Integrand size = 17, antiderivative size = 416 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=-\frac {\sqrt {c} d^2 e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}+\frac {e^3 \log (d+e x)}{c d^4+a e^4}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}-\frac {e^3 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )} \]
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Time = 0.29 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {6857, 1890, 1182, 1176, 631, 210, 1179, 642, 1262, 649, 211, 266} \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=-\frac {\sqrt [4]{c} d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )}+\frac {\sqrt [4]{c} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )}-\frac {\sqrt {c} d^2 e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (a e^4+c d^4\right )}-\frac {e^3 \log \left (a+c x^4\right )}{4 \left (a e^4+c d^4\right )}+\frac {e^3 \log (d+e x)}{a e^4+c d^4} \]
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Rule 210
Rule 211
Rule 266
Rule 631
Rule 642
Rule 649
Rule 1176
Rule 1179
Rule 1182
Rule 1262
Rule 1890
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^4}{\left (c d^4+a e^4\right ) (d+e x)}+\frac {c \left (d^3-d^2 e x+d e^2 x^2-e^3 x^3\right )}{\left (c d^4+a e^4\right ) \left (a+c x^4\right )}\right ) \, dx \\ & = \frac {e^3 \log (d+e x)}{c d^4+a e^4}+\frac {c \int \frac {d^3-d^2 e x+d e^2 x^2-e^3 x^3}{a+c x^4} \, dx}{c d^4+a e^4} \\ & = \frac {e^3 \log (d+e x)}{c d^4+a e^4}+\frac {c \int \left (\frac {d^3+d e^2 x^2}{a+c x^4}+\frac {x \left (-d^2 e-e^3 x^2\right )}{a+c x^4}\right ) \, dx}{c d^4+a e^4} \\ & = \frac {e^3 \log (d+e x)}{c d^4+a e^4}+\frac {c \int \frac {d^3+d e^2 x^2}{a+c x^4} \, dx}{c d^4+a e^4}+\frac {c \int \frac {x \left (-d^2 e-e^3 x^2\right )}{a+c x^4} \, dx}{c d^4+a e^4} \\ & = \frac {e^3 \log (d+e x)}{c d^4+a e^4}+\frac {c \text {Subst}\left (\int \frac {-d^2 e-e^3 x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )}+\frac {\left (d \left (\frac {\sqrt {c} d^2}{\sqrt {a}}-e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )}+\frac {\left (d \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )} \\ & = \frac {e^3 \log (d+e x)}{c d^4+a e^4}-\frac {\left (c d^2 e\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )}-\frac {\left (c e^3\right ) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )}+\frac {\left (d \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )}+\frac {\left (d \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )}-\frac {\left (\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}-\frac {\left (\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )} \\ & = -\frac {\sqrt {c} d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )}+\frac {e^3 \log (d+e x)}{c d^4+a e^4}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}-\frac {e^3 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )}+\frac {\left (\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}-\frac {\left (\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )} \\ & = -\frac {\sqrt {c} d^2 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}+\frac {e^3 \log (d+e x)}{c d^4+a e^4}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}-\frac {e^3 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\frac {-2 \sqrt [4]{c} d \left (\sqrt {2} \sqrt {c} d^2-2 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt [4]{c} d \left (\sqrt {2} \sqrt {c} d^2+2 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+8 a^{3/4} e^3 \log (d+e x)-\sqrt {2} c^{3/4} d^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} \sqrt {a} \sqrt [4]{c} d e^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} c^{3/4} d^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )-\sqrt {2} \sqrt {a} \sqrt [4]{c} d e^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )-2 a^{3/4} e^3 \log \left (a+c x^4\right )}{8 a^{3/4} \left (c d^4+a e^4\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.89 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.51
method | result | size |
risch | \(\frac {e^{3} \ln \left (e x +d \right )}{e^{4} a +d^{4} c}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (a^{4} e^{4}+a^{3} c \,d^{4}\right ) \textit {\_Z}^{4}+4 a^{3} e^{3} \textit {\_Z}^{3}+6 \textit {\_Z}^{2} a^{2} e^{2}+4 \textit {\_Z} a e \right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 e^{6} a^{3}-3 a^{2} d^{4} e^{2} c \right ) \textit {\_R}^{3}+\left (15 a^{2} e^{5}-3 a c \,d^{4} e \right ) \textit {\_R}^{2}+\left (15 e^{4} a -d^{4} c \right ) \textit {\_R} +5 e^{3}\right ) x +\left (6 a^{3} d \,e^{5}-2 a^{2} c \,d^{5} e \right ) \textit {\_R}^{3}+\left (13 a^{2} e^{4} d -a c \,d^{5}\right ) \textit {\_R}^{2}+8 a d \,e^{3} \textit {\_R} +d \,e^{2}\right )\right )}{4}\) | \(211\) |
default | \(\frac {c \left (\frac {d^{3} \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}-\frac {d^{2} e \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{2 \sqrt {a c}}+\frac {d \,e^{2} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}-\frac {e^{3} \ln \left (c \,x^{4}+a \right )}{4 c}\right )}{e^{4} a +d^{4} c}+\frac {e^{3} \ln \left (e x +d \right )}{e^{4} a +d^{4} c}\) | \(289\) |
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Result contains complex when optimal does not.
Time = 97.11 (sec) , antiderivative size = 352864, normalized size of antiderivative = 848.23 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\frac {e^{3} \log \left (e x + d\right )}{c d^{4} + a e^{4}} - \frac {c {\left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{3} - c d^{3} + \sqrt {a} \sqrt {c} d e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{3} + c d^{3} - \sqrt {a} \sqrt {c} d e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {5}{4}} d^{3} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {3}{4}} d e^{2} + 2 \, \sqrt {a} c d^{2} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {5}{4}} d^{3} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {3}{4}} d e^{2} - 2 \, \sqrt {a} c d^{2} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}}\right )}}{8 \, {\left (c d^{4} + a e^{4}\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\frac {e^{4} \log \left ({\left | e x + d \right |}\right )}{c d^{4} e + a e^{5}} - \frac {e^{3} \log \left ({\left | c x^{4} + a \right |}\right )}{4 \, {\left (c d^{4} + a e^{4}\right )}} + \frac {\left (a c^{3}\right )^{\frac {1}{4}} c d \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a c^{2} d^{2} + \sqrt {2} \sqrt {a c} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {1}{4}} a c d e\right )}} + \frac {\left (a c^{3}\right )^{\frac {1}{4}} c d \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a c^{2} d^{2} + \sqrt {2} \sqrt {a c} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {1}{4}} a c d e\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{4 \, {\left (\sqrt {2} a c^{3} d^{4} + \sqrt {2} a^{2} c^{2} e^{4}\right )}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{4 \, {\left (\sqrt {2} a c^{3} d^{4} + \sqrt {2} a^{2} c^{2} e^{4}\right )}} \]
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Time = 9.35 (sec) , antiderivative size = 874, normalized size of antiderivative = 2.10 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\left (\sum _{k=1}^4\ln \left (\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )\,c^4\,e\,\left (d\,e^2+5\,e^3\,x+{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^2\,a^2\,e^5\,x\,240+{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^3\,a^3\,e^6\,x\,320+\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )\,a\,d\,e^3\,32+\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )\,a\,e^4\,x\,60-\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )\,c\,d^4\,x\,4-{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^2\,a\,c\,d^5\,16+{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^2\,a^2\,d\,e^4\,208+{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^3\,a^3\,d\,e^5\,384-{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^3\,a^2\,c\,d^5\,e\,128-{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^3\,a^2\,c\,d^4\,e^2\,x\,192-{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^2\,a\,c\,d^4\,e\,x\,48\right )\right )\,\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )\right )+\frac {e^3\,\ln \left (d+e\,x\right )}{c\,d^4+a\,e^4} \]
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