Optimal. Leaf size=139 \[ \frac{26800085 \sqrt{2 x^2-x+3}}{1719926784 (2 x+5)}-\frac{16295969 \sqrt{2 x^2-x+3}}{71663616 (2 x+5)^2}+\frac{513097 \sqrt{2 x^2-x+3}}{497664 (2 x+5)^3}-\frac{3667 \sqrt{2 x^2-x+3}}{2304 (2 x+5)^4}+\frac{2053207 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{20639121408 \sqrt{2}} \]
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Rubi [A] time = 0.191478, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1650, 806, 724, 206} \[ \frac{26800085 \sqrt{2 x^2-x+3}}{1719926784 (2 x+5)}-\frac{16295969 \sqrt{2 x^2-x+3}}{71663616 (2 x+5)^2}+\frac{513097 \sqrt{2 x^2-x+3}}{497664 (2 x+5)^3}-\frac{3667 \sqrt{2 x^2-x+3}}{2304 (2 x+5)^4}+\frac{2053207 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{20639121408 \sqrt{2}} \]
Antiderivative was successfully verified.
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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)^5 \sqrt{3-x+2 x^2}} \, dx &=-\frac{3667 \sqrt{3-x+2 x^2}}{2304 (5+2 x)^4}-\frac{1}{288} \int \frac{\frac{37027}{16}-\frac{10167 x}{4}+1944 x^2-720 x^3}{(5+2 x)^4 \sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{3667 \sqrt{3-x+2 x^2}}{2304 (5+2 x)^4}+\frac{513097 \sqrt{3-x+2 x^2}}{497664 (5+2 x)^3}+\frac{\int \frac{\frac{2607829}{16}-\frac{295607 x}{2}+77760 x^2}{(5+2 x)^3 \sqrt{3-x+2 x^2}} \, dx}{62208}\\ &=-\frac{3667 \sqrt{3-x+2 x^2}}{2304 (5+2 x)^4}+\frac{513097 \sqrt{3-x+2 x^2}}{497664 (5+2 x)^3}-\frac{16295969 \sqrt{3-x+2 x^2}}{71663616 (5+2 x)^2}-\frac{\int \frac{\frac{19411145}{16}-\frac{6098911 x}{4}}{(5+2 x)^2 \sqrt{3-x+2 x^2}} \, dx}{8957952}\\ &=-\frac{3667 \sqrt{3-x+2 x^2}}{2304 (5+2 x)^4}+\frac{513097 \sqrt{3-x+2 x^2}}{497664 (5+2 x)^3}-\frac{16295969 \sqrt{3-x+2 x^2}}{71663616 (5+2 x)^2}+\frac{26800085 \sqrt{3-x+2 x^2}}{1719926784 (5+2 x)}-\frac{2053207 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{3439853568}\\ &=-\frac{3667 \sqrt{3-x+2 x^2}}{2304 (5+2 x)^4}+\frac{513097 \sqrt{3-x+2 x^2}}{497664 (5+2 x)^3}-\frac{16295969 \sqrt{3-x+2 x^2}}{71663616 (5+2 x)^2}+\frac{26800085 \sqrt{3-x+2 x^2}}{1719926784 (5+2 x)}+\frac{2053207 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{1719926784}\\ &=-\frac{3667 \sqrt{3-x+2 x^2}}{2304 (5+2 x)^4}+\frac{513097 \sqrt{3-x+2 x^2}}{497664 (5+2 x)^3}-\frac{16295969 \sqrt{3-x+2 x^2}}{71663616 (5+2 x)^2}+\frac{26800085 \sqrt{3-x+2 x^2}}{1719926784 (5+2 x)}+\frac{2053207 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{20639121408 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.136726, size = 81, normalized size = 0.58 \[ \frac{24 \sqrt{2 x^2-x+3} \left (214400680 x^3+43592076 x^2-255525906 x-298655447\right )+2053207 \sqrt{2} (2 x+5)^4 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )}{41278242816 (2 x+5)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.07, size = 116, normalized size = 0.8 \begin{align*} -{\frac{16295969}{286654464}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}+{\frac{26800085}{3439853568}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}+{\frac{2053207\,\sqrt{2}}{41278242816}{\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{513097}{3981312}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}}-{\frac{3667}{36864}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{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 = 1.56057, size = 201, normalized size = 1.45 \begin{align*} -\frac{2053207}{41278242816} \, \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{3667 \, \sqrt{2 \, x^{2} - x + 3}}{2304 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac{513097 \, \sqrt{2 \, x^{2} - x + 3}}{497664 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac{16295969 \, \sqrt{2 \, x^{2} - x + 3}}{71663616 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac{26800085 \, \sqrt{2 \, x^{2} - x + 3}}{1719926784 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33975, size = 405, normalized size = 2.91 \begin{align*} \frac{2053207 \, \sqrt{2}{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\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 (214400680 \, x^{3} + 43592076 \, x^{2} - 255525906 \, x - 298655447\right )} \sqrt{2 \, x^{2} - x + 3}}{82556485632 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\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 )^{5} \sqrt{2 x^{2} - x + 3}}\, 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}{\sqrt{2 \, x^{2} - x + 3}{\left (2 \, x + 5\right )}^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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