Optimal. Leaf size=21 \[ -\frac {\sqrt {a+c x^4}}{2 a x^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {4, 270}
\begin {gather*} -\frac {\sqrt {a+c x^4}}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 4
Rule 270
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx &=\int \frac {1}{x^3 \sqrt {a+c x^4}} \, dx\\ &=-\frac {\sqrt {a+c x^4}}{2 a x^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a+c x^4}}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 18, normalized size = 0.86
method | result | size |
gosper | \(-\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}}\) | \(18\) |
default | \(-\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}}\) | \(18\) |
trager | \(-\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}}\) | \(18\) |
risch | \(-\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}}\) | \(18\) |
elliptic | \(-\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 17, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {c x^{4} + a}}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 17, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {c x^{4} + a}}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 20, normalized size = 0.95 \begin {gather*} - \frac {\sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{2 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.54, size = 31, normalized size = 1.48 \begin {gather*} \frac {\sqrt {c}}{{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.51, size = 17, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {c\,x^4+a}}{2\,a\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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