Optimal. Leaf size=120 \[ \frac {1}{16} \sqrt {2 x^2-x+3} (2 x+5)^4-\frac {105}{128} \sqrt {2 x^2-x+3} (2 x+5)^3+\frac {761}{256} \sqrt {2 x^2-x+3} (2 x+5)^2-\frac {(4676 x+19227) \sqrt {2 x^2-x+3}}{2048}-\frac {85429 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4096 \sqrt {2}} \]
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Rubi [A] time = 0.14, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1653, 779, 619, 215} \[ \frac {1}{16} \sqrt {2 x^2-x+3} (2 x+5)^4-\frac {105}{128} \sqrt {2 x^2-x+3} (2 x+5)^3+\frac {761}{256} \sqrt {2 x^2-x+3} (2 x+5)^2-\frac {(4676 x+19227) \sqrt {2 x^2-x+3}}{2048}-\frac {85429 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4096 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 619
Rule 779
Rule 1653
Rubi steps
\begin {align*} \int \frac {(5+2 x) \left (2+x+3 x^2-x^3+5 x^4\right )}{\sqrt {3-x+2 x^2}} \, dx &=\frac {1}{16} (5+2 x)^4 \sqrt {3-x+2 x^2}+\frac {1}{160} \int \frac {(5+2 x) \left (-5055-4390 x-5580 x^2-4200 x^3\right )}{\sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {105}{128} (5+2 x)^3 \sqrt {3-x+2 x^2}+\frac {1}{16} (5+2 x)^4 \sqrt {3-x+2 x^2}+\frac {\int \frac {(5+2 x) \left (327480+105440 x+365280 x^2\right )}{\sqrt {3-x+2 x^2}} \, dx}{10240}\\ &=\frac {761}{256} (5+2 x)^2 \sqrt {3-x+2 x^2}-\frac {105}{128} (5+2 x)^3 \sqrt {3-x+2 x^2}+\frac {1}{16} (5+2 x)^4 \sqrt {3-x+2 x^2}+\frac {\int \frac {(919200-1122240 x) (5+2 x)}{\sqrt {3-x+2 x^2}} \, dx}{245760}\\ &=\frac {761}{256} (5+2 x)^2 \sqrt {3-x+2 x^2}-\frac {105}{128} (5+2 x)^3 \sqrt {3-x+2 x^2}+\frac {1}{16} (5+2 x)^4 \sqrt {3-x+2 x^2}-\frac {(19227+4676 x) \sqrt {3-x+2 x^2}}{2048}+\frac {85429 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{4096}\\ &=\frac {761}{256} (5+2 x)^2 \sqrt {3-x+2 x^2}-\frac {105}{128} (5+2 x)^3 \sqrt {3-x+2 x^2}+\frac {1}{16} (5+2 x)^4 \sqrt {3-x+2 x^2}-\frac {(19227+4676 x) \sqrt {3-x+2 x^2}}{2048}+\frac {85429 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{4096 \sqrt {46}}\\ &=\frac {761}{256} (5+2 x)^2 \sqrt {3-x+2 x^2}-\frac {105}{128} (5+2 x)^3 \sqrt {3-x+2 x^2}+\frac {1}{16} (5+2 x)^4 \sqrt {3-x+2 x^2}-\frac {(19227+4676 x) \sqrt {3-x+2 x^2}}{2048}-\frac {85429 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4096 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 60, normalized size = 0.50 \[ \frac {4 \sqrt {2 x^2-x+3} \left (2048 x^4+7040 x^3+352 x^2-6916 x+2973\right )-85429 \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8192} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 73, normalized size = 0.61 \[ \frac {1}{2048} \, {\left (2048 \, x^{4} + 7040 \, x^{3} + 352 \, x^{2} - 6916 \, x + 2973\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {85429}{16384} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 68, normalized size = 0.57 \[ \frac {1}{2048} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, x + 55\right )} x + 11\right )} x - 1729\right )} x + 2973\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {85429}{8192} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 95, normalized size = 0.79 \[ \sqrt {2 x^{2}-x +3}\, x^{4}+\frac {55 \sqrt {2 x^{2}-x +3}\, x^{3}}{16}+\frac {11 \sqrt {2 x^{2}-x +3}\, x^{2}}{64}-\frac {1729 \sqrt {2 x^{2}-x +3}\, x}{512}+\frac {85429 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8192}+\frac {2973 \sqrt {2 x^{2}-x +3}}{2048} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 96, normalized size = 0.80 \[ \sqrt {2 \, x^{2} - x + 3} x^{4} + \frac {55}{16} \, \sqrt {2 \, x^{2} - x + 3} x^{3} + \frac {11}{64} \, \sqrt {2 \, x^{2} - x + 3} x^{2} - \frac {1729}{512} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {85429}{8192} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {2973}{2048} \, \sqrt {2 \, x^{2} - x + 3} \]
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 \[ \int \frac {\left (2\,x+5\right )\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{\sqrt {2\,x^2-x+3}} \,d x \]
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 \[ \int \frac {\left (2 x + 5\right ) \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\sqrt {2 x^{2} - x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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