Optimal. Leaf size=64 \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{3}{8};2,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a^2 x^3 \sqrt{c+d x^8}} \]
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Rubi [A] time = 0.0561598, antiderivative size = 64, 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} \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{3}{8};2,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a^2 x^3 \sqrt{c+d x^8}} \]
Antiderivative was successfully verified.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{1}{x^4 \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^4 \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{3}{8};2,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a^2 x^3 \sqrt{c+d x^8}}\\ \end{align*}
Mathematica [B] time = 0.270698, size = 226, normalized size = 3.53 \[ \frac{-13 x^8 \left (a+b x^8\right ) \sqrt{\frac{d x^8}{c}+1} \left (8 a^2 d^2-56 a b c d+33 b^2 c^2\right ) F_1\left (\frac{5}{8};\frac{1}{2},1;\frac{13}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+65 a \left (c+d x^8\right ) \left (8 a^2 d-8 a b \left (c-d x^8\right )-11 b^2 c x^8\right )+5 b d x^{16} \left (a+b x^8\right ) \sqrt{\frac{d x^8}{c}+1} (11 b c-8 a d) F_1\left (\frac{13}{8};\frac{1}{2},1;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{1560 a^3 c x^3 \left (a+b x^8\right ) \sqrt{c+d x^8} (b c-a d)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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