Optimal. Leaf size=143 \[ \frac {5}{112} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^4-\frac {823 \left (2 x^2-x+3\right )^{3/2} (2 x+5)^3}{1344}+\frac {11433 \left (2 x^2-x+3\right )^{3/2} (2 x+5)^2}{4480}-\frac {(295276 x+1005757) \left (2 x^2-x+3\right )^{3/2}}{71680}-\frac {51435 (1-4 x) \sqrt {2 x^2-x+3}}{32768}-\frac {1183005 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{65536 \sqrt {2}} \]
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Rubi [A] time = 0.16, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1653, 779, 612, 619, 215} \[ \frac {5}{112} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^4-\frac {823 \left (2 x^2-x+3\right )^{3/2} (2 x+5)^3}{1344}+\frac {11433 \left (2 x^2-x+3\right )^{3/2} (2 x+5)^2}{4480}-\frac {(295276 x+1005757) \left (2 x^2-x+3\right )^{3/2}}{71680}-\frac {51435 (1-4 x) \sqrt {2 x^2-x+3}}{32768}-\frac {1183005 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{65536 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rule 779
Rule 1653
Rubi steps
\begin {align*} \int (5+2 x) \sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx &=\frac {5}{112} (5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}+\frac {1}{224} \int (5+2 x) \sqrt {3-x+2 x^2} \left (-3677-7826 x-10788 x^2-6584 x^3\right ) \, dx\\ &=-\frac {823 (5+2 x)^3 \left (3-x+2 x^2\right )^{3/2}}{1344}+\frac {5}{112} (5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}+\frac {\int (5+2 x) \sqrt {3-x+2 x^2} \left (338328+907872 x+1097568 x^2\right ) \, dx}{21504}\\ &=\frac {11433 (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}}{4480}-\frac {823 (5+2 x)^3 \left (3-x+2 x^2\right )^{3/2}}{1344}+\frac {5}{112} (5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}+\frac {\int (3655008-14173248 x) (5+2 x) \sqrt {3-x+2 x^2} \, dx}{860160}\\ &=\frac {11433 (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}}{4480}-\frac {823 (5+2 x)^3 \left (3-x+2 x^2\right )^{3/2}}{1344}+\frac {5}{112} (5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}-\frac {(1005757+295276 x) \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac {51435 \int \sqrt {3-x+2 x^2} \, dx}{4096}\\ &=-\frac {51435 (1-4 x) \sqrt {3-x+2 x^2}}{32768}+\frac {11433 (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}}{4480}-\frac {823 (5+2 x)^3 \left (3-x+2 x^2\right )^{3/2}}{1344}+\frac {5}{112} (5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}-\frac {(1005757+295276 x) \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac {1183005 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{65536}\\ &=-\frac {51435 (1-4 x) \sqrt {3-x+2 x^2}}{32768}+\frac {11433 (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}}{4480}-\frac {823 (5+2 x)^3 \left (3-x+2 x^2\right )^{3/2}}{1344}+\frac {5}{112} (5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}-\frac {(1005757+295276 x) \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac {\left (51435 \sqrt {\frac {23}{2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{65536}\\ &=-\frac {51435 (1-4 x) \sqrt {3-x+2 x^2}}{32768}+\frac {11433 (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}}{4480}-\frac {823 (5+2 x)^3 \left (3-x+2 x^2\right )^{3/2}}{1344}+\frac {5}{112} (5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}-\frac {(1005757+295276 x) \left (3-x+2 x^2\right )^{3/2}}{71680}-\frac {1183005 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{65536 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 70, normalized size = 0.49 \[ \frac {4 \sqrt {2 x^2-x+3} \left (4915200 x^6+12984320 x^5+1390592 x^4+20304768 x^3+11357024 x^2+14742332 x+6231117\right )-124215525 \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{13762560} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 83, normalized size = 0.58 \[ \frac {1}{3440640} \, {\left (4915200 \, x^{6} + 12984320 \, x^{5} + 1390592 \, x^{4} + 20304768 \, x^{3} + 11357024 \, x^{2} + 14742332 \, x + 6231117\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {1183005}{262144} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 78, normalized size = 0.55 \[ \frac {1}{3440640} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, x + 317\right )} x + 679\right )} x + 158631\right )} x + 354907\right )} x + 3685583\right )} x + 6231117\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {1183005}{131072} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 115, normalized size = 0.80 \[ \frac {5 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x^{4}}{7}+\frac {377 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x^{3}}{168}+\frac {283 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x^{2}}{1120}-\frac {5179 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x}{17920}+\frac {1183005 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{131072}+\frac {51435 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{32768}+\frac {242329 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{215040} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 126, normalized size = 0.88 \[ \frac {5}{7} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + \frac {377}{168} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {283}{1120} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {5179}{17920} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {242329}{215040} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {51435}{8192} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {1183005}{131072} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {51435}{32768} \, \sqrt {2 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.72, size = 170, normalized size = 1.19 \[ \frac {283\,x^2\,{\left (2\,x^2-x+3\right )}^{3/2}}{1120}+\frac {377\,x^3\,{\left (2\,x^2-x+3\right )}^{3/2}}{168}+\frac {5\,x^4\,{\left (2\,x^2-x+3\right )}^{3/2}}{7}+\frac {4478951\,\sqrt {2}\,\ln \left (\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (2\,x-\frac {1}{2}\right )}{2}\right )}{573440}+\frac {194737\,\left (\frac {x}{2}-\frac {1}{8}\right )\,\sqrt {2\,x^2-x+3}}{17920}+\frac {242329\,\sqrt {2\,x^2-x+3}\,\left (32\,x^2-4\,x+45\right )}{3440640}-\frac {5179\,x\,{\left (2\,x^2-x+3\right )}^{3/2}}{17920}+\frac {5573567\,\sqrt {2}\,\ln \left (2\,\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (4\,x-1\right )}{2}\right )}{4587520} \]
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 \[ \int \left (2 x + 5\right ) \sqrt {2 x^{2} - x + 3} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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