Optimal. Leaf size=62 \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{6};1,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a x \sqrt{c+d x^6}} \]
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Rubi [A] time = 0.0503759, 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} \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{6};1,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a x \sqrt{c+d x^6}} \]
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^6\right ) \sqrt{c+d x^6}} \, dx &=\frac{\sqrt{1+\frac{d x^6}{c}} \int \frac{1}{x^2 \left (a+b x^6\right ) \sqrt{1+\frac{d x^6}{c}}} \, dx}{\sqrt{c+d x^6}}\\ &=-\frac{\sqrt{1+\frac{d x^6}{c}} F_1\left (-\frac{1}{6};1,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a x \sqrt{c+d x^6}}\\ \end{align*}
Mathematica [B] time = 0.113662, size = 141, normalized size = 2.27 \[ \frac{10 b d x^{12} \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )-11 x^6 \sqrt{\frac{d x^6}{c}+1} (b c-2 a d) F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )-55 a \left (c+d x^6\right )}{55 a^2 c x \sqrt{c+d x^6}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( b{x}^{6}+a \right ) }{\frac{1}{\sqrt{d{x}^{6}+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^{6} + a\right )} \sqrt{d x^{6} + 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^{6} + c}}{b d x^{14} +{\left (b c + a d\right )} x^{8} + a c x^{2}}, x\right ) \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{1}{x^{2} \left (a + b x^{6}\right ) \sqrt{c + d x^{6}}}\, dx \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^{6} + a\right )} \sqrt{d x^{6} + c} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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