Optimal. Leaf size=45 \[ -\frac {\tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{\sqrt {a}} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1918, 212}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 1918
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx &=-\left (2 \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x (2 a+b x)}{\sqrt {a x^2+b x^3+c x^4}}\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 68, normalized size = 1.51 \begin {gather*} \frac {2 x \sqrt {a+x (b+c x)} \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {x^2 (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 66, normalized size = 1.47
method | result | size |
default | \(-\frac {x \sqrt {c \,x^{2}+b x +a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \sqrt {a}}\) | \(66\) |
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 = 130, normalized size = 2.89 \begin {gather*} \left [\frac {\log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right )}{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 {a x^{2} + b x^{3} + c x^{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.76, size = 59, normalized size = 1.31 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-a}} + \frac {2 \, \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + 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 {c\,x^4+b\,x^3+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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