Optimal. Leaf size=35 \[ -\frac {\log \left (-2 \sqrt {a} \sqrt {a x^2+b x+c}+2 a x+b\right )}{\sqrt {a}} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {621, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c+b x+a x^2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {b+2 a x}{\sqrt {c+b x+a x^2}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {c+b x+a x^2}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 34, normalized size = 0.97 \begin {gather*} \frac {\log \left (2 \sqrt {a} \sqrt {a x^2+b x+c}+2 a x+b\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 35, normalized size = 1.00 \begin {gather*} -\frac {\log \left (b+2 a x-2 \sqrt {a} \sqrt {c+b x+a x^2}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 106, normalized size = 3.03 \begin {gather*} \left [\frac {\log \left (-8 \, a^{2} x^{2} - 8 \, a b x - 4 \, \sqrt {a x^{2} + b x + c} {\left (2 \, a x + b\right )} \sqrt {a} - b^{2} - 4 \, a c\right )}{2 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {a x^{2} + b x + c} {\left (2 \, a x + b\right )} \sqrt {-a}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right )}{a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 36, normalized size = 1.03 \begin {gather*} -\frac {\log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x + c}\right )} \sqrt {a} - b \right |}\right )}{\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 30, normalized size = 0.86
method | result | size |
default | \(\frac {\ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{\sqrt {a}}\) | \(30\) |
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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mupad [B] time = 0.28, size = 29, normalized size = 0.83 \begin {gather*} \frac {\ln \left (\frac {\frac {b}{2}+a\,x}{\sqrt {a}}+\sqrt {a\,x^2+b\,x+c}\right )}{\sqrt {a}} \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 + c}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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