Optimal. Leaf size=59 \[ \frac{x \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{1}{6};2,\frac{1}{2};\frac{7}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a^2 \sqrt{c+d x^6}} \]
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Rubi [A] time = 0.02523, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {430, 429} \[ \frac{x \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{1}{6};2,\frac{1}{2};\frac{7}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a^2 \sqrt{c+d x^6}} \]
Antiderivative was successfully verified.
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Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx &=\frac{\sqrt{1+\frac{d x^6}{c}} \int \frac{1}{\left (a+b x^6\right )^2 \sqrt{1+\frac{d x^6}{c}}} \, dx}{\sqrt{c+d x^6}}\\ &=\frac{x \sqrt{1+\frac{d x^6}{c}} F_1\left (\frac{1}{6};2,\frac{1}{2};\frac{7}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a^2 \sqrt{c+d x^6}}\\ \end{align*}
Mathematica [B] time = 0.28211, size = 329, normalized size = 5.58 \[ \frac{x \left (-2 b d x^6 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{7}{6};\frac{1}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )-\frac{7 a \left (3 b x^6 \left (c+d x^6\right ) \left (2 b c F_1\left (\frac{7}{6};\frac{1}{2},2;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{7}{6};\frac{3}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )+7 a c \left (6 a d-b \left (6 c+d x^6\right )\right ) F_1\left (\frac{1}{6};\frac{1}{2},1;\frac{7}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )}{\left (a+b x^6\right ) \left (3 x^6 \left (2 b c F_1\left (\frac{7}{6};\frac{1}{2},2;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{7}{6};\frac{3}{2},1;\frac{13}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-7 a c F_1\left (\frac{1}{6};\frac{1}{2},1;\frac{7}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )}\right )}{42 a^2 \sqrt{c+d x^6} (a d-b c)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \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}}\,{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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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