Optimal. Leaf size=34 \[ -\frac {2 \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {618, 206} \[ -\frac {2 \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rubi steps
\begin {align*} \int \frac {1}{c+b x+a x^2} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )\right )\\ &=-\frac {2 \tanh ^{-1}\left (\frac {b+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 = 0.01, size = 38, normalized size = 1.12 \[ \frac {2 \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 120, normalized size = 3.53 \[ \left [\frac {\log \left (\frac {2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, a x + b\right )}}{a x^{2} + b x + c}\right )}{\sqrt {b^{2} - 4 \, a c}}, -\frac {2 \, \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right )}{b^{2} - 4 \, a c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.35, size = 34, normalized size = 1.00 \[ \frac {2 \, \arctan \left (\frac {2 \, a x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 1.03 \[ \frac {2 \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 46, normalized size = 1.35 \[ \frac {2\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,a\,x}{\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.21, size = 124, normalized size = 3.65 \[ - \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x + \frac {- 4 a c \sqrt {- \frac {1}{4 a c - b^{2}}} + b^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + b}{2 a} \right )} + \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x + \frac {4 a c \sqrt {- \frac {1}{4 a c - b^{2}}} - b^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + b}{2 a} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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