Optimal. Leaf size=28 \[ \frac {1}{42} \tanh ^{-1}\left (\frac {206+291 x}{84 \sqrt {6+17 x+12 x^2}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {1016, 738, 212}
\begin {gather*} \frac {1}{42} \tanh ^{-1}\left (\frac {291 x+206}{84 \sqrt {12 x^2+17 x+6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 1016
Rubi steps
\begin {align*} \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x) \left (30+31 x-12 x^2\right )} \, dx &=\int \frac {1}{(10-3 x) \sqrt {6+17 x+12 x^2}} \, dx\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{7056-x^2} \, dx,x,\frac {-206-291 x}{\sqrt {6+17 x+12 x^2}}\right )\right )\\ &=\frac {1}{42} \tanh ^{-1}\left (\frac {206+291 x}{84 \sqrt {6+17 x+12 x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 30, normalized size = 1.07 \begin {gather*} \frac {1}{21} \tanh ^{-1}\left (\frac {6 \sqrt {6+17 x+12 x^2}}{7 (2+3 x)}\right ) \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. \(162\) vs.
\(2(22)=44\).
time = 0.13, size = 163, normalized size = 5.82
method | result | size |
trager | \(\frac {\ln \left (-\frac {206+291 x +84 \sqrt {12 x^{2}+17 x +6}}{3 x -10}\right )}{42}\) | \(32\) |
default | \(-\frac {\sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}}{588}-\frac {97 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}\right ) \sqrt {12}}{14112}+\frac {\arctanh \left (\frac {\frac {206}{3}+97 x}{28 \sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}}\right )}{42}-\frac {4 \sqrt {12 \left (x +\frac {3}{4}\right )^{2}-x -\frac {3}{4}}}{49}+\frac {\ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 \left (x +\frac {3}{4}\right )^{2}-x -\frac {3}{4}}\right ) \sqrt {12}}{294}+\frac {\sqrt {12 \left (x +\frac {2}{3}\right )^{2}+x +\frac {2}{3}}}{12}+\frac {\ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 \left (x +\frac {2}{3}\right )^{2}+x +\frac {2}{3}}\right ) \sqrt {12}}{288}\) | \(163\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 53 vs.
\(2 (22) = 44\).
time = 0.37, size = 53, normalized size = 1.89 \begin {gather*} \frac {1}{84} \, \log \left (\frac {291 \, x + 84 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) - \frac {1}{84} \, \log \left (\frac {291 \, x - 84 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 206}{x}\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*} - \int \frac {\sqrt {12 x^{2} + 17 x + 6}}{36 x^{3} - 69 x^{2} - 152 x - 60}\, 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. 63 vs.
\(2 (22) = 44\).
time = 5.63, size = 63, normalized size = 2.25 \begin {gather*} \frac {1}{42} \, \log \left ({\left | -6 \, \sqrt {3} x + 20 \, \sqrt {3} + 3 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 42 \right |}\right ) - \frac {1}{42} \, \log \left ({\left | -6 \, \sqrt {3} x + 20 \, \sqrt {3} + 3 \, \sqrt {12 \, x^{2} + 17 \, x + 6} - 42 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {12\,x^2+17\,x+6}}{\left (3\,x+2\right )\,\left (-12\,x^2+31\,x+30\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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