Optimal. Leaf size=147 \[ \frac{2 (d x)^{3/2} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{3 d \sqrt{a+b x^2+c x^4}} \]
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Rubi [A] time = 0.445494, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 (d x)^{3/2} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{3 d \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d*x]/Sqrt[a + b*x^2 + c*x^4],x]
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Rubi in Sympy [A] time = 31.917, size = 129, normalized size = 0.88 \[ \frac{2 \left (d x\right )^{\frac{3}{2}} \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (\frac{3}{4},\frac{1}{2},\frac{1}{2},\frac{7}{4},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{3 a d \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**(1/2)/(c*x**4+b*x**2+a)**(1/2),x)
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Mathematica [B] time = 0.329395, size = 386, normalized size = 2.63 \[ -\frac{14 a^2 x \sqrt{d x} \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{3 \left (b-\sqrt{b^2-4 a c}\right ) \left (\sqrt{b^2-4 a c}+b\right ) \left (a+b x^2+c x^4\right )^{3/2} \left (x^2 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{7}{4};\frac{1}{2},\frac{3}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{7}{4};\frac{3}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )-7 a F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[d*x]/Sqrt[a + b*x^2 + c*x^4],x]
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Maple [F] time = 0.026, size = 0, normalized size = 0. \[ \int{1\sqrt{dx}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^(1/2)/(c*x^4+b*x^2+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x}}{\sqrt{c x^{4} + b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x)/sqrt(c*x^4 + b*x^2 + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{d x}}{\sqrt{c x^{4} + b x^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x)/sqrt(c*x^4 + b*x^2 + a),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x}}{\sqrt{a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**(1/2)/(c*x**4+b*x**2+a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x}}{\sqrt{c x^{4} + b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x)/sqrt(c*x^4 + b*x^2 + a),x, algorithm="giac")
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