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, 264} \[ -\frac {\sqrt {a+c x^4}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 4
Rule 264
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 = 0.00, size = 21, normalized size = 1.00 \[ -\frac {\sqrt {a+c x^4}}{2 a x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 17, normalized size = 0.81 \[ -\frac {\sqrt {c x^{4} + a}}{2 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 31, normalized size = 1.48 \[ \frac {\sqrt {c}}{{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.86 \[ -\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 17, normalized size = 0.81 \[ -\frac {\sqrt {c x^{4} + a}}{2 \, a x^{2}} \]
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 \[ -\frac {\sqrt {c\,x^4+a}}{2\,a\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.84, size = 20, normalized size = 0.95 \[ - \frac {\sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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