Optimal. Leaf size=161 \[ -\frac{\left (3 x^2-x+2\right )^{7/2}}{26 (2 x+1)^2}+\frac{(134 x+257) \left (3 x^2-x+2\right )^{5/2}}{520 (2 x+1)}+\frac{1}{832} (1227-838 x) \left (3 x^2-x+2\right )^{3/2}+\frac{(21317-10470 x) \sqrt{3 x^2-x+2}}{1536}-\frac{1631}{256} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+\frac{118423 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{3072 \sqrt{3}} \]
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Rubi [A] time = 0.162784, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1650, 812, 814, 843, 619, 215, 724, 206} \[ -\frac{\left (3 x^2-x+2\right )^{7/2}}{26 (2 x+1)^2}+\frac{(134 x+257) \left (3 x^2-x+2\right )^{5/2}}{520 (2 x+1)}+\frac{1}{832} (1227-838 x) \left (3 x^2-x+2\right )^{3/2}+\frac{(21317-10470 x) \sqrt{3 x^2-x+2}}{1536}-\frac{1631}{256} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )+\frac{118423 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{3072 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 812
Rule 814
Rule 843
Rule 619
Rule 215
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right )}{(1+2 x)^3} \, dx &=-\frac{\left (2-x+3 x^2\right )^{7/2}}{26 (1+2 x)^2}-\frac{1}{26} \int \frac{\left (-\frac{29}{2}-67 x\right ) \left (2-x+3 x^2\right )^{5/2}}{(1+2 x)^2} \, dx\\ &=\frac{(257+134 x) \left (2-x+3 x^2\right )^{5/2}}{520 (1+2 x)}-\frac{\left (2-x+3 x^2\right )^{7/2}}{26 (1+2 x)^2}+\frac{1}{208} \int \frac{(793-1676 x) \left (2-x+3 x^2\right )^{3/2}}{1+2 x} \, dx\\ &=\frac{1}{832} (1227-838 x) \left (2-x+3 x^2\right )^{3/2}+\frac{(257+134 x) \left (2-x+3 x^2\right )^{5/2}}{520 (1+2 x)}-\frac{\left (2-x+3 x^2\right )^{7/2}}{26 (1+2 x)^2}-\frac{\int \frac{(-236652+544440 x) \sqrt{2-x+3 x^2}}{1+2 x} \, dx}{19968}\\ &=\frac{(21317-10470 x) \sqrt{2-x+3 x^2}}{1536}+\frac{1}{832} (1227-838 x) \left (2-x+3 x^2\right )^{3/2}+\frac{(257+134 x) \left (2-x+3 x^2\right )^{5/2}}{520 (1+2 x)}-\frac{\left (2-x+3 x^2\right )^{7/2}}{26 (1+2 x)^2}+\frac{\int \frac{42436056-73895952 x}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx}{958464}\\ &=\frac{(21317-10470 x) \sqrt{2-x+3 x^2}}{1536}+\frac{1}{832} (1227-838 x) \left (2-x+3 x^2\right )^{3/2}+\frac{(257+134 x) \left (2-x+3 x^2\right )^{5/2}}{520 (1+2 x)}-\frac{\left (2-x+3 x^2\right )^{7/2}}{26 (1+2 x)^2}-\frac{118423 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{3072}+\frac{21203}{256} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=\frac{(21317-10470 x) \sqrt{2-x+3 x^2}}{1536}+\frac{1}{832} (1227-838 x) \left (2-x+3 x^2\right )^{3/2}+\frac{(257+134 x) \left (2-x+3 x^2\right )^{5/2}}{520 (1+2 x)}-\frac{\left (2-x+3 x^2\right )^{7/2}}{26 (1+2 x)^2}-\frac{21203}{128} \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )-\frac{118423 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{3072 \sqrt{69}}\\ &=\frac{(21317-10470 x) \sqrt{2-x+3 x^2}}{1536}+\frac{1}{832} (1227-838 x) \left (2-x+3 x^2\right )^{3/2}+\frac{(257+134 x) \left (2-x+3 x^2\right )^{5/2}}{520 (1+2 x)}-\frac{\left (2-x+3 x^2\right )^{7/2}}{26 (1+2 x)^2}+\frac{118423 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{3072 \sqrt{3}}-\frac{1631}{256} \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.108018, size = 113, normalized size = 0.7 \[ \frac{\frac{6 \sqrt{3 x^2-x+2} \left (27648 x^6-22464 x^5+83616 x^4-76200 x^3+256564 x^2+464446 x+142057\right )}{(2 x+1)^2}-293580 \sqrt{13} \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )-592115 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{46080} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 199, normalized size = 1.2 \begin{align*}{\frac{1631}{6760} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{-19+114\,x}{676} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{19}{338} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{1}{2}} \right ) ^{-1}}-{\frac{1}{104} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{1}{2}} \right ) ^{-2}}-{\frac{-1745+10470\,x}{1536}\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}-{\frac{-419+2514\,x}{2496} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{1631\,\sqrt{13}}{256}{\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{118423\,\sqrt{3}}{9216}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }+{\frac{1631}{256}\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}+{\frac{1631}{1248} \left ( 3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59254, size = 232, normalized size = 1.44 \begin{align*} \frac{67}{520} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} - \frac{{\left (3 \, x^{2} - x + 2\right )}^{\frac{7}{2}}}{26 \,{\left (4 \, x^{2} + 4 \, x + 1\right )}} - \frac{419}{416} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x + \frac{1227}{832} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} + \frac{19 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}}}{52 \,{\left (2 \, x + 1\right )}} - \frac{1745}{256} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{118423}{9216} \, \sqrt{3} \operatorname{arsinh}\left (\frac{6}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{1631}{256} \, \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{21317}{1536} \, \sqrt{3 \, x^{2} - x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49135, size = 487, normalized size = 3.02 \begin{align*} \frac{592115 \, \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 ) + 293580 \, \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 ) + 12 \,{\left (27648 \, x^{6} - 22464 \, x^{5} + 83616 \, x^{4} - 76200 \, x^{3} + 256564 \, x^{2} + 464446 \, x + 142057\right )} \sqrt{3 \, x^{2} - x + 2}}{92160 \,{\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{\left (3 x^{2} - x + 2\right )^{\frac{5}{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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