Optimal. Leaf size=49 \[ -\frac {\tanh ^{-1}\left (\frac {x \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x^2+b x^4+c x^6}}\right )}{2 \sqrt {a}} \]
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Rubi [A]
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2021, 1918,
212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x^2+b x^4+c x^6}}\right )}{2 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 1918
Rule 2021
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x^2 \left (a+b x^2+c x^4\right )}} \, dx &=\int \frac {1}{\sqrt {a x^2+b x^4+c x^6}} \, dx\\ &=-\text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x \left (2 a+b x^2\right )}{\sqrt {a x^2+b x^4+c x^6}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {x \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x^2+b x^4+c x^6}}\right )}{2 \sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 78, normalized size = 1.59 \begin {gather*} \frac {x \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {x^2 \left (a+b x^2+c x^4\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 72, normalized size = 1.47
method | result | size |
default | \(-\frac {x \sqrt {c \,x^{4}+b \,x^{2}+a}\, \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2 \sqrt {x^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}\, \sqrt {a}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 135, normalized size = 2.76 \begin {gather*} \left [\frac {\log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{6} + b x^{4} + a x^{2}} {\left (b x^{2} + 2 \, a\right )} \sqrt {a}}{x^{5}}\right )}{4 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {c x^{6} + b x^{4} + a x^{2}} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{5} + a b x^{3} + a^{2} x\right )}}\right )}{2 \, a}\right ] \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}{\sqrt {x^{2} \left (a + b x^{2} + c x^{4}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.01, size = 62, normalized size = 1.27 \begin {gather*} -\frac {\arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-a}} + \frac {\arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x^2\,\left (c\,x^4+b\,x^2+a\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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