Optimal. Leaf size=317 \[ -\frac {35-9 i \sqrt {7}}{56 x^2}-\frac {35+9 i \sqrt {7}}{56 x^2}+\frac {3 \left (7-11 i \sqrt {7}\right )}{56 x}+\frac {3 \left (7+11 i \sqrt {7}\right )}{56 x}+\frac {\left (355-73 i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {i-\sqrt {7}+8 i x}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{8 \sqrt {14 \left (35-i \sqrt {7}\right )}}-\frac {\left (355+73 i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {i+\sqrt {7}+8 i x}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{8 \sqrt {14 \left (35+i \sqrt {7}\right )}}-\frac {1}{16} \left (35-9 i \sqrt {7}\right ) \log (x)-\frac {1}{16} \left (35+9 i \sqrt {7}\right ) \log (x)+\frac {1}{32} \left (35-9 i \sqrt {7}\right ) \log \left (4 i+\left (i-\sqrt {7}\right ) x+4 i x^2\right )+\frac {1}{32} \left (35+9 i \sqrt {7}\right ) \log \left (4 i+\left (i+\sqrt {7}\right ) x+4 i x^2\right ) \]
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Rubi [A]
time = 0.39, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2112, 814,
648, 632, 212, 642} \begin {gather*} -\frac {35+9 i \sqrt {7}}{56 x^2}-\frac {35-9 i \sqrt {7}}{56 x^2}+\frac {1}{32} \left (35-9 i \sqrt {7}\right ) \log \left (4 i x^2+\left (-\sqrt {7}+i\right ) x+4 i\right )+\frac {1}{32} \left (35+9 i \sqrt {7}\right ) \log \left (4 i x^2+\left (\sqrt {7}+i\right ) x+4 i\right )+\frac {3 \left (7+11 i \sqrt {7}\right )}{56 x}+\frac {3 \left (7-11 i \sqrt {7}\right )}{56 x}-\frac {1}{16} \left (35+9 i \sqrt {7}\right ) \log (x)-\frac {1}{16} \left (35-9 i \sqrt {7}\right ) \log (x)+\frac {\left (355-73 i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {8 i x-\sqrt {7}+i}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{8 \sqrt {14 \left (35-i \sqrt {7}\right )}}-\frac {\left (355+73 i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {8 i x+\sqrt {7}+i}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{8 \sqrt {14 \left (35+i \sqrt {7}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 814
Rule 2112
Rubi steps
\begin {align*} \int \frac {5+x+3 x^2+2 x^3}{x^3 \left (2+x+5 x^2+x^3+2 x^4\right )} \, dx &=\frac {i \int \frac {9-5 i \sqrt {7}+\left (10-2 i \sqrt {7}\right ) x}{x^3 \left (4+\left (1-i \sqrt {7}\right ) x+4 x^2\right )} \, dx}{\sqrt {7}}-\frac {i \int \frac {9+5 i \sqrt {7}+\left (10+2 i \sqrt {7}\right ) x}{x^3 \left (4+\left (1+i \sqrt {7}\right ) x+4 x^2\right )} \, dx}{\sqrt {7}}\\ &=-\frac {i \int \left (\frac {9+5 i \sqrt {7}}{4 x^3}+\frac {3 \left (11-i \sqrt {7}\right )}{8 x^2}-\frac {7 i \left (-9 i+5 \sqrt {7}\right )}{16 x}+\frac {-223 i-61 \sqrt {7}+14 \left (9 i-5 \sqrt {7}\right ) x}{8 \left (4 i+\left (i-\sqrt {7}\right ) x+4 i x^2\right )}\right ) \, dx}{\sqrt {7}}+\frac {i \int \left (\frac {9-5 i \sqrt {7}}{4 x^3}+\frac {3 \left (11+i \sqrt {7}\right )}{8 x^2}+\frac {7 i \left (9 i+5 \sqrt {7}\right )}{16 x}+\frac {-223 i+61 \sqrt {7}+14 \left (9 i+5 \sqrt {7}\right ) x}{8 \left (4 i+\left (i+\sqrt {7}\right ) x+4 i x^2\right )}\right ) \, dx}{\sqrt {7}}\\ &=-\frac {35-9 i \sqrt {7}}{56 x^2}-\frac {35+9 i \sqrt {7}}{56 x^2}+\frac {3 \left (7-11 i \sqrt {7}\right )}{56 x}+\frac {3 \left (7+11 i \sqrt {7}\right )}{56 x}-\frac {1}{16} \left (35-9 i \sqrt {7}\right ) \log (x)-\frac {1}{16} \left (35+9 i \sqrt {7}\right ) \log (x)-\frac {i \int \frac {-223 i-61 \sqrt {7}+14 \left (9 i-5 \sqrt {7}\right ) x}{4 i+\left (i-\sqrt {7}\right ) x+4 i x^2} \, dx}{8 \sqrt {7}}+\frac {i \int \frac {-223 i+61 \sqrt {7}+14 \left (9 i+5 \sqrt {7}\right ) x}{4 i+\left (i+\sqrt {7}\right ) x+4 i x^2} \, dx}{8 \sqrt {7}}\\ &=-\frac {35-9 i \sqrt {7}}{56 x^2}-\frac {35+9 i \sqrt {7}}{56 x^2}+\frac {3 \left (7-11 i \sqrt {7}\right )}{56 x}+\frac {3 \left (7+11 i \sqrt {7}\right )}{56 x}-\frac {1}{16} \left (35-9 i \sqrt {7}\right ) \log (x)-\frac {1}{16} \left (35+9 i \sqrt {7}\right ) \log (x)+\frac {1}{112} \left (511 i-355 \sqrt {7}\right ) \int \frac {1}{4 i+\left (i-\sqrt {7}\right ) x+4 i x^2} \, dx-\frac {1}{32} \left (-35+9 i \sqrt {7}\right ) \int \frac {i-\sqrt {7}+8 i x}{4 i+\left (i-\sqrt {7}\right ) x+4 i x^2} \, dx+\frac {1}{32} \left (35+9 i \sqrt {7}\right ) \int \frac {i+\sqrt {7}+8 i x}{4 i+\left (i+\sqrt {7}\right ) x+4 i x^2} \, dx+\frac {1}{112} \left (511 i+355 \sqrt {7}\right ) \int \frac {1}{4 i+\left (i+\sqrt {7}\right ) x+4 i x^2} \, dx\\ &=-\frac {35-9 i \sqrt {7}}{56 x^2}-\frac {35+9 i \sqrt {7}}{56 x^2}+\frac {3 \left (7-11 i \sqrt {7}\right )}{56 x}+\frac {3 \left (7+11 i \sqrt {7}\right )}{56 x}-\frac {1}{16} \left (35-9 i \sqrt {7}\right ) \log (x)-\frac {1}{16} \left (35+9 i \sqrt {7}\right ) \log (x)+\frac {1}{32} \left (35-9 i \sqrt {7}\right ) \log \left (4 i+\left (i-\sqrt {7}\right ) x+4 i x^2\right )+\frac {1}{32} \left (35+9 i \sqrt {7}\right ) \log \left (4 i+\left (i+\sqrt {7}\right ) x+4 i x^2\right )+\frac {1}{56} \left (-511 i+355 \sqrt {7}\right ) \text {Subst}\left (\int \frac {1}{2 \left (35-i \sqrt {7}\right )-x^2} \, dx,x,i-\sqrt {7}+8 i x\right )-\frac {1}{56} \left (511 i+355 \sqrt {7}\right ) \text {Subst}\left (\int \frac {1}{2 \left (35+i \sqrt {7}\right )-x^2} \, dx,x,i+\sqrt {7}+8 i x\right )\\ &=-\frac {35-9 i \sqrt {7}}{56 x^2}-\frac {35+9 i \sqrt {7}}{56 x^2}+\frac {3 \left (7-11 i \sqrt {7}\right )}{56 x}+\frac {3 \left (7+11 i \sqrt {7}\right )}{56 x}+\frac {\left (355-73 i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {i-\sqrt {7}+8 i x}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{8 \sqrt {14 \left (35-i \sqrt {7}\right )}}-\frac {\left (355+73 i \sqrt {7}\right ) \tanh ^{-1}\left (\frac {i+\sqrt {7}+8 i x}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{8 \sqrt {14 \left (35+i \sqrt {7}\right )}}-\frac {1}{16} \left (35-9 i \sqrt {7}\right ) \log (x)-\frac {1}{16} \left (35+9 i \sqrt {7}\right ) \log (x)+\frac {1}{32} \left (35-9 i \sqrt {7}\right ) \log \left (4 i+\left (i-\sqrt {7}\right ) x+4 i x^2\right )+\frac {1}{32} \left (35+9 i \sqrt {7}\right ) \log \left (4 i+\left (i+\sqrt {7}\right ) x+4 i x^2\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.01, size = 116, normalized size = 0.37 \begin {gather*} -\frac {5}{4 x^2}+\frac {3}{4 x}-\frac {35 \log (x)}{8}+\frac {1}{8} \text {RootSum}\left [2+\text {$\#$1}+5 \text {$\#$1}^2+\text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {61 \log (x-\text {$\#$1})+141 \log (x-\text {$\#$1}) \text {$\#$1}+47 \log (x-\text {$\#$1}) \text {$\#$1}^2+70 \log (x-\text {$\#$1}) \text {$\#$1}^3}{1+10 \text {$\#$1}+3 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.03, size = 77, normalized size = 0.24
method | result | size |
risch | \(\frac {\frac {3 x}{4}-\frac {5}{4}}{x^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (686 \textit {\_Z}^{4}-12005 \textit {\_Z}^{3}+73696 \textit {\_Z}^{2}-50176 \textit {\_Z} +65536\right )}{\sum }\textit {\_R} \ln \left (-2261742 \textit {\_R}^{3}+41411909 \textit {\_R}^{2}-249593568 \textit {\_R} +154597376 x +130505728\right )\right )}{4}-\frac {35 \ln \left (x \right )}{8}\) | \(62\) |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )}{\sum }\frac {\left (70 \textit {\_R}^{3}+47 \textit {\_R}^{2}+141 \textit {\_R} +61\right ) \ln \left (x -\textit {\_R} \right )}{8 \textit {\_R}^{3}+3 \textit {\_R}^{2}+10 \textit {\_R} +1}\right )}{8}-\frac {5}{4 x^{2}}+\frac {3}{4 x}-\frac {35 \ln \left (x \right )}{8}\) | \(77\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1274 vs. \(2 (196) = 392\).
time = 1.17, size = 1274, normalized size = 4.02 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.76, size = 70, normalized size = 0.22 \begin {gather*} - \frac {35 \log {\left (x \right )}}{8} + \operatorname {RootSum} {\left (2744 t^{4} - 12005 t^{3} + 18424 t^{2} - 3136 t + 1024, \left ( t \mapsto t \log {\left (- \frac {20101387287723 t^{4}}{91907904361586} + \frac {944515214496 t^{3}}{45953952180793} + \frac {16572327093911939 t^{2}}{5882105879141504} - \frac {4564471749800865 t}{735263234892688} + x + \frac {70084064010625}{91907904361586} \right )} \right )\right )} + \frac {3 x - 5}{4 x^{2}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.25, size = 246, normalized size = 0.78 \begin {gather*} \left (\sum _{k=1}^4\ln \left (-\frac {8939\,\mathrm {root}\left (z^4-\frac {35\,z^3}{8}+\frac {47\,z^2}{7}-\frac {8\,z}{7}+\frac {128}{343},z,k\right )}{128}-\frac {69\,x}{8}+\frac {\mathrm {root}\left (z^4-\frac {35\,z^3}{8}+\frac {47\,z^2}{7}-\frac {8\,z}{7}+\frac {128}{343},z,k\right )\,x\,14945}{128}-\frac {{\mathrm {root}\left (z^4-\frac {35\,z^3}{8}+\frac {47\,z^2}{7}-\frac {8\,z}{7}+\frac {128}{343},z,k\right )}^2\,x\,269991}{1024}-\frac {{\mathrm {root}\left (z^4-\frac {35\,z^3}{8}+\frac {47\,z^2}{7}-\frac {8\,z}{7}+\frac {128}{343},z,k\right )}^3\,x\,1393}{8}+\frac {{\mathrm {root}\left (z^4-\frac {35\,z^3}{8}+\frac {47\,z^2}{7}-\frac {8\,z}{7}+\frac {128}{343},z,k\right )}^4\,x\,3675}{32}-\frac {35697\,{\mathrm {root}\left (z^4-\frac {35\,z^3}{8}+\frac {47\,z^2}{7}-\frac {8\,z}{7}+\frac {128}{343},z,k\right )}^2}{512}-\frac {18487\,{\mathrm {root}\left (z^4-\frac {35\,z^3}{8}+\frac {47\,z^2}{7}-\frac {8\,z}{7}+\frac {128}{343},z,k\right )}^3}{256}-\frac {441\,{\mathrm {root}\left (z^4-\frac {35\,z^3}{8}+\frac {47\,z^2}{7}-\frac {8\,z}{7}+\frac {128}{343},z,k\right )}^4}{32}+\frac {245}{8}\right )\,\mathrm {root}\left (z^4-\frac {35\,z^3}{8}+\frac {47\,z^2}{7}-\frac {8\,z}{7}+\frac {128}{343},z,k\right )\right )-\frac {35\,\ln \left (x\right )}{8}+\frac {\frac {3\,x}{4}-\frac {5}{4}}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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