Optimal. Leaf size=195 \[ -\frac{14087245 \left (2 x^2-x+3\right )^{5/2}}{71663616 (2 x+5)^4}+\frac{182165 \left (2 x^2-x+3\right )^{5/2}}{248832 (2 x+5)^5}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}-\frac{(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{13759414272 (2 x+5)^3}+\frac{(27596573612 x+151764102421) \sqrt{2 x^2-x+3}}{55037657088 (2 x+5)}-\frac{1903976002333 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{660451885056 \sqrt{2}}+\frac{369 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}} \]
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Rubi [A] time = 0.267546, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1650, 810, 812, 843, 619, 215, 724, 206} \[ -\frac{14087245 \left (2 x^2-x+3\right )^{5/2}}{71663616 (2 x+5)^4}+\frac{182165 \left (2 x^2-x+3\right )^{5/2}}{248832 (2 x+5)^5}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{3456 (2 x+5)^6}-\frac{(6793718806 x+9802984711) \left (2 x^2-x+3\right )^{3/2}}{13759414272 (2 x+5)^3}+\frac{(27596573612 x+151764102421) \sqrt{2 x^2-x+3}}{55037657088 (2 x+5)}-\frac{1903976002333 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{660451885056 \sqrt{2}}+\frac{369 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 810
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)^7} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}-\frac{1}{432} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{68375}{16}-\frac{28085 x}{4}+2916 x^2-1080 x^3\right )}{(5+2 x)^6} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}+\frac{\int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{10561025}{16}-1010880 x+194400 x^2\right )}{(5+2 x)^5} \, dx}{155520}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac{14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}-\frac{\int \frac{\left (\frac{420053845}{16}-\frac{182410625 x}{4}\right ) \left (3-x+2 x^2\right )^{3/2}}{(5+2 x)^4} \, dx}{44789760}\\ &=-\frac{(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac{14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}+\frac{\int \frac{\left (-\frac{206718515415}{8}+\frac{103487151045 x}{2}\right ) \sqrt{3-x+2 x^2}}{(5+2 x)^2} \, dx}{51597803520}\\ &=\frac{(151764102421+27596573612 x) \sqrt{3-x+2 x^2}}{55037657088 (5+2 x)}-\frac{(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac{14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}-\frac{\int \frac{-\frac{4760153161395}{4}+2379948687360 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{412782428160}\\ &=\frac{(151764102421+27596573612 x) \sqrt{3-x+2 x^2}}{55037657088 (5+2 x)}-\frac{(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac{14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}-\frac{369}{128} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx+\frac{1903976002333 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{110075314176}\\ &=\frac{(151764102421+27596573612 x) \sqrt{3-x+2 x^2}}{55037657088 (5+2 x)}-\frac{(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac{14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}-\frac{1903976002333 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{55037657088}-\frac{369 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{128 \sqrt{46}}\\ &=\frac{(151764102421+27596573612 x) \sqrt{3-x+2 x^2}}{55037657088 (5+2 x)}-\frac{(9802984711+6793718806 x) \left (3-x+2 x^2\right )^{3/2}}{13759414272 (5+2 x)^3}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{3456 (5+2 x)^6}+\frac{182165 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^5}-\frac{14087245 \left (3-x+2 x^2\right )^{5/2}}{71663616 (5+2 x)^4}+\frac{369 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}}-\frac{1903976002333 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{660451885056 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.258643, size = 108, normalized size = 0.55 \[ \frac{\frac{24 \sqrt{2 x^2-x+3} \left (275188285440 x^6+11854023276320 x^5+103803827945872 x^4+422554114856528 x^3+910256842473992 x^2+1011372787716826 x+458411625354581\right )}{(2 x+5)^6}-1903976002333 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )+1903958949888 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1320903770112} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 246, normalized size = 1.3 \begin{align*}{\frac{182165}{7962624} \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{-82772668391+331090673564\,x}{3962711310336}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{3607708597}{2972033482752} \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{1903976002333\,\sqrt{2}}{1320903770112}{\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}{221184} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-6}}-{\frac{369\,\sqrt{2}}{256}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{1903976002333}{213986410758144} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}+{\frac{1903976002333}{3962711310336}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{149610673}{41278242816} \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{-125860542215+503442168860\,x}{213986410758144} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}+{\frac{125860542215}{106993205379072} \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{14087245}{1146617856} \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 = 1.5413, size = 401, normalized size = 2.06 \begin{align*} \frac{3607708597}{1486016741376} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3456 \,{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} + \frac{182165 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{248832 \,{\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} - \frac{14087245 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{71663616 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac{149610673 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{5159780352 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac{3607708597 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{743008370688 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac{82772668391}{990677827584} \, \sqrt{2 \, x^{2} - x + 3} x - \frac{369}{256} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{1903976002333}{1320903770112} \, \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{165562389227}{330225942528} \, \sqrt{2 \, x^{2} - x + 3} + \frac{125860542215 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2972033482752 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44627, size = 830, normalized size = 4.26 \begin{align*} \frac{1903958949888 \, \sqrt{2}{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )} \log \left (4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 1903976002333 \, \sqrt{2}{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\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 (275188285440 \, x^{6} + 11854023276320 \, x^{5} + 103803827945872 \, x^{4} + 422554114856528 \, x^{3} + 910256842473992 \, x^{2} + 1011372787716826 \, x + 458411625354581\right )} \sqrt{2 \, x^{2} - x + 3}}{2641807540224 \,{\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\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 )^{7}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26596, size = 610, normalized size = 3.13 \begin{align*} \frac{369}{256} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac{1903976002333}{1320903770112} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{1903976002333}{1320903770112} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{5}{64} \, \sqrt{2 \, x^{2} - x + 3} + \frac{\sqrt{2}{\left (159278433934432 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{11} + 6347903280912544 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{10} + 48544526840833424 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{9} + 305716670132783088 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{8} + 88313821135911024 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{7} - 2423668581998843376 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{6} - 397211131697032056 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} + 11708897232532299576 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} - 12803484860728491138 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} + 12593033197867577234 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} - 3042533760672408875 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 589526263249780195\right )}}{110075314176 \,{\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 )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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