Optimal. Leaf size=18 \[ \frac {\sqrt {a+c x^4}}{2 c} \]
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Rubi [A] time = 0.00, antiderivative size = 18, 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, 261} \begin {gather*} \frac {\sqrt {a+c x^4}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 4
Rule 261
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx &=\int \frac {x^3}{\sqrt {a+c x^4}} \, dx\\ &=\frac {\sqrt {a+c x^4}}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a+c x^4}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a+c x^4}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 14, normalized size = 0.78 \begin {gather*} \frac {\sqrt {c x^{4} + a}}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 14, normalized size = 0.78 \begin {gather*} \frac {\sqrt {c x^{4} + a}}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 15, normalized size = 0.83 \begin {gather*} \frac {\sqrt {c \,x^{4}+a}}{2 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 14, normalized size = 0.78 \begin {gather*} \frac {\sqrt {c x^{4} + a}}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.66, size = 14, normalized size = 0.78 \begin {gather*} \frac {\sqrt {c\,x^4+a}}{2\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.89, size = 22, normalized size = 1.22 \begin {gather*} \begin {cases} \frac {\sqrt {a + c x^{4}}}{2 c} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4 \sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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