Optimal. Leaf size=62 \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{1}{8};2,\frac{1}{2};\frac{7}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a^2 x \sqrt{c+d x^8}} \]
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Rubi [A] time = 0.0544516, 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, Rules used = {511, 510} \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{1}{8};2,\frac{1}{2};\frac{7}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a^2 x \sqrt{c+d x^8}} \]
Antiderivative was successfully verified.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx &=\frac{\sqrt{1+\frac{d x^8}{c}} \int \frac{1}{x^2 \left (a+b x^8\right )^2 \sqrt{1+\frac{d x^8}{c}}} \, dx}{\sqrt{c+d x^8}}\\ &=-\frac{\sqrt{1+\frac{d x^8}{c}} 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}}\\ \end{align*}
Mathematica [B] time = 0.263178, size = 226, normalized size = 3.65 \[ \frac{-5 x^8 \left (a+b x^8\right ) \sqrt{\frac{d x^8}{c}+1} \left (24 a^2 d^2-40 a b c d+9 b^2 c^2\right ) F_1\left (\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+35 a \left (c+d x^8\right ) \left (8 a^2 d-8 a b \left (c-d x^8\right )-9 b^2 c x^8\right )+7 b d x^{16} \left (a+b x^8\right ) \sqrt{\frac{d x^8}{c}+1} (9 b c-8 a d) F_1\left (\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{280 a^3 c x \left (a+b x^8\right ) \sqrt{c+d x^8} (b c-a d)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.068, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x^{8} + c}}{b^{2} d x^{26} +{\left (b^{2} c + 2 \, a b d\right )} x^{18} +{\left (2 \, a b c + a^{2} d\right )} x^{10} + a^{2} c x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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