Optimal. Leaf size=19 \[ \frac {g x}{\sqrt {a+b x^2+c x^4}} \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1588} \[ \frac {g x}{\sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin {align*} \int \frac {a g-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac {g x}{\sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [A] time = 0.71, size = 17, normalized size = 0.89 \[ \frac {g x}{\sqrt {c x^{4} + b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.91, size = 60, normalized size = 3.16 \[ \frac {{\left (b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g\right )} x}{\sqrt {c x^{4} + b x^{2} + a} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.95 \[ \frac {g x}{\sqrt {c \,x^{4}+b \,x^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 17, normalized size = 0.89 \[ \frac {g x}{\sqrt {c x^{4} + b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 17, normalized size = 0.89 \[ \frac {g\,x}{\sqrt {c\,x^4+b\,x^2+a}} \]
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 \[ - g \left (\int \left (- \frac {a}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx + \int \frac {c x^{4}}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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