Optimal. Leaf size=69 \[ \frac{2 \sqrt{e x} \sqrt{c+d x^4} F_1\left (\frac{1}{8};1,-\frac{1}{2};\frac{9}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{\frac{d x^4}{c}+1}} \]
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Rubi [A] time = 0.223926, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{2 \sqrt{e x} \sqrt{c+d x^4} F_1\left (\frac{1}{8};1,-\frac{1}{2};\frac{9}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{\frac{d x^4}{c}+1}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x^4]/(Sqrt[e*x]*(a + b*x^4)),x]
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Rubi in Sympy [A] time = 50.2075, size = 56, normalized size = 0.81 \[ \frac{2 \sqrt{e x} \sqrt{c + d x^{4}} \operatorname{appellf_{1}}{\left (\frac{1}{8},- \frac{1}{2},1,\frac{9}{8},- \frac{d x^{4}}{c},- \frac{b x^{4}}{a} \right )}}{a e \sqrt{1 + \frac{d x^{4}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**4+c)**(1/2)/(e*x)**(1/2)/(b*x**4+a),x)
[Out]
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Mathematica [B] time = 0.263671, size = 168, normalized size = 2.43 \[ \frac{18 a c x \sqrt{c+d x^4} F_1\left (\frac{1}{8};-\frac{1}{2},1;\frac{9}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\sqrt{e x} \left (a+b x^4\right ) \left (4 x^4 \left (a d F_1\left (\frac{9}{8};\frac{1}{2},1;\frac{17}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-2 b c F_1\left (\frac{9}{8};-\frac{1}{2},2;\frac{17}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )+9 a c F_1\left (\frac{1}{8};-\frac{1}{2},1;\frac{9}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[c + d*x^4]/(Sqrt[e*x]*(a + b*x^4)),x]
[Out]
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Maple [F] time = 0.078, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{4}+a}\sqrt{d{x}^{4}+c}{\frac{1}{\sqrt{ex}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^4+c)^(1/2)/(e*x)^(1/2)/(b*x^4+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} \sqrt{e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*sqrt(e*x)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*sqrt(e*x)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{4}}}{\sqrt{e x} \left (a + b x^{4}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**4+c)**(1/2)/(e*x)**(1/2)/(b*x**4+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} \sqrt{e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*sqrt(e*x)),x, algorithm="giac")
[Out]