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 = 0.0777517, antiderivative size = 84, normalized size of antiderivative = 1., 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} \[ -\frac{388 x+275}{98 (10-3 x) \sqrt{12 x^2+17 x+6}}+\frac{3137 \sqrt{12 x^2+17 x+6}}{38416 (10-3 x)}+\frac{97 \tanh ^{-1}\left (\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right )}{3226944} \]
Antiderivative was successfully verified.
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Rule 1002
Rule 740
Rule 806
Rule 724
Rule 206
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*}
Mathematica [A] time = 0.243785, size = 114, normalized size = 1.36 \[ \frac{\sqrt{12 x^2+17 x+6} \left (97 \left (36 x^3-69 x^2-152 x-60\right ) \tanh ^{-1}\left (\frac{7 \sqrt{3 x+2}}{6 \sqrt{4 x+3}}\right )-42 \sqrt{3 x+2} \sqrt{4 x+3} \left (37644 x^2-98767 x-88978\right )\right )}{1613472 (3 x-10) (3 x+2)^{3/2} (4 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.064, size = 245, normalized size = 2.9 \begin{align*}{\frac{32}{2401} \left ( 12\, \left ( x+3/4 \right ) ^{2}-x-{\frac{3}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{4}} \right ) ^{-2}}+{\frac{384}{117649}\sqrt{12\, \left ( x+3/4 \right ) ^{2}-x-{\frac{3}{4}}}}-{\frac{16\,\sqrt{12}}{117649}\ln \left ({\frac{\sqrt{12}}{12} \left ({\frac{17}{2}}+12\,x \right ) }+\sqrt{12\, \left ( x+3/4 \right ) ^{2}-x-{\frac{3}{4}}} \right ) }-{\frac{1}{72} \left ( 12\, \left ( x+2/3 \right ) ^{2}+x+{\frac{2}{3}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{2}{3}} \right ) ^{-2}}+{\frac{1}{288}\sqrt{12\, \left ( x+2/3 \right ) ^{2}+x+{\frac{2}{3}}}}+{\frac{\sqrt{12}}{6912}\ln \left ({\frac{\sqrt{12}}{12} \left ({\frac{17}{2}}+12\,x \right ) }+\sqrt{12\, \left ( x+2/3 \right ) ^{2}+x+{\frac{2}{3}}} \right ) }-{\frac{97}{45177216}\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}}}-{\frac{7057\,\sqrt{12}}{813189888}\ln \left ({\frac{\sqrt{12}}{12} \left ({\frac{17}{2}}+12\,x \right ) }+\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}} \right ) }+{\frac{97}{3226944}{\it Artanh} \left ({\frac{1}{28} \left ({\frac{206}{3}}+97\,x \right ){\frac{1}{\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}}}}} \right ) }-{\frac{1}{67765824} \left ( 12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}} \right ) ^{{\frac{3}{2}}} \left ( x-{\frac{10}{3}} \right ) ^{-1}}+{\frac{17+24\,x}{135531648}\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58619, size = 370, normalized size = 4.4 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22682, size = 215, normalized size = 2.56 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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