Optimal. Leaf size=147 \[ -\frac {14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac {1331 (7409+116368 x)}{101568 \sqrt {3-x+2 x^2}}-\frac {1308645 \sqrt {3-x+2 x^2}}{4096}+\frac {526075 x \sqrt {3-x+2 x^2}}{3072}+\frac {38375}{384} x^2 \sqrt {3-x+2 x^2}+\frac {625}{32} x^3 \sqrt {3-x+2 x^2}+\frac {16955197 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8192 \sqrt {2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1674, 1675,
654, 633, 221} \begin {gather*} \frac {38375}{384} \sqrt {2 x^2-x+3} x^2+\frac {526075 \sqrt {2 x^2-x+3} x}{3072}-\frac {1308645 \sqrt {2 x^2-x+3}}{4096}+\frac {1331 (116368 x+7409)}{101568 \sqrt {2 x^2-x+3}}-\frac {14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}+\frac {625}{32} \sqrt {2 x^2-x+3} x^3+\frac {16955197 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8192 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 633
Rule 654
Rule 1674
Rule 1675
Rubi steps
\begin {align*} \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=-\frac {14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac {2}{69} \int \frac {\frac {3839123}{256}-\frac {1983543 x}{128}-\frac {1464801 x^2}{64}+\frac {430905 x^3}{32}+\frac {639975 x^4}{16}+\frac {250125 x^5}{8}+\frac {43125 x^6}{4}}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac {14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac {1331 (7409+116368 x)}{101568 \sqrt {3-x+2 x^2}}+\frac {4 \int \frac {-\frac {141812733}{256}-\frac {1880595 x}{16}+\frac {15512925 x^2}{64}+\frac {3372375 x^3}{16}+\frac {991875 x^4}{16}}{\sqrt {3-x+2 x^2}} \, dx}{1587}\\ &=-\frac {14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac {1331 (7409+116368 x)}{101568 \sqrt {3-x+2 x^2}}+\frac {625}{32} x^3 \sqrt {3-x+2 x^2}+\frac {\int \frac {-\frac {141812733}{32}-\frac {1880595 x}{2}+\frac {22098975 x^2}{16}+\frac {60901125 x^3}{32}}{\sqrt {3-x+2 x^2}} \, dx}{3174}\\ &=-\frac {14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac {1331 (7409+116368 x)}{101568 \sqrt {3-x+2 x^2}}+\frac {38375}{384} x^2 \sqrt {3-x+2 x^2}+\frac {625}{32} x^3 \sqrt {3-x+2 x^2}+\frac {\int \frac {-\frac {425438199}{16}-\frac {272971935 x}{16}+\frac {834881025 x^2}{64}}{\sqrt {3-x+2 x^2}} \, dx}{19044}\\ &=-\frac {14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac {1331 (7409+116368 x)}{101568 \sqrt {3-x+2 x^2}}+\frac {526075 x \sqrt {3-x+2 x^2}}{3072}+\frac {38375}{384} x^2 \sqrt {3-x+2 x^2}+\frac {625}{32} x^3 \sqrt {3-x+2 x^2}+\frac {\int \frac {-\frac {9311654259}{64}-\frac {6230458845 x}{128}}{\sqrt {3-x+2 x^2}} \, dx}{76176}\\ &=-\frac {14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac {1331 (7409+116368 x)}{101568 \sqrt {3-x+2 x^2}}-\frac {1308645 \sqrt {3-x+2 x^2}}{4096}+\frac {526075 x \sqrt {3-x+2 x^2}}{3072}+\frac {38375}{384} x^2 \sqrt {3-x+2 x^2}+\frac {625}{32} x^3 \sqrt {3-x+2 x^2}-\frac {16955197 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{8192}\\ &=-\frac {14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac {1331 (7409+116368 x)}{101568 \sqrt {3-x+2 x^2}}-\frac {1308645 \sqrt {3-x+2 x^2}}{4096}+\frac {526075 x \sqrt {3-x+2 x^2}}{3072}+\frac {38375}{384} x^2 \sqrt {3-x+2 x^2}+\frac {625}{32} x^3 \sqrt {3-x+2 x^2}-\frac {16955197 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{8192 \sqrt {46}}\\ &=-\frac {14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac {1331 (7409+116368 x)}{101568 \sqrt {3-x+2 x^2}}-\frac {1308645 \sqrt {3-x+2 x^2}}{4096}+\frac {526075 x \sqrt {3-x+2 x^2}}{3072}+\frac {38375}{384} x^2 \sqrt {3-x+2 x^2}+\frac {625}{32} x^3 \sqrt {3-x+2 x^2}+\frac {16955197 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8192 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 1.06, size = 85, normalized size = 0.58 \begin {gather*} \frac {-18974698519+49883864262 x-36481630395 x^2+39848900984 x^3-5076781260 x^4+3504730800 x^5+2090608000 x^6+507840000 x^7}{6500352 \left (3-x+2 x^2\right )^{3/2}}+\frac {16955197 \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{8192 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 214, normalized size = 1.46
method | result | size |
risch | \(\frac {507840000 x^{7}+2090608000 x^{6}+3504730800 x^{5}-5076781260 x^{4}+39848900984 x^{3}-36481630395 x^{2}+49883864262 x -18974698519}{6500352 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {16955197 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{16384}\) | \(65\) |
trager | \(\frac {507840000 x^{7}+2090608000 x^{6}+3504730800 x^{5}-5076781260 x^{4}+39848900984 x^{3}-36481630395 x^{2}+49883864262 x -18974698519}{6500352 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {16955197 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -\RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{16384}\) | \(92\) |
default | \(\frac {\frac {992926033 x}{3250176}-\frac {992926033}{13000704}}{\sqrt {2 x^{2}-x +3}}+\frac {138025 x^{5}}{256 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {799745 x^{4}}{1024 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {16955197 x^{3}}{12288 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {67488035 x^{2}}{16384 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {55167267 x}{131072 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {625 x^{7}}{8 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {30875 x^{6}}{96 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {16955197 x}{8192 \sqrt {2 x^{2}-x +3}}+\frac {16955197}{32768 \sqrt {2 x^{2}-x +3}}-\frac {2149616639}{524288 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {\frac {5141612725 x}{9043968}-\frac {5141612725}{36175872}}{\left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {16955197 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{16384}\) | \(214\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 253 vs.
\(2 (118) = 236\).
time = 0.51, size = 253, normalized size = 1.72 \begin {gather*} \frac {625 \, x^{7}}{8 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {30875 \, x^{6}}{96 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {138025 \, x^{5}}{256 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {799745 \, x^{4}}{1024 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {16955197}{13000704} \, x {\left (\frac {284 \, x}{\sqrt {2 \, x^{2} - x + 3}} - \frac {3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {71}{\sqrt {2 \, x^{2} - x + 3}} + \frac {805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} - \frac {16955197}{16384} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {1203818987}{6500352} \, \sqrt {2 \, x^{2} - x + 3} + \frac {3536205583 \, x}{3250176 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {2638851 \, x^{2}}{512 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {257773037}{1083392 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {29484067 \, x}{23552 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {374445479}{70656 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.82, size = 132, normalized size = 0.90 \begin {gather*} \frac {26907897639 \, \sqrt {2} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8 \, {\left (507840000 \, x^{7} + 2090608000 \, x^{6} + 3504730800 \, x^{5} - 5076781260 \, x^{4} + 39848900984 \, x^{3} - 36481630395 \, x^{2} + 49883864262 \, x - 18974698519\right )} \sqrt {2 \, x^{2} - x + 3}}{52002816 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (5 x^{2} + 3 x + 2\right )^{4}}{\left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.79, size = 81, normalized size = 0.55 \begin {gather*} \frac {16955197}{16384} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {{\left ({\left (4 \, {\left (2645 \, {\left (20 \, {\left (40 \, {\left (60 \, x + 247\right )} x + 16563\right )} x - 479847\right )} x + 9962225246\right )} x - 36481630395\right )} x + 49883864262\right )} x - 18974698519}{6500352 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (5\,x^2+3\,x+2\right )}^4}{{\left (2\,x^2-x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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