Optimal. Leaf size=22 \[ \frac {\sqrt {b x^2+c x^4}}{c x} \]
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Rubi [A]
time = 0.01, 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, 1602}
\begin {gather*} \frac {\sqrt {b x^2+c x^4}}{c x} \end {gather*}
Antiderivative was successfully verified.
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Rule 3
Rule 1602
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 = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^2 \left (b+c x^2\right )}}{c x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 26, normalized size = 1.18
method | result | size |
trager | \(\frac {\sqrt {c \,x^{4}+b \,x^{2}}}{c x}\) | \(21\) |
gosper | \(\frac {\left (c \,x^{2}+b \right ) x}{c \sqrt {c \,x^{4}+b \,x^{2}}}\) | \(26\) |
default | \(\frac {\left (c \,x^{2}+b \right ) x}{c \sqrt {c \,x^{4}+b \,x^{2}}}\) | \(26\) |
risch | \(\frac {x \left (c \,x^{2}+b \right )}{\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, c}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 13, normalized size = 0.59 \begin {gather*} \frac {\sqrt {c x^{2} + b}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 20, normalized size = 0.91 \begin {gather*} \frac {\sqrt {c x^{4} + b x^{2}}}{c x} \end {gather*}
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 \begin {gather*} \int \frac {x^{2}}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.48, size = 28, normalized size = 1.27 \begin {gather*} -\frac {\sqrt {b} \mathrm {sgn}\left (x\right )}{c} + \frac {\sqrt {c x^{2} + b}}{c \mathrm {sgn}\left (x\right )} \end {gather*}
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 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2}}{c\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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