Optimal. Leaf size=59 \[ \frac{x \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{1}{8};1,\frac{1}{2};\frac{9}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a \sqrt{c+d x^8}} \]
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Rubi [A] time = 0.0923407, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{x \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{1}{8};1,\frac{1}{2};\frac{9}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a \sqrt{c+d x^8}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^8)*Sqrt[c + d*x^8]),x]
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Rubi in Sympy [A] time = 21.9986, size = 48, normalized size = 0.81 \[ \frac{x \sqrt{c + d x^{8}} \operatorname{appellf_{1}}{\left (\frac{1}{8},\frac{1}{2},1,\frac{9}{8},- \frac{d x^{8}}{c},- \frac{b x^{8}}{a} \right )}}{a c \sqrt{1 + \frac{d x^{8}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**8+a)/(d*x**8+c)**(1/2),x)
[Out]
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Mathematica [B] time = 0.272379, size = 161, normalized size = 2.73 \[ -\frac{9 a c x F_1\left (\frac{1}{8};\frac{1}{2},1;\frac{9}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{\left (a+b x^8\right ) \sqrt{c+d x^8} \left (4 x^8 \left (2 b c F_1\left (\frac{9}{8};\frac{1}{2},2;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{9}{8};\frac{3}{2},1;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-9 a c F_1\left (\frac{1}{8};\frac{1}{2},1;\frac{9}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((a + b*x^8)*Sqrt[c + d*x^8]),x]
[Out]
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Maple [F] time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{8}+a}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^8+a)/(d*x^8+c)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{8} + a\right )} \sqrt{d x^{8} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{8}\right ) \sqrt{c + d x^{8}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**8+a)/(d*x**8+c)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{8} + a\right )} \sqrt{d x^{8} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="giac")
[Out]