Optimal. Leaf size=195 \[ -\frac{3730507 \left (2 x^2-x+3\right )^{5/2}}{11943936 (2 x+5)^3}+\frac{158527 \left (2 x^2-x+3\right )^{5/2}}{165888 (2 x+5)^4}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{2880 (2 x+5)^5}+\frac{(44773976 x+246012435) \left (2 x^2-x+3\right )^{3/2}}{95551488 (2 x+5)^2}-\frac{(1028823716 x+5658774871) \sqrt{2 x^2-x+3}}{127401984 (2 x+5)}+\frac{70991525167 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1528823808 \sqrt{2}}-\frac{23775 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{512 \sqrt{2}} \]
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Rubi [A] time = 0.262875, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {1650, 812, 843, 619, 215, 724, 206} \[ -\frac{3730507 \left (2 x^2-x+3\right )^{5/2}}{11943936 (2 x+5)^3}+\frac{158527 \left (2 x^2-x+3\right )^{5/2}}{165888 (2 x+5)^4}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{2880 (2 x+5)^5}+\frac{(44773976 x+246012435) \left (2 x^2-x+3\right )^{3/2}}{95551488 (2 x+5)^2}-\frac{(1028823716 x+5658774871) \sqrt{2 x^2-x+3}}{127401984 (2 x+5)}+\frac{70991525167 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1528823808 \sqrt{2}}-\frac{23775 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{512 \sqrt{2}} \]
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{\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^6} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}-\frac{1}{360} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{60035}{16}-6615 x+2430 x^2-900 x^3\right )}{(5+2 x)^5} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}+\frac{\int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{8114455}{16}-\frac{3488315 x}{4}+129600 x^2\right )}{(5+2 x)^4} \, dx}{103680}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac{3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}-\frac{\int \frac{\left (\frac{332138325}{16}-\frac{83951205 x}{2}\right ) \left (3-x+2 x^2\right )^{3/2}}{(5+2 x)^3} \, dx}{22394880}\\ &=\frac{(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac{3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}+\frac{\int \frac{\left (\frac{7719844365}{4}-3858088935 x\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^2} \, dx}{238878720}\\ &=-\frac{(5658774871+1028823716 x) \sqrt{3-x+2 x^2}}{127401984 (5+2 x)}+\frac{(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac{3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}-\frac{\int \frac{\frac{177475757505}{2}-177479424000 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{1911029760}\\ &=-\frac{(5658774871+1028823716 x) \sqrt{3-x+2 x^2}}{127401984 (5+2 x)}+\frac{(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac{3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}+\frac{23775}{512} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx-\frac{70991525167 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{254803968}\\ &=-\frac{(5658774871+1028823716 x) \sqrt{3-x+2 x^2}}{127401984 (5+2 x)}+\frac{(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac{3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}+\frac{70991525167 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{127401984}+\frac{23775 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{512 \sqrt{46}}\\ &=-\frac{(5658774871+1028823716 x) \sqrt{3-x+2 x^2}}{127401984 (5+2 x)}+\frac{(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac{158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac{3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}-\frac{23775 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{512 \sqrt{2}}+\frac{70991525167 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{1528823808 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.247036, size = 108, normalized size = 0.55 \[ \frac{\frac{24 \sqrt{2 x^2-x+3} \left (1592524800 x^6-30496849920 x^5-1023534029552 x^4-7117092892448 x^3-21590439797064 x^2-30872393829992 x-17093312738327\right )}{(2 x+5)^5}+354957625835 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )-354958848000 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{15288238080} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 225, normalized size = 1.2 \begin{align*} -{\frac{3667}{92160} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-5}}+{\frac{-3086715581+12346862324\,x}{9172942848}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{134077495}{6879707136} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}+{\frac{70991525167\,\sqrt{2}}{3057647616}{\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{23775\,\sqrt{2}}{1024}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{70991525167}{495338913792} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}-{\frac{70991525167}{9172942848}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{3730507}{95551488} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}}+{\frac{-4698578717+18794314868\,x}{495338913792} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}-{\frac{4698578717}{247669456896} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}+{\frac{158527}{2654208} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65331, size = 339, normalized size = 1.74 \begin{align*} -\frac{134077495}{3439853568} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{2880 \,{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} + \frac{158527 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{165888 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} - \frac{3730507 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{11943936 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac{134077495 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1719926784 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac{3086715581}{2293235712} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{23775}{1024} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{70991525167}{3057647616} \, \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{6173186729}{764411904} \, \sqrt{2 \, x^{2} - x + 3} - \frac{4698578717 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{6879707136 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52147, size = 747, normalized size = 3.83 \begin{align*} \frac{354958848000 \, \sqrt{2}{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 354957625835 \, \sqrt{2}{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\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 (1592524800 \, x^{6} - 30496849920 \, x^{5} - 1023534029552 \, x^{4} - 7117092892448 \, x^{3} - 21590439797064 \, x^{2} - 30872393829992 \, x - 17093312738327\right )} \sqrt{2 \, x^{2} - x + 3}}{30576476160 \,{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\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 (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30233, size = 548, normalized size = 2.81 \begin{align*} \frac{1}{256} \, \sqrt{2 \, x^{2} - x + 3}{\left (20 \, x - 633\right )} - \frac{23775}{1024} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{70991525167}{3057647616} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{70991525167}{3057647616} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{\sqrt{2}{\left (8281387393360 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{9} + 275661428628240 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{8} + 1560382703345760 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{7} + 4938646760855520 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{6} - 9673562837036232 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} - 30647310393849000 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} + 70060241036847960 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} - 97730658088823880 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 30180638363071845 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 7096913381268319\right )}}{1274019840 \,{\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 )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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