Optimal. Leaf size=172 \[ \frac {(141051019-23482924 x) \sqrt {3-x+2 x^2}}{65536}+\frac {(500141-123060 x) \left (3-x+2 x^2\right )^{3/2}}{12288}+\frac {3505}{896} \left (3-x+2 x^2\right )^{5/2}-\frac {311}{448} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac {5}{112} (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}+\frac {1622009981 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{131072 \sqrt {2}}-\frac {99009 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{8 \sqrt {2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 172, normalized size of antiderivative = 1.00, 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 = {1667, 828,
857, 633, 221, 738, 212} \begin {gather*} \frac {5}{112} (2 x+5)^2 \left (2 x^2-x+3\right )^{5/2}-\frac {311}{448} (2 x+5) \left (2 x^2-x+3\right )^{5/2}+\frac {3505}{896} \left (2 x^2-x+3\right )^{5/2}+\frac {(500141-123060 x) \left (2 x^2-x+3\right )^{3/2}}{12288}+\frac {(141051019-23482924 x) \sqrt {2 x^2-x+3}}{65536}-\frac {99009 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{8 \sqrt {2}}+\frac {1622009981 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{131072 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 221
Rule 633
Rule 738
Rule 828
Rule 857
Rule 1667
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} \, dx &=\frac {5}{112} (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}+\frac {1}{224} \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (573-9926 x-14508 x^2-7464 x^3\right )}{5+2 x} \, dx\\ &=-\frac {311}{448} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac {5}{112} (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}+\frac {\int \frac {\left (3-x+2 x^2\right )^{3/2} \left (-430152+2062560 x+1682400 x^2\right )}{5+2 x} \, dx}{21504}\\ &=\frac {3505}{896} \left (3-x+2 x^2\right )^{5/2}-\frac {311}{448} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac {5}{112} (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}+\frac {\int \frac {(24853920-68913600 x) \left (3-x+2 x^2\right )^{3/2}}{5+2 x} \, dx}{860160}\\ &=\frac {(500141-123060 x) \left (3-x+2 x^2\right )^{3/2}}{12288}+\frac {3505}{896} \left (3-x+2 x^2\right )^{5/2}-\frac {311}{448} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac {5}{112} (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}-\frac {\int \frac {(-29846322240+78902624640 x) \sqrt {3-x+2 x^2}}{5+2 x} \, dx}{55050240}\\ &=\frac {(141051019-23482924 x) \sqrt {3-x+2 x^2}}{65536}+\frac {(500141-123060 x) \left (3-x+2 x^2\right )^{3/2}}{12288}+\frac {3505}{896} \left (3-x+2 x^2\right )^{5/2}-\frac {311}{448} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac {5}{112} (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}+\frac {\int \frac {21812190368640-43599628289280 x}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{1761607680}\\ &=\frac {(141051019-23482924 x) \sqrt {3-x+2 x^2}}{65536}+\frac {(500141-123060 x) \left (3-x+2 x^2\right )^{3/2}}{12288}+\frac {3505}{896} \left (3-x+2 x^2\right )^{5/2}-\frac {311}{448} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac {5}{112} (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}-\frac {1622009981 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{131072}+\frac {297027}{4} \int \frac {1}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx\\ &=\frac {(141051019-23482924 x) \sqrt {3-x+2 x^2}}{65536}+\frac {(500141-123060 x) \left (3-x+2 x^2\right )^{3/2}}{12288}+\frac {3505}{896} \left (3-x+2 x^2\right )^{5/2}-\frac {311}{448} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac {5}{112} (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}-\frac {297027}{2} \text {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )-\frac {1622009981 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{131072 \sqrt {46}}\\ &=\frac {(141051019-23482924 x) \sqrt {3-x+2 x^2}}{65536}+\frac {(500141-123060 x) \left (3-x+2 x^2\right )^{3/2}}{12288}+\frac {3505}{896} \left (3-x+2 x^2\right )^{5/2}-\frac {311}{448} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac {5}{112} (5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}+\frac {1622009981 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{131072 \sqrt {2}}-\frac {99009 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.67, size = 113, normalized size = 0.66 \begin {gather*} \frac {4 \sqrt {3-x+2 x^2} \left (3149403255-609499532 x+159973408 x^2-46476672 x^3+14493696 x^4-3710976 x^5+983040 x^6\right )+68130865152 \sqrt {2} \tanh ^{-1}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )+34062209601 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{5505024} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 183, normalized size = 1.06 \[\frac {5 x^{2} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{28}-\frac {111 x \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{224}+\frac {1395 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{896}-\frac {10255 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{4096}-\frac {707595 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{65536}-\frac {1622009981 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{262144}+\frac {3667 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{96}-\frac {40337 \left (4 x -1\right ) \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{512}+\frac {33003 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{16}-\frac {99009 \sqrt {2}\, \arctanh \left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{16}\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 157, normalized size = 0.91 \begin {gather*} \frac {5}{28} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} - \frac {111}{224} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {1395}{896} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} - \frac {10255}{1024} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {500141}{12288} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {5870731}{16384} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {1622009981}{262144} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {99009}{16} \, \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 {141051019}{65536} \, \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 = 0.38, size = 135, normalized size = 0.78 \begin {gather*} \frac {1}{1376256} \, {\left (983040 \, x^{6} - 3710976 \, x^{5} + 14493696 \, x^{4} - 46476672 \, x^{3} + 159973408 \, x^{2} - 609499532 \, x + 3149403255\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {1622009981}{524288} \, \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 ) + \frac {99009}{32} \, \sqrt {2} \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 ) \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 (2 x^{2} - x + 3\right )^{\frac {3}{2}} \cdot \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{2 x + 5}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.94, size = 139, normalized size = 0.81 \begin {gather*} \frac {1}{1376256} \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (16 \, {\left (4 \, {\left (40 \, x - 151\right )} x + 2359\right )} x - 121033\right )} x + 4999169\right )} x - 152374883\right )} x + 3149403255\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {1622009981}{262144} \, \sqrt {2} \log \left (-4 \, \sqrt {2} x + \sqrt {2} + 4 \, \sqrt {2 \, x^{2} - x + 3}\right ) - \frac {99009}{16} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {99009}{16} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\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 (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{2\,x+5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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