Optimal. Leaf size=103 \[ -\frac {1}{20} \left (12 x^2+17 x+6\right )^{5/2}+\frac {97}{768} (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}-\frac {97 (24 x+17) \sqrt {12 x^2+17 x+6}}{24576}+\frac {97 \tanh ^{-1}\left (\frac {24 x+17}{4 \sqrt {3} \sqrt {12 x^2+17 x+6}}\right )}{98304 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1002, 640, 612, 621, 206} \[ -\frac {1}{20} \left (12 x^2+17 x+6\right )^{5/2}+\frac {97}{768} (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}-\frac {97 (24 x+17) \sqrt {12 x^2+17 x+6}}{24576}+\frac {97 \tanh ^{-1}\left (\frac {24 x+17}{4 \sqrt {3} \sqrt {12 x^2+17 x+6}}\right )}{98304 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1002
Rubi steps
\begin {align*} \int (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt {6+17 x+12 x^2} \, dx &=\int (10-3 x) \left (6+17 x+12 x^2\right )^{3/2} \, dx\\ &=-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97}{8} \int \left (6+17 x+12 x^2\right )^{3/2} \, dx\\ &=\frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}-\frac {97}{512} \int \sqrt {6+17 x+12 x^2} \, dx\\ &=-\frac {97 (17+24 x) \sqrt {6+17 x+12 x^2}}{24576}+\frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97 \int \frac {1}{\sqrt {6+17 x+12 x^2}} \, dx}{49152}\\ &=-\frac {97 (17+24 x) \sqrt {6+17 x+12 x^2}}{24576}+\frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97 \operatorname {Subst}\left (\int \frac {1}{48-x^2} \, dx,x,\frac {17+24 x}{\sqrt {6+17 x+12 x^2}}\right )}{24576}\\ &=-\frac {97 (17+24 x) \sqrt {6+17 x+12 x^2}}{24576}+\frac {97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac {1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac {97 \tanh ^{-1}\left (\frac {17+24 x}{4 \sqrt {3} \sqrt {6+17 x+12 x^2}}\right )}{98304 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 72, normalized size = 0.70 \[ \frac {485 \sqrt {3} \tanh ^{-1}\left (\frac {24 x+17}{4 \sqrt {36 x^2+51 x+18}}\right )+12 \sqrt {12 x^2+17 x+6} \left (-884736 x^4+1963008 x^3+6837888 x^2+5455144 x+1353611\right )}{1474560} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 73, normalized size = 0.71 \[ -\frac {1}{122880} \, {\left (884736 \, x^{4} - 1963008 \, x^{3} - 6837888 \, x^{2} - 5455144 \, x - 1353611\right )} \sqrt {12 \, x^{2} + 17 \, x + 6} + \frac {97}{589824} \, \sqrt {3} \log \left (8 \, \sqrt {3} \sqrt {12 \, x^{2} + 17 \, x + 6} {\left (24 \, x + 17\right )} + 1152 \, x^{2} + 1632 \, x + 577\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 70, normalized size = 0.68 \[ -\frac {1}{122880} \, {\left (8 \, {\left (48 \, {\left (72 \, {\left (32 \, x - 71\right )} x - 17807\right )} x - 681893\right )} x - 1353611\right )} \sqrt {12 \, x^{2} + 17 \, x + 6} - \frac {97}{294912} \, \sqrt {3} \log \left ({\left | -4 \, \sqrt {3} {\left (2 \, \sqrt {3} x - \sqrt {12 \, x^{2} + 17 \, x + 6}\right )} - 17 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 96, normalized size = 0.93 \[ -\frac {3 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}} x^{2}}{5}+\frac {349 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}} x}{160}+\frac {97 \sqrt {12}\, \ln \left (\frac {\left (12 x +\frac {17}{2}\right ) \sqrt {12}}{12}+\sqrt {12 x^{2}+17 x +6}\right )}{589824}+\frac {7093 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{3840}-\frac {97 \left (24 x +17\right ) \sqrt {12 x^{2}+17 x +6}}{24576} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 104, normalized size = 1.01 \[ -\frac {3}{5} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{2} + \frac {349}{160} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x + \frac {7093}{3840} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} - \frac {97}{1024} \, \sqrt {12 \, x^{2} + 17 \, x + 6} x + \frac {97}{294912} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {12 \, x^{2} + 17 \, x + 6} + 24 \, x + 17\right ) - \frac {1649}{24576} \, \sqrt {12 \, x^{2} + 17 \, x + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.69, size = 136, normalized size = 1.32 \[ \frac {3753\,\left (\frac {x}{2}+\frac {17}{48}\right )\,\sqrt {12\,x^2+17\,x+6}}{80}-\frac {417\,\sqrt {12}\,\ln \left (\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (12\,x+\frac {17}{2}\right )}{12}\right )}{10240}-\frac {3\,x^2\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{5}+\frac {7093\,\sqrt {12\,x^2+17\,x+6}\,\left (1152\,x^2+408\,x-291\right )}{368640}+\frac {349\,x\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{160}+\frac {120581\,\sqrt {12}\,\ln \left (2\,\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (24\,x+17\right )}{12}\right )}{2949120} \]
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 \[ - \int \left (- 152 x \sqrt {12 x^{2} + 17 x + 6}\right )\, dx - \int \left (- 69 x^{2} \sqrt {12 x^{2} + 17 x + 6}\right )\, dx - \int 36 x^{3} \sqrt {12 x^{2} + 17 x + 6}\, dx - \int \left (- 60 \sqrt {12 x^{2} + 17 x + 6}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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