Optimal. Leaf size=19 \[ -\frac {\sqrt {a+b x^2}}{a x} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, 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 = {5, 270}
\begin {gather*} -\frac {\sqrt {a+b x^2}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 5
Rule 270
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx\\ &=-\frac {\sqrt {a+b x^2}}{a x}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 19, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a+b x^2}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 18, normalized size = 0.95
method | result | size |
gosper | \(-\frac {\sqrt {b \,x^{2}+a}}{a x}\) | \(18\) |
default | \(-\frac {\sqrt {b \,x^{2}+a}}{a x}\) | \(18\) |
trager | \(-\frac {\sqrt {b \,x^{2}+a}}{a x}\) | \(18\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}}{a x}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 17, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {b x^{2} + a}}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 17, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {b x^{2} + a}}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 19, normalized size = 1.00 \begin {gather*} - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.80, size = 30, normalized size = 1.58 \begin {gather*} \frac {2 \, \sqrt {b}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 17, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {b\,x^2+a}}{a\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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