Optimal. Leaf size=64 \[ -\frac {\sqrt {1+\frac {d x^6}{c}} F_1\left (-\frac {1}{3};1,\frac {1}{2};\frac {2}{3};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{2 a x^2 \sqrt {c+d x^6}} \]
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Rubi [A]
time = 0.06, antiderivative size = 64, normalized size of antiderivative = 1.00, 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 = {476, 525, 524}
\begin {gather*} -\frac {\sqrt {\frac {d x^6}{c}+1} F_1\left (-\frac {1}{3};1,\frac {1}{2};\frac {2}{3};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{2 a x^2 \sqrt {c+d x^6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 476
Rule 524
Rule 525
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right ) \sqrt {c+d x^3}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {1+\frac {d x^6}{c}} \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right ) \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};1,\frac {1}{2};\frac {2}{3};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{2 a x^2 \sqrt {c+d x^6}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(64)=128\).
time = 10.08, size = 141, normalized size = 2.20 \begin {gather*} \frac {-20 a \left (c+d x^6\right )+5 (-2 b c+a d) x^6 \sqrt {1+\frac {d x^6}{c}} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+2 b d x^{12} \sqrt {1+\frac {d x^6}{c}} F_1\left (\frac {5}{3};\frac {1}{2},1;\frac {8}{3};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )}{40 a^2 c x^2 \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 \[\int \frac {1}{x^{3} \left (b \,x^{6}+a \right ) \sqrt {d \,x^{6}+c}}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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^{3} \left (a + b x^{6}\right ) \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^3\,\left (b\,x^6+a\right )\,\sqrt {d\,x^6+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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