Optimal. Leaf size=185 \[ \frac {16493087661 \sqrt {3-x+2 x^2}}{29360128}+\frac {1572007407 x \sqrt {3-x+2 x^2}}{7340032}-\frac {15428243 x^2 \sqrt {3-x+2 x^2}}{131072}-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {2899366573 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8388608 \sqrt {2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1675, 654,
633, 221} \begin {gather*} -\frac {15428243 \sqrt {2 x^2-x+3} x^2}{131072}+\frac {1572007407 \sqrt {2 x^2-x+3} x}{7340032}+\frac {16493087661 \sqrt {2 x^2-x+3}}{29360128}+\frac {625}{16} \sqrt {2 x^2-x+3} x^7+\frac {57375}{448} \sqrt {2 x^2-x+3} x^6+\frac {2116475 \sqrt {2 x^2-x+3} x^5}{10752}+\frac {686531 \sqrt {2 x^2-x+3} x^4}{6144}-\frac {19750457 \sqrt {2 x^2-x+3} x^3}{229376}+\frac {2899366573 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8388608 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 633
Rule 654
Rule 1675
Rubi steps
\begin {align*} \int \frac {\left (2+3 x+5 x^2\right )^4}{\sqrt {3-x+2 x^2}} \, dx &=\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {1}{16} \int \frac {256+1536 x+6016 x^2+14976 x^3+28176 x^4+37440 x^5+24475 x^6+\frac {57375 x^7}{2}}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {1}{224} \int \frac {3584+21504 x+84224 x^2+209664 x^3+394464 x^4+7785 x^5+\frac {2116475 x^6}{4}}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {\int \frac {43008+258048 x+1010688 x^2+2515968 x^3-\frac {12812853 x^4}{4}+\frac {24028585 x^5}{8}}{\sqrt {3-x+2 x^2}} \, dx}{2688}\\ &=\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {\int \frac {430080+2580480 x+10106880 x^2-\frac {21766395 x^3}{2}-\frac {296256855 x^4}{16}}{\sqrt {3-x+2 x^2}} \, dx}{26880}\\ &=-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {\int \frac {3440640+20643840 x+\frac {3959992335 x^2}{16}-\frac {4859896545 x^3}{32}}{\sqrt {3-x+2 x^2}} \, dx}{215040}\\ &=-\frac {15428243 x^2 \sqrt {3-x+2 x^2}}{131072}-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {\int \frac {20643840+\frac {16561498275 x}{16}+\frac {70740333315 x^2}{64}}{\sqrt {3-x+2 x^2}} \, dx}{1290240}\\ &=\frac {1572007407 x \sqrt {3-x+2 x^2}}{7340032}-\frac {15428243 x^2 \sqrt {3-x+2 x^2}}{131072}-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {\int \frac {-\frac {206936176905}{64}+\frac {742188944745 x}{128}}{\sqrt {3-x+2 x^2}} \, dx}{5160960}\\ &=\frac {16493087661 \sqrt {3-x+2 x^2}}{29360128}+\frac {1572007407 x \sqrt {3-x+2 x^2}}{7340032}-\frac {15428243 x^2 \sqrt {3-x+2 x^2}}{131072}-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}-\frac {2899366573 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{8388608}\\ &=\frac {16493087661 \sqrt {3-x+2 x^2}}{29360128}+\frac {1572007407 x \sqrt {3-x+2 x^2}}{7340032}-\frac {15428243 x^2 \sqrt {3-x+2 x^2}}{131072}-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}-\frac {2899366573 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{8388608 \sqrt {46}}\\ &=\frac {16493087661 \sqrt {3-x+2 x^2}}{29360128}+\frac {1572007407 x \sqrt {3-x+2 x^2}}{7340032}-\frac {15428243 x^2 \sqrt {3-x+2 x^2}}{131072}-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {2899366573 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8388608 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 85, normalized size = 0.46 \begin {gather*} \frac {4 \sqrt {3-x+2 x^2} \left (49479262983+18864088884 x-10367779296 x^2-7584175488 x^3+9842108416 x^4+17338163200 x^5+11280384000 x^6+3440640000 x^7\right )+60886698033 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{352321536} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 147, normalized size = 0.79
method | result | size |
risch | \(\frac {\left (3440640000 x^{7}+11280384000 x^{6}+17338163200 x^{5}+9842108416 x^{4}-7584175488 x^{3}-10367779296 x^{2}+18864088884 x +49479262983\right ) \sqrt {2 x^{2}-x +3}}{88080384}-\frac {2899366573 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{16777216}\) | \(65\) |
trager | \(\left (\frac {625}{16} x^{7}+\frac {57375}{448} x^{6}+\frac {2116475}{10752} x^{5}+\frac {686531}{6144} x^{4}-\frac {19750457}{229376} x^{3}-\frac {15428243}{131072} x^{2}+\frac {1572007407}{7340032} x +\frac {16493087661}{29360128}\right ) \sqrt {2 x^{2}-x +3}+\frac {2899366573 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}+\RootOf \left (\textit {\_Z}^{2}-2\right )\right )}{16777216}\) | \(89\) |
default | \(\frac {16493087661 \sqrt {2 x^{2}-x +3}}{29360128}-\frac {2899366573 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{16777216}+\frac {2116475 x^{5} \sqrt {2 x^{2}-x +3}}{10752}+\frac {57375 x^{6} \sqrt {2 x^{2}-x +3}}{448}+\frac {625 x^{7} \sqrt {2 x^{2}-x +3}}{16}+\frac {1572007407 x \sqrt {2 x^{2}-x +3}}{7340032}-\frac {15428243 x^{2} \sqrt {2 x^{2}-x +3}}{131072}-\frac {19750457 x^{3} \sqrt {2 x^{2}-x +3}}{229376}+\frac {686531 x^{4} \sqrt {2 x^{2}-x +3}}{6144}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 148, normalized size = 0.80 \begin {gather*} \frac {625}{16} \, \sqrt {2 \, x^{2} - x + 3} x^{7} + \frac {57375}{448} \, \sqrt {2 \, x^{2} - x + 3} x^{6} + \frac {2116475}{10752} \, \sqrt {2 \, x^{2} - x + 3} x^{5} + \frac {686531}{6144} \, \sqrt {2 \, x^{2} - x + 3} x^{4} - \frac {19750457}{229376} \, \sqrt {2 \, x^{2} - x + 3} x^{3} - \frac {15428243}{131072} \, \sqrt {2 \, x^{2} - x + 3} x^{2} + \frac {1572007407}{7340032} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {2899366573}{16777216} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {16493087661}{29360128} \, \sqrt {2 \, x^{2} - x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.33, size = 88, normalized size = 0.48 \begin {gather*} \frac {1}{88080384} \, {\left (3440640000 \, x^{7} + 11280384000 \, x^{6} + 17338163200 \, x^{5} + 9842108416 \, x^{4} - 7584175488 \, x^{3} - 10367779296 \, x^{2} + 18864088884 \, x + 49479262983\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {2899366573}{33554432} \, \sqrt {2} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\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}}{\sqrt {2 x^{2} - x + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.18, size = 83, normalized size = 0.45 \begin {gather*} \frac {1}{88080384} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, {\left (120 \, {\left (140 \, x + 459\right )} x + 84659\right )} x + 4805717\right )} x - 59251371\right )} x - 323993103\right )} x + 4716022221\right )} x + 49479262983\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {2899366573}{16777216} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \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}{\sqrt {2\,x^2-x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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