Optimal. Leaf size=230 \[ \frac {1}{28} \left (7+5 i \sqrt {7}\right ) \log \left (4 x^2+\left (1-i \sqrt {7}\right ) x+4\right )+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) \log \left (4 x^2+\left (1+i \sqrt {7}\right ) x+4\right )+\frac {1}{14} \left (7+5 i \sqrt {7}\right ) x+\frac {1}{14} \left (7-5 i \sqrt {7}\right ) x-\frac {\left (7 \sqrt {7}+19 i\right ) \tan ^{-1}\left (\frac {8 x-i \sqrt {7}+1}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{\sqrt {14 \left (35+i \sqrt {7}\right )}}+\frac {\left (-7 \sqrt {7}+19 i\right ) \tan ^{-1}\left (\frac {8 x+i \sqrt {7}+1}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{\sqrt {14 \left (35-i \sqrt {7}\right )}} \]
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Rubi [A] time = 0.36, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2087, 773, 634, 618, 204, 628} \[ \frac {1}{28} \left (7+5 i \sqrt {7}\right ) \log \left (4 x^2+\left (1-i \sqrt {7}\right ) x+4\right )+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) \log \left (4 x^2+\left (1+i \sqrt {7}\right ) x+4\right )+\frac {1}{14} \left (7+5 i \sqrt {7}\right ) x+\frac {1}{14} \left (7-5 i \sqrt {7}\right ) x-\frac {\left (7 \sqrt {7}+19 i\right ) \tan ^{-1}\left (\frac {8 x-i \sqrt {7}+1}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{\sqrt {14 \left (35+i \sqrt {7}\right )}}+\frac {\left (-7 \sqrt {7}+19 i\right ) \tan ^{-1}\left (\frac {8 x+i \sqrt {7}+1}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{\sqrt {14 \left (35-i \sqrt {7}\right )}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 773
Rule 2087
Rubi steps
\begin {align*} \int \frac {x \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx &=\frac {i \int \frac {x \left (9-5 i \sqrt {7}+\left (10-2 i \sqrt {7}\right ) x\right )}{4+\left (1-i \sqrt {7}\right ) x+4 x^2} \, dx}{\sqrt {7}}-\frac {i \int \frac {x \left (9+5 i \sqrt {7}+\left (10+2 i \sqrt {7}\right ) x\right )}{4+\left (1+i \sqrt {7}\right ) x+4 x^2} \, dx}{\sqrt {7}}\\ &=\frac {1}{14} \left (7-5 i \sqrt {7}\right ) x+\frac {1}{14} \left (7+5 i \sqrt {7}\right ) x+\frac {i \int \frac {-4 \left (10-2 i \sqrt {7}\right )+\left (-\left (1-i \sqrt {7}\right ) \left (10-2 i \sqrt {7}\right )+4 \left (9-5 i \sqrt {7}\right )\right ) x}{4+\left (1-i \sqrt {7}\right ) x+4 x^2} \, dx}{4 \sqrt {7}}-\frac {i \int \frac {-4 \left (10+2 i \sqrt {7}\right )+\left (-\left (1+i \sqrt {7}\right ) \left (10+2 i \sqrt {7}\right )+4 \left (9+5 i \sqrt {7}\right )\right ) x}{4+\left (1+i \sqrt {7}\right ) x+4 x^2} \, dx}{4 \sqrt {7}}\\ &=\frac {1}{14} \left (7-5 i \sqrt {7}\right ) x+\frac {1}{14} \left (7+5 i \sqrt {7}\right ) x-\frac {1}{28} \left (-7+5 i \sqrt {7}\right ) \int \frac {1+i \sqrt {7}+8 x}{4+\left (1+i \sqrt {7}\right ) x+4 x^2} \, dx+\frac {1}{28} \left (7+5 i \sqrt {7}\right ) \int \frac {1-i \sqrt {7}+8 x}{4+\left (1-i \sqrt {7}\right ) x+4 x^2} \, dx+\frac {1}{14} \left (-49+19 i \sqrt {7}\right ) \int \frac {1}{4+\left (1+i \sqrt {7}\right ) x+4 x^2} \, dx-\frac {1}{14} \left (49+19 i \sqrt {7}\right ) \int \frac {1}{4+\left (1-i \sqrt {7}\right ) x+4 x^2} \, dx\\ &=\frac {1}{14} \left (7-5 i \sqrt {7}\right ) x+\frac {1}{14} \left (7+5 i \sqrt {7}\right ) x+\frac {1}{28} \left (7+5 i \sqrt {7}\right ) \log \left (4+\left (1-i \sqrt {7}\right ) x+4 x^2\right )+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) \log \left (4+\left (1+i \sqrt {7}\right ) x+4 x^2\right )+\frac {1}{7} \left (49-19 i \sqrt {7}\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (35-i \sqrt {7}\right )-x^2} \, dx,x,1+i \sqrt {7}+8 x\right )+\frac {1}{7} \left (49+19 i \sqrt {7}\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (35+i \sqrt {7}\right )-x^2} \, dx,x,1-i \sqrt {7}+8 x\right )\\ &=\frac {1}{14} \left (7-5 i \sqrt {7}\right ) x+\frac {1}{14} \left (7+5 i \sqrt {7}\right ) x-\frac {\left (19 i+7 \sqrt {7}\right ) \tan ^{-1}\left (\frac {1-i \sqrt {7}+8 x}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{\sqrt {14 \left (35+i \sqrt {7}\right )}}+\frac {\left (19 i-7 \sqrt {7}\right ) \tan ^{-1}\left (\frac {1+i \sqrt {7}+8 x}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{\sqrt {14 \left (35-i \sqrt {7}\right )}}+\frac {1}{28} \left (7+5 i \sqrt {7}\right ) \log \left (4+\left (1-i \sqrt {7}\right ) x+4 x^2\right )+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) \log \left (4+\left (1+i \sqrt {7}\right ) x+4 x^2\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 94, normalized size = 0.41 \[ 2 \text {RootSum}\left [2 \text {$\#$1}^4+\text {$\#$1}^3+5 \text {$\#$1}^2+\text {$\#$1}+2\& ,\frac {\text {$\#$1}^3 \log (x-\text {$\#$1})-2 \text {$\#$1}^2 \log (x-\text {$\#$1})+2 \text {$\#$1} \log (x-\text {$\#$1})-\log (x-\text {$\#$1})}{8 \text {$\#$1}^3+3 \text {$\#$1}^2+10 \text {$\#$1}+1}\& \right ]+x \]
Antiderivative was successfully verified.
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fricas [B] time = 3.07, size = 1190, normalized size = 5.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, x^{3} + 3 \, x^{2} + x + 5\right )} x}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 62, normalized size = 0.27 \[ x +\frac {2 \left (\RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{3}-2 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{2}+2 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )-1\right ) \ln \left (-\RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )+x \right )}{8 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{3}+3 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )^{2}+10 \RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ x + 2 \, \int \frac {x^{3} - 2 \, x^{2} + 2 \, x - 1}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 183, normalized size = 0.80 \[ x+\left (\sum _{k=1}^4\ln \left (\frac {115\,\mathrm {root}\left (z^4-z^3+\frac {6\,z^2}{7}-\frac {48\,z}{49}+\frac {128}{343},z,k\right )}{8}+15\,x-\frac {\mathrm {root}\left (z^4-z^3+\frac {6\,z^2}{7}-\frac {48\,z}{49}+\frac {128}{343},z,k\right )\,x\,137}{8}+\frac {{\mathrm {root}\left (z^4-z^3+\frac {6\,z^2}{7}-\frac {48\,z}{49}+\frac {128}{343},z,k\right )}^2\,x\,133}{8}-\frac {{\mathrm {root}\left (z^4-z^3+\frac {6\,z^2}{7}-\frac {48\,z}{49}+\frac {128}{343},z,k\right )}^3\,x\,147}{4}-\frac {189\,{\mathrm {root}\left (z^4-z^3+\frac {6\,z^2}{7}-\frac {48\,z}{49}+\frac {128}{343},z,k\right )}^2}{16}+\frac {49\,{\mathrm {root}\left (z^4-z^3+\frac {6\,z^2}{7}-\frac {48\,z}{49}+\frac {128}{343},z,k\right )}^3}{16}-4\right )\,\mathrm {root}\left (z^4-z^3+\frac {6\,z^2}{7}-\frac {48\,z}{49}+\frac {128}{343},z,k\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.95, size = 48, normalized size = 0.21 \[ x + \operatorname {RootSum} {\left (343 t^{4} - 343 t^{3} + 294 t^{2} - 336 t + 128, \left (t \mapsto t \log {\left (\frac {3773 t^{3}}{304} - \frac {1029 t^{2}}{304} + \frac {1001 t}{152} + x - \frac {121}{19} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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