Optimal. Leaf size=23 \[ -\frac {\sqrt {b x^2+c x^4}}{b x^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 23, 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, 2039}
\begin {gather*} -\frac {\sqrt {b x^2+c x^4}}{b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3
Rule 2039
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {2+2 a-2 (1+a)+b x^2+c x^4}} \, dx &=\int \frac {1}{x \sqrt {b x^2+c x^4}} \, dx\\ &=-\frac {\sqrt {b x^2+c x^4}}{b x^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {x^2 \left (b+c x^2\right )}}{b x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 26, normalized size = 1.13
method | result | size |
trager | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}}}{b \,x^{2}}\) | \(22\) |
gosper | \(-\frac {c \,x^{2}+b}{b \sqrt {c \,x^{4}+b \,x^{2}}}\) | \(26\) |
default | \(-\frac {c \,x^{2}+b}{b \sqrt {c \,x^{4}+b \,x^{2}}}\) | \(26\) |
risch | \(-\frac {c \,x^{2}+b}{\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, b}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 21, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}}}{b x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 21, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}}}{b x^{2}} \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 {1}{x \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 = 4.14, size = 34, normalized size = 1.48 \begin {gather*} \frac {2 \, \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.31, size = 21, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {c\,x^4+b\,x^2}}{b\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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