Optimal. Leaf size=84 \[ \frac {122691 x}{128}-\frac {28747 x^2}{128}-\frac {21229 x^3}{96}+\frac {6245 x^4}{64}+\frac {1855 x^5}{8}+\frac {3625 x^6}{24}+\frac {625 x^7}{14}+\frac {1156639 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{256 \sqrt {23}}+\frac {307461}{512} \log \left (3-x+2 x^2\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1671, 648, 632,
210, 642} \begin {gather*} \frac {1156639 \text {ArcTan}\left (\frac {1-4 x}{\sqrt {23}}\right )}{256 \sqrt {23}}+\frac {625 x^7}{14}+\frac {3625 x^6}{24}+\frac {1855 x^5}{8}+\frac {6245 x^4}{64}-\frac {21229 x^3}{96}-\frac {28747 x^2}{128}+\frac {307461}{512} \log \left (2 x^2-x+3\right )+\frac {122691 x}{128} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1671
Rubi steps
\begin {align*} \int \frac {\left (2+3 x+5 x^2\right )^4}{3-x+2 x^2} \, dx &=\int \left (\frac {122691}{128}-\frac {28747 x}{64}-\frac {21229 x^2}{32}+\frac {6245 x^3}{16}+\frac {9275 x^4}{8}+\frac {3625 x^5}{4}+\frac {625 x^6}{2}-\frac {14641 (25-21 x)}{128 \left (3-x+2 x^2\right )}\right ) \, dx\\ &=\frac {122691 x}{128}-\frac {28747 x^2}{128}-\frac {21229 x^3}{96}+\frac {6245 x^4}{64}+\frac {1855 x^5}{8}+\frac {3625 x^6}{24}+\frac {625 x^7}{14}-\frac {14641}{128} \int \frac {25-21 x}{3-x+2 x^2} \, dx\\ &=\frac {122691 x}{128}-\frac {28747 x^2}{128}-\frac {21229 x^3}{96}+\frac {6245 x^4}{64}+\frac {1855 x^5}{8}+\frac {3625 x^6}{24}+\frac {625 x^7}{14}+\frac {307461}{512} \int \frac {-1+4 x}{3-x+2 x^2} \, dx-\frac {1156639}{512} \int \frac {1}{3-x+2 x^2} \, dx\\ &=\frac {122691 x}{128}-\frac {28747 x^2}{128}-\frac {21229 x^3}{96}+\frac {6245 x^4}{64}+\frac {1855 x^5}{8}+\frac {3625 x^6}{24}+\frac {625 x^7}{14}+\frac {307461}{512} \log \left (3-x+2 x^2\right )+\frac {1156639}{256} \text {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )\\ &=\frac {122691 x}{128}-\frac {28747 x^2}{128}-\frac {21229 x^3}{96}+\frac {6245 x^4}{64}+\frac {1855 x^5}{8}+\frac {3625 x^6}{24}+\frac {625 x^7}{14}+\frac {1156639 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{256 \sqrt {23}}+\frac {307461}{512} \log \left (3-x+2 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 72, normalized size = 0.86 \begin {gather*} \frac {x \left (2576511-603687 x-594412 x^2+262290 x^3+623280 x^4+406000 x^5+120000 x^6\right )}{2688}-\frac {1156639 \tan ^{-1}\left (\frac {-1+4 x}{\sqrt {23}}\right )}{256 \sqrt {23}}+\frac {307461}{512} \log \left (3-x+2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 64, normalized size = 0.76
method | result | size |
default | \(\frac {625 x^{7}}{14}+\frac {3625 x^{6}}{24}+\frac {1855 x^{5}}{8}+\frac {6245 x^{4}}{64}-\frac {21229 x^{3}}{96}-\frac {28747 x^{2}}{128}+\frac {122691 x}{128}+\frac {307461 \ln \left (2 x^{2}-x +3\right )}{512}-\frac {1156639 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{5888}\) | \(64\) |
risch | \(\frac {625 x^{7}}{14}+\frac {3625 x^{6}}{24}+\frac {1855 x^{5}}{8}+\frac {6245 x^{4}}{64}-\frac {21229 x^{3}}{96}-\frac {28747 x^{2}}{128}+\frac {122691 x}{128}+\frac {307461 \ln \left (16 x^{2}-8 x +24\right )}{512}-\frac {1156639 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{5888}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 63, normalized size = 0.75 \begin {gather*} \frac {625}{14} \, x^{7} + \frac {3625}{24} \, x^{6} + \frac {1855}{8} \, x^{5} + \frac {6245}{64} \, x^{4} - \frac {21229}{96} \, x^{3} - \frac {28747}{128} \, x^{2} - \frac {1156639}{5888} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {122691}{128} \, x + \frac {307461}{512} \, \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 = 1.72, size = 63, normalized size = 0.75 \begin {gather*} \frac {625}{14} \, x^{7} + \frac {3625}{24} \, x^{6} + \frac {1855}{8} \, x^{5} + \frac {6245}{64} \, x^{4} - \frac {21229}{96} \, x^{3} - \frac {28747}{128} \, x^{2} - \frac {1156639}{5888} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {122691}{128} \, x + \frac {307461}{512} \, \log \left (2 \, x^{2} - x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 87, normalized size = 1.04 \begin {gather*} \frac {625 x^{7}}{14} + \frac {3625 x^{6}}{24} + \frac {1855 x^{5}}{8} + \frac {6245 x^{4}}{64} - \frac {21229 x^{3}}{96} - \frac {28747 x^{2}}{128} + \frac {122691 x}{128} + \frac {307461 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{512} - \frac {1156639 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{5888} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.12, size = 63, normalized size = 0.75 \begin {gather*} \frac {625}{14} \, x^{7} + \frac {3625}{24} \, x^{6} + \frac {1855}{8} \, x^{5} + \frac {6245}{64} \, x^{4} - \frac {21229}{96} \, x^{3} - \frac {28747}{128} \, x^{2} - \frac {1156639}{5888} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {122691}{128} \, x + \frac {307461}{512} \, \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 = 3.44, size = 65, normalized size = 0.77 \begin {gather*} \frac {122691\,x}{128}+\frac {307461\,\ln \left (2\,x^2-x+3\right )}{512}-\frac {1156639\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{5888}-\frac {28747\,x^2}{128}-\frac {21229\,x^3}{96}+\frac {6245\,x^4}{64}+\frac {1855\,x^5}{8}+\frac {3625\,x^6}{24}+\frac {625\,x^7}{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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