Optimal. Leaf size=84 \[ -\frac {388 x+275}{98 (10-3 x) \sqrt {12 x^2+17 x+6}}+\frac {3137 \sqrt {12 x^2+17 x+6}}{38416 (10-3 x)}+\frac {97 \tanh ^{-1}\left (\frac {291 x+206}{84 \sqrt {12 x^2+17 x+6}}\right )}{3226944} \]
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Rubi [A] time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {1002, 740, 806, 724, 206} \begin {gather*} -\frac {388 x+275}{98 (10-3 x) \sqrt {12 x^2+17 x+6}}+\frac {3137 \sqrt {12 x^2+17 x+6}}{38416 (10-3 x)}+\frac {97 \tanh ^{-1}\left (\frac {291 x+206}{84 \sqrt {12 x^2+17 x+6}}\right )}{3226944} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 740
Rule 806
Rule 1002
Rubi steps
\begin {align*} \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x)^2 \left (30+31 x-12 x^2\right )^2} \, dx &=\int \frac {1}{(10-3 x)^2 \left (6+17 x+12 x^2\right )^{3/2}} \, dx\\ &=-\frac {275+388 x}{98 (10-3 x) \sqrt {6+17 x+12 x^2}}-\frac {1}{882} \int \frac {-\frac {14859}{2}-10476 x}{(10-3 x)^2 \sqrt {6+17 x+12 x^2}} \, dx\\ &=-\frac {275+388 x}{98 (10-3 x) \sqrt {6+17 x+12 x^2}}+\frac {3137 \sqrt {6+17 x+12 x^2}}{38416 (10-3 x)}+\frac {97 \int \frac {1}{(10-3 x) \sqrt {6+17 x+12 x^2}} \, dx}{76832}\\ &=-\frac {275+388 x}{98 (10-3 x) \sqrt {6+17 x+12 x^2}}+\frac {3137 \sqrt {6+17 x+12 x^2}}{38416 (10-3 x)}-\frac {97 \operatorname {Subst}\left (\int \frac {1}{7056-x^2} \, dx,x,\frac {-206-291 x}{\sqrt {6+17 x+12 x^2}}\right )}{38416}\\ &=-\frac {275+388 x}{98 (10-3 x) \sqrt {6+17 x+12 x^2}}+\frac {3137 \sqrt {6+17 x+12 x^2}}{38416 (10-3 x)}+\frac {97 \tanh ^{-1}\left (\frac {206+291 x}{84 \sqrt {6+17 x+12 x^2}}\right )}{3226944}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 114, normalized size = 1.36 \begin {gather*} \frac {\sqrt {12 x^2+17 x+6} \left (97 \left (36 x^3-69 x^2-152 x-60\right ) \tanh ^{-1}\left (\frac {7 \sqrt {3 x+2}}{6 \sqrt {4 x+3}}\right )-42 \sqrt {3 x+2} \sqrt {4 x+3} \left (37644 x^2-98767 x-88978\right )\right )}{1613472 (3 x-10) (3 x+2)^{3/2} (4 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 80, normalized size = 0.95 \begin {gather*} \frac {\sqrt {12 x^2+17 x+6} \left (-37644 x^2+98767 x+88978\right )}{38416 (3 x-10) (3 x+2) (4 x+3)}+\frac {97 \tanh ^{-1}\left (\frac {6 \sqrt {12 x^2+17 x+6}}{7 (3 x+2)}\right )}{1613472} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 126, normalized size = 1.50 \begin {gather*} \frac {97 \, {\left (36 \, x^{3} - 69 \, x^{2} - 152 \, x - 60\right )} \log \left (\frac {291 \, x + 84 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) - 97 \, {\left (36 \, x^{3} - 69 \, x^{2} - 152 \, x - 60\right )} \log \left (\frac {291 \, x - 84 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) - 168 \, {\left (37644 \, x^{2} - 98767 \, x - 88978\right )} \sqrt {12 \, x^{2} + 17 \, x + 6}}{6453888 \, {\left (36 \, x^{3} - 69 \, x^{2} - 152 \, x - 60\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 159, normalized size = 1.89 \begin {gather*} \frac {1}{9680832} \, \sqrt {3} {\left (\sqrt {3} {\left (175672 \, \sqrt {3} + 97 \, \log \left (\frac {7 \, \sqrt {3} - 12}{7 \, \sqrt {3} + 12}\right )\right )} \mathrm {sgn}\left (\frac {1}{3 \, x + 2}\right ) - {\left (97 \, \sqrt {3} \log \left (\frac {{\left | -28 \, \sqrt {3} + 24 \, \sqrt {\frac {1}{3 \, x + 2} + 4} \right |}}{4 \, {\left (7 \, \sqrt {3} + 6 \, \sqrt {\frac {1}{3 \, x + 2} + 4}\right )}}\right ) + 134456 \, \sqrt {\frac {1}{3 \, x + 2} + 4} + \frac {28 \, {\left (\frac {221183}{3 \, x + 2} - 18436\right )}}{12 \, {\left (\frac {1}{3 \, x + 2} + 4\right )}^{\frac {3}{2}} - 49 \, \sqrt {\frac {1}{3 \, x + 2} + 4}}\right )} \mathrm {sgn}\left (\frac {1}{3 \, x + 2}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 245, normalized size = 2.92 \begin {gather*} \frac {97 \arctanh \left (\frac {97 x +\frac {206}{3}}{28 \sqrt {97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}}}\right )}{3226944}-\frac {7057 \sqrt {12}\, \ln \left (\frac {\left (12 x +\frac {17}{2}\right ) \sqrt {12}}{12}+\sqrt {97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}}\right )}{813189888}+\frac {\sqrt {12}\, \ln \left (\frac {\left (12 x +\frac {17}{2}\right ) \sqrt {12}}{12}+\sqrt {x +12 \left (x +\frac {2}{3}\right )^{2}+\frac {2}{3}}\right )}{6912}-\frac {16 \sqrt {12}\, \ln \left (\frac {\left (12 x +\frac {17}{2}\right ) \sqrt {12}}{12}+\sqrt {-x +12 \left (x +\frac {3}{4}\right )^{2}-\frac {3}{4}}\right )}{117649}+\frac {384 \sqrt {-x +12 \left (x +\frac {3}{4}\right )^{2}-\frac {3}{4}}}{117649}+\frac {\sqrt {x +12 \left (x +\frac {2}{3}\right )^{2}+\frac {2}{3}}}{288}-\frac {97 \sqrt {97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}}}{45177216}-\frac {\left (x +12 \left (x +\frac {2}{3}\right )^{2}+\frac {2}{3}\right )^{\frac {3}{2}}}{72 \left (x +\frac {2}{3}\right )^{2}}+\frac {32 \left (-x +12 \left (x +\frac {3}{4}\right )^{2}-\frac {3}{4}\right )^{\frac {3}{2}}}{2401 \left (x +\frac {3}{4}\right )^{2}}-\frac {\left (97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}\right )^{\frac {3}{2}}}{67765824 \left (x -\frac {10}{3}\right )}+\frac {\left (24 x +17\right ) \sqrt {97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}}}{135531648} \end {gather*}
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*} \int \frac {\sqrt {12 \, x^{2} + 17 \, x + 6}}{{\left (12 \, x^{2} - 31 \, x - 30\right )}^{2} {\left (3 \, x + 2\right )}^{2}}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {\sqrt {12\,x^2+17\,x+6}}{{\left (3\,x+2\right )}^2\,{\left (-12\,x^2+31\,x+30\right )}^2} \,d x \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 {\left (3 x + 2\right ) \left (4 x + 3\right )}}{\left (3 x - 10\right )^{2} \left (3 x + 2\right )^{2} \left (4 x + 3\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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