Optimal. Leaf size=22 \[ \frac {\sqrt {b x^2+c x^4}}{c x} \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3, 1588} \[ \frac {\sqrt {b x^2+c x^4}}{c x} \]
Antiderivative was successfully verified.
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Rule 3
Rule 1588
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {2+2 a-2 (1+a)+b x^2+c x^4}} \, dx &=\int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {\sqrt {b x^2+c x^4}}{c x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ \frac {\sqrt {x^2 \left (b+c x^2\right )}}{c x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 20, normalized size = 0.91 \[ \frac {\sqrt {c x^{4} + b x^{2}}}{c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 31, normalized size = 1.41 \[ -\frac {2 \, \sqrt {b}}{{\left (\sqrt {c + \frac {b}{x^{2}}} - \frac {\sqrt {b}}{x}\right )}^{2} - c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 1.18 \[ \frac {\left (c \,x^{2}+b \right ) x}{\sqrt {c \,x^{4}+b \,x^{2}}\, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 13, normalized size = 0.59 \[ \frac {\sqrt {c x^{2} + b}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.37, size = 20, normalized size = 0.91 \[ \frac {\sqrt {c\,x^4+b\,x^2}}{c\,x} \]
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 \[ \int \frac {x^{2}}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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