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 = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 510
Rule 511
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*}
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Mathematica [B] time = 0.13, size = 141, normalized size = 2.27 \[ \frac {-11 x^6 \sqrt {\frac {d x^6}{c}+1} (b c-2 a d) F_1\left (\frac {5}{6};\frac {1}{2},1;\frac {11}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+10 b d x^{12} \sqrt {\frac {d x^6}{c}+1} F_1\left (\frac {11}{6};\frac {1}{2},1;\frac {17}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )-55 a \left (c+d x^6\right )}{55 a^2 c x \sqrt {c+d x^6}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x^{6} + c}}{b d x^{14} + {\left (b c + a d\right )} x^{8} + a c x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{6}+a \right ) \sqrt {d \,x^{6}+c}\, x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^2\,\left (b\,x^6+a\right )\,\sqrt {d\,x^6+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a + b x^{6}\right ) \sqrt {c + d x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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