Optimal. Leaf size=36 \[ \frac {2 \tanh ^{-1}\left (\frac {b+\frac {2 a}{x}}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
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Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1366, 632, 212}
\begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\frac {2 a}{x}+b}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 1366
Rubi steps
\begin {align*} \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right ) x^2} \, dx &=-\text {Subst}\left (\int \frac {1}{c+b x+a x^2} \, dx,x,\frac {1}{x}\right )\\ &=2 \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+\frac {2 a}{x}\right )\\ &=\frac {2 \tanh ^{-1}\left (\frac {b+\frac {2 a}{x}}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 38, normalized size = 1.06 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 35, normalized size = 0.97
method | result | size |
default | \(\frac {2 \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\) | \(35\) |
risch | \(-\frac {\ln \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}{\sqrt {-4 a c +b^{2}}}+\frac {\ln \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right )}{\sqrt {-4 a c +b^{2}}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 120, normalized size = 3.33 \begin {gather*} \left [\frac {\log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{\sqrt {b^{2} - 4 \, a c}}, -\frac {2 \, \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{b^{2} - 4 \, a c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs.
\(2 (32) = 64\).
time = 0.09, size = 124, normalized size = 3.44 \begin {gather*} - \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x + \frac {- 4 a c \sqrt {- \frac {1}{4 a c - b^{2}}} + b^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + b}{2 c} \right )} + \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x + \frac {4 a c \sqrt {- \frac {1}{4 a c - b^{2}}} - b^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + b}{2 c} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.10, size = 34, normalized size = 0.94 \begin {gather*} \frac {2 \, \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 46, normalized size = 1.28 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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