Optimal. Leaf size=115 \[ -\frac{\left (3 x^2-x+2\right )^{3/2}}{26 (2 x+1)^2}+\frac{11 (10 x+7) \sqrt{3 x^2-x+2}}{104 (2 x+1)}-\frac{803 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{208 \sqrt{13}}+\frac{11 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8 \sqrt{3}} \]
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Rubi [A] time = 0.116874, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {1650, 812, 843, 619, 215, 724, 206} \[ -\frac{\left (3 x^2-x+2\right )^{3/2}}{26 (2 x+1)^2}+\frac{11 (10 x+7) \sqrt{3 x^2-x+2}}{104 (2 x+1)}-\frac{803 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{208 \sqrt{13}}+\frac{11 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 812
Rule 843
Rule 619
Rule 215
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{2-x+3 x^2} \left (1+3 x+4 x^2\right )}{(1+2 x)^3} \, dx &=-\frac{\left (2-x+3 x^2\right )^{3/2}}{26 (1+2 x)^2}-\frac{1}{26} \int \frac{\left (-\frac{33}{2}-55 x\right ) \sqrt{2-x+3 x^2}}{(1+2 x)^2} \, dx\\ &=\frac{11 (7+10 x) \sqrt{2-x+3 x^2}}{104 (1+2 x)}-\frac{\left (2-x+3 x^2\right )^{3/2}}{26 (1+2 x)^2}+\frac{1}{208} \int \frac{517-572 x}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=\frac{11 (7+10 x) \sqrt{2-x+3 x^2}}{104 (1+2 x)}-\frac{\left (2-x+3 x^2\right )^{3/2}}{26 (1+2 x)^2}-\frac{11}{8} \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx+\frac{803}{208} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=\frac{11 (7+10 x) \sqrt{2-x+3 x^2}}{104 (1+2 x)}-\frac{\left (2-x+3 x^2\right )^{3/2}}{26 (1+2 x)^2}-\frac{803}{104} \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )-\frac{11 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{8 \sqrt{69}}\\ &=\frac{11 (7+10 x) \sqrt{2-x+3 x^2}}{104 (1+2 x)}-\frac{\left (2-x+3 x^2\right )^{3/2}}{26 (1+2 x)^2}+\frac{11 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8 \sqrt{3}}-\frac{803 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )}{208 \sqrt{13}}\\ \end{align*}
Mathematica [A] time = 0.0815547, size = 93, normalized size = 0.81 \[ \frac{\frac{78 \sqrt{3 x^2-x+2} \left (208 x^2+268 x+69\right )}{(2 x+1)^2}-2409 \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-3718 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{8112} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 125, normalized size = 1.1 \begin{align*}{\frac{11}{338} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{1}{2}} \right ) ^{-1}}+{\frac{803}{2704}\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}-{\frac{11\,\sqrt{3}}{24}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }-{\frac{803\,\sqrt{13}}{2704}{\it Artanh} \left ({\frac{2\,\sqrt{13}}{13} \left ({\frac{9}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}}} \right ) }-{\frac{-11+66\,x}{676}\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}-{\frac{1}{104} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{1}{2}} \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49728, size = 154, normalized size = 1.34 \begin{align*} -\frac{11}{24} \, \sqrt{3} \operatorname{arsinh}\left (\frac{6}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{803}{2704} \, \sqrt{13} \operatorname{arsinh}\left (\frac{8 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 1 \right |}} - \frac{9 \, \sqrt{23}}{23 \,{\left | 2 \, x + 1 \right |}}\right ) + \frac{55}{104} \, \sqrt{3 \, x^{2} - x + 2} - \frac{{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}}{26 \,{\left (4 \, x^{2} + 4 \, x + 1\right )}} + \frac{11 \, \sqrt{3 \, x^{2} - x + 2}}{52 \,{\left (2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65216, size = 405, normalized size = 3.52 \begin{align*} \frac{3718 \, \sqrt{3}{\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 2409 \, \sqrt{13}{\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (-\frac{4 \, \sqrt{13} \sqrt{3 \, x^{2} - x + 2}{\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 156 \,{\left (208 \, x^{2} + 268 \, x + 69\right )} \sqrt{3 \, x^{2} - x + 2}}{16224 \,{\left (4 \, x^{2} + 4 \, x + 1\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{3 x^{2} - x + 2} \left (4 x^{2} + 3 x + 1\right )}{\left (2 x + 1\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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