Optimal. Leaf size=84 \[ \frac {125 x}{8}-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}+\frac {165099 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8464 \sqrt {23}}+\frac {825}{32} \log \left (3-x+2 x^2\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1674, 1671,
648, 632, 210, 642} \begin {gather*} \frac {165099 \text {ArcTan}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8464 \sqrt {23}}+\frac {121 (21193-12828 x)}{33856 \left (2 x^2-x+3\right )}-\frac {1331 (17-45 x)}{1472 \left (2 x^2-x+3\right )^2}+\frac {825}{32} \log \left (2 x^2-x+3\right )+\frac {125 x}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1671
Rule 1674
Rubi steps
\begin {align*} \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^3} \, dx &=-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {1}{46} \int \frac {-\frac {40885}{32}-\frac {19067 x}{8}+\frac {22195 x^2}{4}+\frac {13225 x^3}{2}+2875 x^4}{\left (3-x+2 x^2\right )^2} \, dx\\ &=-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}+\frac {\int \frac {\frac {23997}{2}+92575 x+\frac {66125 x^2}{2}}{3-x+2 x^2} \, dx}{1058}\\ &=-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}+\frac {\int \left (\frac {66125}{4}-\frac {33 (4557-13225 x)}{4 \left (3-x+2 x^2\right )}\right ) \, dx}{1058}\\ &=\frac {125 x}{8}-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}-\frac {33 \int \frac {4557-13225 x}{3-x+2 x^2} \, dx}{4232}\\ &=\frac {125 x}{8}-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}-\frac {165099 \int \frac {1}{3-x+2 x^2} \, dx}{16928}+\frac {825}{32} \int \frac {-1+4 x}{3-x+2 x^2} \, dx\\ &=\frac {125 x}{8}-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}+\frac {825}{32} \log \left (3-x+2 x^2\right )+\frac {165099 \text {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )}{8464}\\ &=\frac {125 x}{8}-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}+\frac {165099 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8464 \sqrt {23}}+\frac {825}{32} \log \left (3-x+2 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 84, normalized size = 1.00 \begin {gather*} \frac {125 x}{8}+\frac {1331 (-17+45 x)}{1472 \left (3-x+2 x^2\right )^2}-\frac {121 (-21193+12828 x)}{33856 \left (3-x+2 x^2\right )}-\frac {165099 \tan ^{-1}\left (\frac {-1+4 x}{\sqrt {23}}\right )}{8464 \sqrt {23}}+\frac {825}{32} \log \left (3-x+2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 63, normalized size = 0.75
method | result | size |
default | \(\frac {125 x}{8}+\frac {-\frac {388047}{4232} x^{3}+\frac {3340447}{16928} x^{2}-\frac {1460833}{8464} x +\frac {3586319}{16928}}{\left (2 x^{2}-x +3\right )^{2}}+\frac {825 \ln \left (2 x^{2}-x +3\right )}{32}-\frac {165099 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{194672}\) | \(63\) |
risch | \(\frac {125 x}{8}+\frac {-\frac {388047}{4232} x^{3}+\frac {3340447}{16928} x^{2}-\frac {1460833}{8464} x +\frac {3586319}{16928}}{\left (2 x^{2}-x +3\right )^{2}}+\frac {825 \ln \left (16 x^{2}-8 x +24\right )}{32}-\frac {165099 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{194672}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 72, normalized size = 0.86 \begin {gather*} -\frac {165099}{194672} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {125}{8} \, x - \frac {121 \, {\left (12828 \, x^{3} - 27607 \, x^{2} + 24146 \, x - 29639\right )}}{16928 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} + \frac {825}{32} \, \log \left (2 \, x^{2} - x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.91, size = 118, normalized size = 1.40 \begin {gather*} \frac {24334000 \, x^{5} - 24334000 \, x^{4} + 43385176 \, x^{3} - 330198 \, \sqrt {23} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 40329281 \, x^{2} + 10037775 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x^{2} - x + 3\right ) - 12446818 \, x + 82485337}{389344 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 82, normalized size = 0.98 \begin {gather*} \frac {125 x}{8} + \frac {- 1552188 x^{3} + 3340447 x^{2} - 2921666 x + 3586319}{67712 x^{4} - 67712 x^{3} + 220064 x^{2} - 101568 x + 152352} + \frac {825 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{32} - \frac {165099 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{194672} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.00, size = 62, normalized size = 0.74 \begin {gather*} -\frac {165099}{194672} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {125}{8} \, x - \frac {121 \, {\left (12828 \, x^{3} - 27607 \, x^{2} + 24146 \, x - 29639\right )}}{16928 \, {\left (2 \, x^{2} - x + 3\right )}^{2}} + \frac {825}{32} \, \log \left (2 \, x^{2} - x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 72, normalized size = 0.86 \begin {gather*} \frac {125\,x}{8}+\frac {825\,\ln \left (2\,x^2-x+3\right )}{32}-\frac {165099\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{194672}-\frac {\frac {388047\,x^3}{16928}-\frac {3340447\,x^2}{67712}+\frac {1460833\,x}{33856}-\frac {3586319}{67712}}{x^4-x^3+\frac {13\,x^2}{4}-\frac {3\,x}{2}+\frac {9}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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