Optimal. Leaf size=62 \[ -\frac {\sqrt {1+\frac {d x^6}{c}} F_1\left (-\frac {1}{6};2,\frac {1}{2};\frac {5}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a^2 x \sqrt {c+d x^6}} \]
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Rubi [A]
time = 0.03, 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 = {525, 524}
\begin {gather*} -\frac {\sqrt {\frac {d x^6}{c}+1} F_1\left (-\frac {1}{6};2,\frac {1}{2};\frac {5}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a^2 x \sqrt {c+d x^6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^6\right )^2 \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 )^2 \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};2,\frac {1}{2};\frac {5}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a^2 x \sqrt {c+d x^6}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(226\) vs. \(2(62)=124\).
time = 10.22, size = 226, normalized size = 3.65 \begin {gather*} \frac {55 a \left (c+d x^6\right ) \left (6 a^2 d-7 b^2 c x^6-6 a b \left (c-d x^6\right )\right )-11 \left (7 b^2 c^2-24 a b c d+12 a^2 d^2\right ) x^6 \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} 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 (7 b c-6 a d) x^{12} \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} F_1\left (\frac {11}{6};\frac {1}{2},1;\frac {17}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )}{330 a^3 c (b c-a d) x \left (a+b x^6\right ) \sqrt {c+d x^6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{2} \left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^2\,{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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