Optimal. Leaf size=22 \[ \frac {\sinh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{\sqrt {c}} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {633, 221}
\begin {gather*} \frac {\sinh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{\sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 633
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\frac {b^2+4 c}{4 c}+b x+c x^2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 c}}} \, dx,x,b+2 c x\right )}{2 c}\\ &=\frac {\sinh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{\sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 44, normalized size = 2.00 \begin {gather*} -\frac {\log \left (b+2 c x-\sqrt {c} \sqrt {4+\frac {b^2}{c}+4 b x+4 c x^2}\right )}{\sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs.
\(2(16)=32\).
time = 0.68, size = 51, normalized size = 2.32
method | result | size |
default | \(\frac {\ln \left (\frac {\left (4 c x +2 b \right ) \sqrt {4}}{4 \sqrt {c}}+\sqrt {\frac {b^{2}+4 c}{c}+4 b x +4 c \,x^{2}}\right ) \sqrt {4}}{2 \sqrt {c}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 16, normalized size = 0.73 \begin {gather*} \frac {\operatorname {arsinh}\left (\frac {2 \, c x + b}{2 \, \sqrt {c}}\right )}{\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (16) = 32\).
time = 2.12, size = 137, normalized size = 6.23 \begin {gather*} \left [\frac {\log \left (-4 \, c^{2} x^{2} - 4 \, b c x - b^{2} - {\left (2 \, c x + b\right )} \sqrt {c} \sqrt {\frac {4 \, c^{2} x^{2} + 4 \, b c x + b^{2} + 4 \, c}{c}} - 2 \, c\right )}{2 \, \sqrt {c}}, -\frac {\sqrt {-c} \arctan \left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c} \sqrt {\frac {4 \, c^{2} x^{2} + 4 \, b c x + b^{2} + 4 \, c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2} + 4 \, c}\right )}{c}\right ] \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*} 2 \int \frac {1}{\sqrt {\frac {b^{2}}{c} + 4 b x + 4 c x^{2} + 4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (16) = 32\).
time = 0.75, size = 60, normalized size = 2.73 \begin {gather*} -\frac {\log \left ({\left | -b c^{2} - {\left (2 \, \sqrt {c^{3}} x - \sqrt {4 \, c^{3} x^{2} + 4 \, b c^{2} x + b^{2} c + 4 \, c^{2}}\right )} \sqrt {c} {\left | c \right |} \right |}\right )}{\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.42, size = 40, normalized size = 1.82 \begin {gather*} \frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+\sqrt {\frac {b^2+4\,c}{c}+4\,b\,x+4\,c\,x^2}\right )}{\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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