*c - b*c*x^4),x]
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Rubi in Sympy [A] time = 22.7375, size = 94, normalized size = 0.91 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a + b x^{4}}} \right )}}{4 \sqrt [4]{a} \sqrt [4]{b} c} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a + b x^{4}}} \right )}}{4 \sqrt [4]{a} \sqrt [4]{b} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**(1/2)/(-b*c*x**4+a*c),x)
[Out]
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Mathematica [C] time = 0.242136, size = 155, normalized size = 1.5 \[ \frac{5 a x \sqrt{a+b x^4} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right )}{c \left (a-b x^4\right ) \left (2 b x^4 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right )+F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right )\right )+5 a F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[a + b*x^4]/(a*c - b*c*x^4),x]
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Maple [A] time = 0.024, size = 103, normalized size = 1. \[ -{\frac{\sqrt{2}}{4\,c}\arctan \left ({\frac{\sqrt{2}}{2\,x}\sqrt{b{x}^{4}+a}{\frac{1}{\sqrt [4]{ab}}}} \right ){\frac{1}{\sqrt [4]{ab}}}}+{\frac{\sqrt{2}}{8\,c}\ln \left ({1 \left ({\frac{\sqrt{2}}{2\,x}\sqrt{b{x}^{4}+a}}+\sqrt [4]{ab} \right ) \left ({\frac{\sqrt{2}}{2\,x}\sqrt{b{x}^{4}+a}}-\sqrt [4]{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^(1/2)/(-b*c*x^4+a*c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{b x^{4} + a}}{b c x^{4} - a c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(b*x^4 + a)/(b*c*x^4 - a*c),x, algorithm="maxima")
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Fricas [A] time = 0.592124, size = 450, normalized size = 4.37 \[ -\left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{2 \,{\left (2 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} - \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}}\right )}}{\sqrt{b x^{4} + a}{\left (a c^{2} \sqrt{\frac{1}{a b c^{4}}} - x^{2}\right )} - \frac{b x^{4} - a}{\sqrt{b}}}\right ) + \frac{1}{4} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \log \left (\frac{4 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a}{\left (a c^{2} \sqrt{\frac{1}{a b c^{4}}} + x^{2}\right )}}{b x^{4} - a}\right ) - \frac{1}{4} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{4 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} - \sqrt{b x^{4} + a}{\left (a c^{2} \sqrt{\frac{1}{a b c^{4}}} + x^{2}\right )}}{b x^{4} - a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(b*x^4 + a)/(b*c*x^4 - a*c),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{\int \frac{\sqrt{a + b x^{4}}}{- a + b x^{4}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)**(1/2)/(-b*c*x**4+a*c),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{b x^{4} + a}}{b c x^{4} - a c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(b*x^4 + a)/(b*c*x^4 - a*c),x, algorithm="giac")
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