Optimal. Leaf size=64 \[ \frac{x^3 \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{3}{8};1,\frac{1}{2};\frac{11}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a \sqrt{c+d x^8}} \]
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Rubi [A] time = 0.053032, 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} \[ \frac{x^3 \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{3}{8};1,\frac{1}{2};\frac{11}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a \sqrt{c+d x^8}} \]
Antiderivative was successfully verified.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx &=\frac{\sqrt{1+\frac{d x^8}{c}} \int \frac{x^2}{\left (a+b x^8\right ) \sqrt{1+\frac{d x^8}{c}}} \, dx}{\sqrt{c+d x^8}}\\ &=\frac{x^3 \sqrt{1+\frac{d x^8}{c}} F_1\left (\frac{3}{8};1,\frac{1}{2};\frac{11}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a \sqrt{c+d x^8}}\\ \end{align*}
Mathematica [A] time = 0.037005, size = 65, normalized size = 1.02 \[ \frac{x^3 \sqrt{\frac{c+d x^8}{c}} F_1\left (\frac{3}{8};\frac{1}{2},1;\frac{11}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{3 a \sqrt{c+d x^8}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{b{x}^{8}+a}{\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{x^{2}}{{\left (b x^{8} + a\right )} \sqrt{d x^{8} + c}}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (a + b x^{8}\right ) \sqrt{c + d x^{8}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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