Optimal. Leaf size=101 \[ -\frac {11373 \sqrt {3-x+2 x^2}}{1024}+\frac {3443}{768} x \sqrt {3-x+2 x^2}+\frac {655}{96} x^2 \sqrt {3-x+2 x^2}+\frac {25}{8} x^3 \sqrt {3-x+2 x^2}+\frac {30725 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2048 \sqrt {2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {655}{96} \sqrt {2 x^2-x+3} x^2+\frac {3443}{768} \sqrt {2 x^2-x+3} x-\frac {11373 \sqrt {2 x^2-x+3}}{1024}+\frac {25}{8} \sqrt {2 x^2-x+3} x^3+\frac {30725 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2048 \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 )^2}{\sqrt {3-x+2 x^2}} \, dx &=\frac {25}{8} x^3 \sqrt {3-x+2 x^2}+\frac {1}{8} \int \frac {32+96 x+7 x^2+\frac {655 x^3}{2}}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {655}{96} x^2 \sqrt {3-x+2 x^2}+\frac {25}{8} x^3 \sqrt {3-x+2 x^2}+\frac {1}{48} \int \frac {192-1389 x+\frac {3443 x^2}{4}}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {3443}{768} x \sqrt {3-x+2 x^2}+\frac {655}{96} x^2 \sqrt {3-x+2 x^2}+\frac {25}{8} x^3 \sqrt {3-x+2 x^2}+\frac {1}{192} \int \frac {-\frac {7257}{4}-\frac {34119 x}{8}}{\sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {11373 \sqrt {3-x+2 x^2}}{1024}+\frac {3443}{768} x \sqrt {3-x+2 x^2}+\frac {655}{96} x^2 \sqrt {3-x+2 x^2}+\frac {25}{8} x^3 \sqrt {3-x+2 x^2}-\frac {30725 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{2048}\\ &=-\frac {11373 \sqrt {3-x+2 x^2}}{1024}+\frac {3443}{768} x \sqrt {3-x+2 x^2}+\frac {655}{96} x^2 \sqrt {3-x+2 x^2}+\frac {25}{8} x^3 \sqrt {3-x+2 x^2}-\frac {30725 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{2048 \sqrt {46}}\\ &=-\frac {11373 \sqrt {3-x+2 x^2}}{1024}+\frac {3443}{768} x \sqrt {3-x+2 x^2}+\frac {655}{96} x^2 \sqrt {3-x+2 x^2}+\frac {25}{8} x^3 \sqrt {3-x+2 x^2}+\frac {30725 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2048 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 65, normalized size = 0.64 \begin {gather*} \frac {4 \sqrt {3-x+2 x^2} \left (-34119+13772 x+20960 x^2+9600 x^3\right )+92175 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{12288} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 79, normalized size = 0.78
method | result | size |
risch | \(\frac {\left (9600 x^{3}+20960 x^{2}+13772 x -34119\right ) \sqrt {2 x^{2}-x +3}}{3072}-\frac {30725 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4096}\) | \(45\) |
trager | \(\left (\frac {25}{8} x^{3}+\frac {655}{96} x^{2}+\frac {3443}{768} x -\frac {11373}{1024}\right ) \sqrt {2 x^{2}-x +3}+\frac {30725 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}+\RootOf \left (\textit {\_Z}^{2}-2\right )\right )}{4096}\) | \(69\) |
default | \(\frac {25 x^{3} \sqrt {2 x^{2}-x +3}}{8}+\frac {655 x^{2} \sqrt {2 x^{2}-x +3}}{96}+\frac {3443 x \sqrt {2 x^{2}-x +3}}{768}-\frac {11373 \sqrt {2 x^{2}-x +3}}{1024}-\frac {30725 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4096}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 80, normalized size = 0.79 \begin {gather*} \frac {25}{8} \, \sqrt {2 \, x^{2} - x + 3} x^{3} + \frac {655}{96} \, \sqrt {2 \, x^{2} - x + 3} x^{2} + \frac {3443}{768} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {30725}{4096} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {11373}{1024} \, \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 = 2.10, size = 68, normalized size = 0.67 \begin {gather*} \frac {1}{3072} \, {\left (9600 \, x^{3} + 20960 \, x^{2} + 13772 \, x - 34119\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {30725}{8192} \, \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 )^{2}}{\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 = 4.84, size = 63, normalized size = 0.62 \begin {gather*} \frac {1}{3072} \, {\left (4 \, {\left (40 \, {\left (60 \, x + 131\right )} x + 3443\right )} x - 34119\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {30725}{4096} \, \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 )}^2}{\sqrt {2\,x^2-x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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