Optimal. Leaf size=62 \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{1}{8};2,\frac{1}{2};\frac{7}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a^2 x \sqrt{c+d x^8}} \]
[Out]
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Rubi [A] time = 0.204087, antiderivative size = 62, 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 \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{1}{8};2,\frac{1}{2};\frac{7}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a^2 x \sqrt{c+d x^8}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]
[Out]
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Rubi in Sympy [A] time = 25.2912, size = 53, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{8}} \operatorname{appellf_{1}}{\left (- \frac{1}{8},\frac{1}{2},2,\frac{7}{8},- \frac{d x^{8}}{c},- \frac{b x^{8}}{a} \right )}}{a^{2} c x \sqrt{1 + \frac{d x^{8}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)
[Out]
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Mathematica [B] time = 1.45121, size = 399, normalized size = 6.44 \[ \frac{-\frac{75 a x^8 \left (24 a^2 d^2-40 a b c d+9 b^2 c^2\right ) F_1\left (\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{4 x^8 \left (2 b c F_1\left (\frac{15}{8};\frac{1}{2},2;\frac{23}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{15}{8};\frac{3}{2},1;\frac{23}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-15 a c F_1\left (\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}+\frac{35 \left (c+d x^8\right ) \left (-8 a^2 d+8 a b \left (c-d x^8\right )+9 b^2 c x^8\right )}{c}+\frac{161 a b d x^{16} (9 b c-8 a d) F_1\left (\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{4 x^8 \left (2 b c F_1\left (\frac{23}{8};\frac{1}{2},2;\frac{31}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{23}{8};\frac{3}{2},1;\frac{31}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-23 a c F_1\left (\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}}{280 a^2 x \left (a+b x^8\right ) \sqrt{c+d x^8} (a d-b c)} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^2*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]
[Out]
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Maple [F] time = 0.119, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{18} + 2 \, a b x^{10} + a^{2} x^{2}\right )} \sqrt{d x^{8} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^2),x, algorithm="giac")
[Out]