Optimal. Leaf size=38 \[ \frac {2 \tan ^{-1}\left (\frac {2 b x+c}{\sqrt {4 a b-c^2}}\right )}{\sqrt {4 a b-c^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 38, 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, 204} \[ \frac {2 \tan ^{-1}\left (\frac {2 b x+c}{\sqrt {4 a b-c^2}}\right )}{\sqrt {4 a b-c^2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rubi steps
\begin {align*} \int \frac {1}{a+c x+b x^2} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{-4 a b+c^2-x^2} \, dx,x,c+2 b x\right )\right )\\ &=\frac {2 \tan ^{-1}\left (\frac {c+2 b x}{\sqrt {4 a b-c^2}}\right )}{\sqrt {4 a b-c^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 1.00 \[ \frac {2 \tan ^{-1}\left (\frac {2 b x+c}{\sqrt {4 a b-c^2}}\right )}{\sqrt {4 a b-c^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 113, normalized size = 2.97 \[ \left [-\frac {\sqrt {-4 \, a b + c^{2}} \log \left (\frac {2 \, b^{2} x^{2} + 2 \, b c x - 2 \, a b + c^{2} - \sqrt {-4 \, a b + c^{2}} {\left (2 \, b x + c\right )}}{b x^{2} + c x + a}\right )}{4 \, a b - c^{2}}, -\frac {2 \, \arctan \left (-\frac {2 \, b x + c}{\sqrt {4 \, a b - c^{2}}}\right )}{\sqrt {4 \, a b - c^{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 34, normalized size = 0.89 \[ \frac {2 \, \arctan \left (\frac {2 \, b x + c}{\sqrt {4 \, a b - c^{2}}}\right )}{\sqrt {4 \, a b - c^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 35, normalized size = 0.92 \[ \frac {2 \arctan \left (\frac {2 b x +c}{\sqrt {4 a b -c^{2}}}\right )}{\sqrt {4 a b -c^{2}}} \]
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 = 0.23, size = 46, normalized size = 1.21 \[ \frac {2\,\mathrm {atan}\left (\frac {c}{\sqrt {4\,a\,b-c^2}}+\frac {2\,b\,x}{\sqrt {4\,a\,b-c^2}}\right )}{\sqrt {4\,a\,b-c^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.22, size = 124, normalized size = 3.26 \[ - \sqrt {- \frac {1}{4 a b - c^{2}}} \log {\left (x + \frac {- 4 a b \sqrt {- \frac {1}{4 a b - c^{2}}} + c^{2} \sqrt {- \frac {1}{4 a b - c^{2}}} + c}{2 b} \right )} + \sqrt {- \frac {1}{4 a b - c^{2}}} \log {\left (x + \frac {4 a b \sqrt {- \frac {1}{4 a b - c^{2}}} - c^{2} \sqrt {- \frac {1}{4 a b - c^{2}}} + c}{2 b} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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