Optimal. Leaf size=73 \[ \frac {\left (b^2-4 a c\right ) \log \left (-2 \sqrt {a} \sqrt {a x^2+b x+c}+2 a x+b\right )}{8 a^{3/2}}+\frac {(2 a x+b) \sqrt {a x^2+b x+c}}{4 a} \]
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Rubi [A] time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {612, 621, 206} \begin {gather*} \frac {(2 a x+b) \sqrt {a x^2+b x+c}}{4 a}-\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{8 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rubi steps
\begin {align*} \int \sqrt {c+b x+a x^2} \, dx &=\frac {(b+2 a x) \sqrt {c+b x+a x^2}}{4 a}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c+b x+a x^2}} \, dx}{8 a}\\ &=\frac {(b+2 a x) \sqrt {c+b x+a x^2}}{4 a}-\frac {\left (b^2-4 a c\right ) \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 )}{4 a}\\ &=\frac {(b+2 a x) \sqrt {c+b x+a x^2}}{4 a}-\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {c+b x+a x^2}}\right )}{8 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 71, normalized size = 0.97 \begin {gather*} \frac {(2 a x+b) \sqrt {x (a x+b)+c}}{4 a}-\frac {\left (b^2-4 a c\right ) \log \left (2 \sqrt {a} \sqrt {x (a x+b)+c}+2 a x+b\right )}{8 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 77, normalized size = 1.05 \begin {gather*} \frac {(b+2 a x) \sqrt {c+b x+a x^2}}{4 a}+\frac {\left (b^2-4 a c\right ) \log \left (a b+2 a^2 x-2 a^{3/2} \sqrt {c+b x+a x^2}\right )}{8 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 177, normalized size = 2.42 \begin {gather*} \left [-\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {a} \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 ) - 4 \, {\left (2 \, a^{2} x + a b\right )} \sqrt {a x^{2} + b x + c}}{16 \, a^{2}}, \frac {{\left (b^{2} - 4 \, a c\right )} \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 ) + 2 \, {\left (2 \, a^{2} x + a b\right )} \sqrt {a x^{2} + b x + c}}{8 \, a^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 68, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, \sqrt {a x^{2} + b x + c} {\left (2 \, x + \frac {b}{a}\right )} + \frac {{\left (b^{2} - 4 \, a c\right )} \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x + c}\right )} \sqrt {a} - b \right |}\right )}{8 \, a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 65, normalized size = 0.89
method | result | size |
default | \(\frac {\left (2 a x +b \right ) \sqrt {a \,x^{2}+b x +c}}{4 a}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{8 a^{\frac {3}{2}}}\) | \(65\) |
risch | \(\frac {\left (2 a x +b \right ) \sqrt {a \,x^{2}+b x +c}}{4 a}+\frac {\ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c}{2 \sqrt {a}}-\frac {\ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) b^{2}}{8 a^{\frac {3}{2}}}\) | \(89\) |
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 = 0.12, size = 63, normalized size = 0.86 \begin {gather*} \left (\frac {x}{2}+\frac {b}{4\,a}\right )\,\sqrt {a\,x^2+b\,x+c}+\frac {\ln \left (\frac {\frac {b}{2}+a\,x}{\sqrt {a}}+\sqrt {a\,x^2+b\,x+c}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,a^{3/2}} \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 \sqrt {a x^{2} + b x + c}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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