Optimal. Leaf size=110 \[ -\frac{1255878-62021 x}{24681024 \sqrt{2 x^2-x+3}}-\frac{3667 \sqrt{2 x^2-x+3}}{186624 (2 x+5)}+\frac{2203 x+9897}{357696 \left (2 x^2-x+3\right )^{3/2}}-\frac{2821 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{2239488 \sqrt{2}} \]
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Rubi [A] time = 0.152912, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1646, 806, 724, 206} \[ -\frac{1255878-62021 x}{24681024 \sqrt{2 x^2-x+3}}-\frac{3667 \sqrt{2 x^2-x+3}}{186624 (2 x+5)}+\frac{2203 x+9897}{357696 \left (2 x^2-x+3\right )^{3/2}}-\frac{2821 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{2239488 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1646
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}} \, dx &=\frac{9897+2203 x}{357696 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{119353}{20736}+\frac{481765 x}{10368}+\frac{113983 x^2}{1296}}{(5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=\frac{9897+2203 x}{357696 \left (3-x+2 x^2\right )^{3/2}}-\frac{1255878-62021 x}{24681024 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{\frac{10109719}{124416}-\frac{4961491 x}{62208}}{(5+2 x)^2 \sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=\frac{9897+2203 x}{357696 \left (3-x+2 x^2\right )^{3/2}}-\frac{1255878-62021 x}{24681024 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{186624 (5+2 x)}+\frac{2821 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{373248}\\ &=\frac{9897+2203 x}{357696 \left (3-x+2 x^2\right )^{3/2}}-\frac{1255878-62021 x}{24681024 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{186624 (5+2 x)}-\frac{2821 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{186624}\\ &=\frac{9897+2203 x}{357696 \left (3-x+2 x^2\right )^{3/2}}-\frac{1255878-62021 x}{24681024 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{186624 (5+2 x)}-\frac{2821 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{2239488 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.406562, size = 92, normalized size = 0.84 \[ \frac{-\frac{12 \sqrt{2} \left (6767036 x^4+10350004 x^3+63941915 x^2-18840090 x+79153407\right )}{529 (2 x+5) \left (2 x^2-x+3\right )^{3/2}}-2821 \log \left (12 \sqrt{4 x^2-2 x+6}-22 x+17\right )+2821 \log (2 x+5)}{2239488 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.058, size = 194, normalized size = 1.8 \begin{align*} -{\frac{5\,x}{16} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{203}{192} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{-3173+12692\,x}{4416} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{-3173+12692\,x}{6348}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{2821}{124416} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}-{\frac{-2081161+8324644\,x}{2861568} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}-{\frac{-199077743+796310972\,x}{394896384}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{2821}{746496}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{2821\,\sqrt{2}}{4478976}{\it Artanh} \left ({\frac{\sqrt{2}}{12} \left ({\frac{17}{2}}-11\,x \right ){\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}} \right ) }-{\frac{3667}{1152} \left ( x+{\frac{5}{2}} \right ) ^{-1} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.55799, size = 171, normalized size = 1.55 \begin{align*} \frac{2821}{4478976} \, \sqrt{2} \operatorname{arsinh}\left (\frac{22 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 5 \right |}} - \frac{17 \, \sqrt{23}}{23 \,{\left | 2 \, x + 5 \right |}}\right ) - \frac{1691759 \, x}{98724096 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{265339}{32908032 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{248617 \, x}{715392 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{3667}{576 \,{\left (2 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 5 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{259621}{238464 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31711, size = 425, normalized size = 3.86 \begin{align*} \frac{1492309 \, \sqrt{2}{\left (8 \, x^{5} + 12 \, x^{4} + 6 \, x^{3} + 53 \, x^{2} - 12 \, x + 45\right )} \log \left (-\frac{24 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) - 48 \,{\left (6767036 \, x^{4} + 10350004 \, x^{3} + 63941915 \, x^{2} - 18840090 \, x + 79153407\right )} \sqrt{2 \, x^{2} - x + 3}}{4738756608 \,{\left (8 \, x^{5} + 12 \, x^{4} + 6 \, x^{3} + 53 \, x^{2} - 12 \, x + 45\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{5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x + 5\right )^{2} \left (2 x^{2} - x + 3\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 \, x^{4} - x^{3} + 3 \, x^{2} + x + 2}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}{\left (2 \, x + 5\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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