Optimal. Leaf size=112 \[ \frac{65991-8779 x}{4292352 \sqrt{2 x^2-x+3}}+\frac{115369 \sqrt{2 x^2-x+3}}{1492992 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{20736 (2 x+5)^2}-\frac{52631 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{5971968 \sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.145885, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1646, 1650, 806, 724, 206} \[ \frac{65991-8779 x}{4292352 \sqrt{2 x^2-x+3}}+\frac{115369 \sqrt{2 x^2-x+3}}{1492992 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{20736 (2 x+5)^2}-\frac{52631 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{5971968 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1646
Rule 1650
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)^3 \left (3-x+2 x^2\right )^{3/2}} \, dx &=\frac{65991-8779 x}{4292352 \sqrt{3-x+2 x^2}}+\frac{2}{23} \int \frac{\frac{5168261}{746496}+\frac{3637795 x}{186624}+\frac{5620625 x^2}{186624}}{(5+2 x)^3 \sqrt{3-x+2 x^2}} \, dx\\ &=\frac{65991-8779 x}{4292352 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{20736 (5+2 x)^2}-\frac{\int \frac{\frac{842237}{1296}-\frac{4102487 x}{2592}}{(5+2 x)^2 \sqrt{3-x+2 x^2}} \, dx}{1656}\\ &=\frac{65991-8779 x}{4292352 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{20736 (5+2 x)^2}+\frac{115369 \sqrt{3-x+2 x^2}}{1492992 (5+2 x)}+\frac{52631 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{995328}\\ &=\frac{65991-8779 x}{4292352 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{20736 (5+2 x)^2}+\frac{115369 \sqrt{3-x+2 x^2}}{1492992 (5+2 x)}-\frac{52631 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{497664}\\ &=\frac{65991-8779 x}{4292352 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{20736 (5+2 x)^2}+\frac{115369 \sqrt{3-x+2 x^2}}{1492992 (5+2 x)}-\frac{52631 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{5971968 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.296961, size = 84, normalized size = 0.75 \[ \frac{\frac{12 \left (3444340 x^3+3263288 x^2+5842933 x+11594283\right )}{23 (2 x+5)^2 \sqrt{x^2-\frac{x}{2}+\frac{3}{2}}}-52631 \log \left (12 \sqrt{4 x^2-2 x+6}-22 x+17\right )+52631 \log (2 x+5)}{5971968 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.058, size = 144, normalized size = 1.3 \begin{align*} -{\frac{5}{16}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{-149+596\,x}{368}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{3667}{4608} \left ( x+{\frac{5}{2}} \right ) ^{-2}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{196043}{165888} \left ( x+{\frac{5}{2}} \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{52631}{1990656}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{-19399069+77596276\,x}{45785088}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{52631\,\sqrt{2}}{11943936}{\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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.51339, size = 201, normalized size = 1.79 \begin{align*} \frac{52631}{11943936} \, \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{861085 \, x}{11446272 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{1163201}{3815424 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{3667}{1152 \,{\left (4 \, \sqrt{2 \, x^{2} - x + 3} x^{2} + 20 \, \sqrt{2 \, x^{2} - x + 3} x + 25 \, \sqrt{2 \, x^{2} - x + 3}\right )}} + \frac{196043}{82944 \,{\left (2 \, \sqrt{2 \, x^{2} - x + 3} x + 5 \, \sqrt{2 \, x^{2} - x + 3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.3407, size = 379, normalized size = 3.38 \begin{align*} \frac{1210513 \, \sqrt{2}{\left (8 \, x^{4} + 36 \, x^{3} + 42 \, x^{2} + 35 \, x + 75\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 (3444340 \, x^{3} + 3263288 \, x^{2} + 5842933 \, x + 11594283\right )} \sqrt{2 \, x^{2} - x + 3}}{549421056 \,{\left (8 \, x^{4} + 36 \, x^{3} + 42 \, x^{2} + 35 \, x + 75\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )^{3} \left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.15733, size = 297, normalized size = 2.65 \begin{align*} -\frac{52631}{11943936} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{52631}{11943936} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{8779 \, x - 65991}{4292352 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{\sqrt{2}{\left (3594214 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} + 19874490 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} - 30140067 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 19989859\right )}}{2985984 \,{\left (2 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 11\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]