Optimal. Leaf size=64 \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{3};2,\frac{1}{2};\frac{2}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{2 a^2 x^2 \sqrt{c+d x^6}} \]
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Rubi [A] time = 0.0875388, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {465, 511, 510} \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{3};2,\frac{1}{2};\frac{2}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{2 a^2 x^2 \sqrt{c+d x^6}} \]
Antiderivative was successfully verified.
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Rule 465
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^3\right )^2 \sqrt{c+d x^3}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{1+\frac{d x^6}{c}} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^3\right )^2 \sqrt{1+\frac{d x^3}{c}}} \, dx,x,x^2\right )}{2 \sqrt{c+d x^6}}\\ &=-\frac{\sqrt{1+\frac{d x^6}{c}} F_1\left (-\frac{1}{3};2,\frac{1}{2};\frac{2}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{2 a^2 x^2 \sqrt{c+d x^6}}\\ \end{align*}
Mathematica [B] time = 0.24826, size = 226, normalized size = 3.53 \[ \frac{-5 x^6 \left (a+b x^6\right ) \sqrt{\frac{d x^6}{c}+1} \left (3 a^2 d^2-15 a b c d+8 b^2 c^2\right ) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+20 a \left (c+d x^6\right ) \left (3 a^2 d-3 a b \left (c-d x^6\right )-4 b^2 c x^6\right )+2 b d x^{12} \left (a+b x^6\right ) \sqrt{\frac{d x^6}{c}+1} (4 b c-3 a d) F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{120 a^3 c x^2 \left (a+b x^6\right ) \sqrt{c+d x^6} (b c-a d)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( b{x}^{6}+a \right ) ^{2}}{\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 )}^{2} \sqrt{d x^{6} + c} x^{3}}\,{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^{6} + a\right )}^{2} \sqrt{d x^{6} + c} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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