Integrand size = 20, antiderivative size = 417 \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\frac {\sqrt {a} d^3 \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {b} \left (b c^4+a d^4\right )}-\frac {c \left (\sqrt {b} c^2-\sqrt {a} d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^4+a d^4\right )}+\frac {c \left (\sqrt {b} c^2-\sqrt {a} d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^4+a d^4\right )}+\frac {c^2 d \log (c+d x)}{b c^4+a d^4}+\frac {c \left (\sqrt {b} c^2+\sqrt {a} d^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^4+a d^4\right )}-\frac {c \left (\sqrt {b} c^2+\sqrt {a} d^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^4+a d^4\right )}-\frac {c^2 d \log \left (a+b x^4\right )}{4 \left (b c^4+a d^4\right )} \]
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Time = 0.37 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6857, 1475, 1182, 1176, 631, 210, 1179, 642, 1262, 649, 211, 266} \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\frac {\sqrt {a} d^3 \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {b} \left (a d^4+b c^4\right )}-\frac {c \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} c^2-\sqrt {a} d^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}+\frac {c \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {b} c^2-\sqrt {a} d^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}-\frac {c^2 d \log \left (a+b x^4\right )}{4 \left (a d^4+b c^4\right )}+\frac {c^2 d \log (c+d x)}{a d^4+b c^4}+\frac {c \left (\sqrt {a} d^2+\sqrt {b} c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )}-\frac {c \left (\sqrt {a} d^2+\sqrt {b} c^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^4+b c^4\right )} \]
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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 1475
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^2 d^2}{\left (b c^4+a d^4\right ) (c+d x)}+\frac {(c-d x) \left (-a d^2+b c^2 x^2\right )}{\left (b c^4+a d^4\right ) \left (a+b x^4\right )}\right ) \, dx \\ & = \frac {c^2 d \log (c+d x)}{b c^4+a d^4}+\frac {\int \frac {(c-d x) \left (-a d^2+b c^2 x^2\right )}{a+b x^4} \, dx}{b c^4+a d^4} \\ & = \frac {c^2 d \log (c+d x)}{b c^4+a d^4}+\frac {c \int \frac {-a d^2+b c^2 x^2}{a+b x^4} \, dx}{b c^4+a d^4}-\frac {d \int \frac {x \left (-a d^2+b c^2 x^2\right )}{a+b x^4} \, dx}{b c^4+a d^4} \\ & = \frac {c^2 d \log (c+d x)}{b c^4+a d^4}-\frac {d \text {Subst}\left (\int \frac {-a d^2+b c^2 x}{a+b x^2} \, dx,x,x^2\right )}{2 \left (b c^4+a d^4\right )}+\frac {\left (c \left (c^2-\frac {\sqrt {a} d^2}{\sqrt {b}}\right )\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{2 \left (b c^4+a d^4\right )}-\frac {\left (c \left (c^2+\frac {\sqrt {a} d^2}{\sqrt {b}}\right )\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{2 \left (b c^4+a d^4\right )} \\ & = \frac {c^2 d \log (c+d x)}{b c^4+a d^4}-\frac {\left (b c^2 d\right ) \text {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,x^2\right )}{2 \left (b c^4+a d^4\right )}+\frac {\left (a d^3\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{2 \left (b c^4+a d^4\right )}+\frac {\left (c \left (c^2-\frac {\sqrt {a} d^2}{\sqrt {b}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 \left (b c^4+a d^4\right )}+\frac {\left (c \left (c^2-\frac {\sqrt {a} d^2}{\sqrt {b}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 \left (b c^4+a d^4\right )}+\frac {\left (\sqrt [4]{b} c \left (c^2+\frac {\sqrt {a} d^2}{\sqrt {b}}\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{a} \left (b c^4+a d^4\right )}+\frac {\left (\sqrt [4]{b} c \left (c^2+\frac {\sqrt {a} d^2}{\sqrt {b}}\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{a} \left (b c^4+a d^4\right )} \\ & = \frac {\sqrt {a} d^3 \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {b} \left (b c^4+a d^4\right )}+\frac {c^2 d \log (c+d x)}{b c^4+a d^4}+\frac {\sqrt [4]{b} c \left (c^2+\frac {\sqrt {a} d^2}{\sqrt {b}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (b c^4+a d^4\right )}-\frac {\sqrt [4]{b} c \left (c^2+\frac {\sqrt {a} d^2}{\sqrt {b}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (b c^4+a d^4\right )}-\frac {c^2 d \log \left (a+b x^4\right )}{4 \left (b c^4+a d^4\right )}+\frac {\left (\sqrt [4]{b} c \left (c^2-\frac {\sqrt {a} d^2}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (b c^4+a d^4\right )}-\frac {\left (\sqrt [4]{b} c \left (c^2-\frac {\sqrt {a} d^2}{\sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (b c^4+a d^4\right )} \\ & = \frac {\sqrt {a} d^3 \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {b} \left (b c^4+a d^4\right )}-\frac {\sqrt [4]{b} c \left (c^2-\frac {\sqrt {a} d^2}{\sqrt {b}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (b c^4+a d^4\right )}+\frac {\sqrt [4]{b} c \left (c^2-\frac {\sqrt {a} d^2}{\sqrt {b}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (b c^4+a d^4\right )}+\frac {c^2 d \log (c+d x)}{b c^4+a d^4}+\frac {\sqrt [4]{b} c \left (c^2+\frac {\sqrt {a} d^2}{\sqrt {b}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (b c^4+a d^4\right )}-\frac {\sqrt [4]{b} c \left (c^2+\frac {\sqrt {a} d^2}{\sqrt {b}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (b c^4+a d^4\right )}-\frac {c^2 d \log \left (a+b x^4\right )}{4 \left (b c^4+a d^4\right )} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\frac {-2 \left (\sqrt {2} b^{3/4} c^3-\sqrt {2} \sqrt {a} \sqrt [4]{b} c d^2+2 a^{3/4} d^3\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt {2} b^{3/4} c^3-\sqrt {2} \sqrt {a} \sqrt [4]{b} c d^2-2 a^{3/4} d^3\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt [4]{b} c \left (8 \sqrt [4]{a} \sqrt [4]{b} c d \log (c+d x)+\sqrt {2} \left (\sqrt {b} c^2+\sqrt {a} d^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )-\sqrt {2} \sqrt {b} c^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )-\sqrt {2} \sqrt {a} d^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )-2 \sqrt [4]{a} \sqrt [4]{b} c d \log \left (a+b x^4\right )\right )}{8 \sqrt [4]{a} \sqrt {b} \left (b c^4+a d^4\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.84 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.54
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (a^{2} b^{2} d^{4}+a \,b^{3} c^{4}\right ) \textit {\_Z}^{4}+4 a \,b^{2} c^{2} d \,\textit {\_Z}^{3}+2 a \,\textit {\_Z}^{2} d^{2} b \right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 a^{2} b^{2} d^{6}-3 a \,b^{3} c^{4} d^{2}\right ) \textit {\_R}^{4}+10 \textit {\_R}^{3} a \,b^{2} c^{2} d^{3}+\left (9 a b \,d^{4}+b^{2} c^{4}\right ) \textit {\_R}^{2}-5 \textit {\_R} b \,c^{2} d +4 d^{2}\right ) x +\left (6 a^{2} b^{2} c \,d^{5}-2 a \,b^{3} c^{5} d \right ) \textit {\_R}^{4}+6 a \,b^{2} c^{3} d^{2} \textit {\_R}^{3}+8 a b c \,d^{3} \textit {\_R}^{2}-b \,c^{3} \textit {\_R} +4 c d \right )\right )}{4}+\frac {c^{2} d \ln \left (d x +c \right )}{a \,d^{4}+b \,c^{4}}\) | \(226\) |
default | \(\frac {-\frac {d^{2} c \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {a \,d^{3} \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}+\frac {c^{3} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {c^{2} d \ln \left (b \,x^{4}+a \right )}{4}}{a \,d^{4}+b \,c^{4}}+\frac {c^{2} d \ln \left (d x +c \right )}{a \,d^{4}+b \,c^{4}}\) | \(281\) |
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Result contains complex when optimal does not.
Time = 23.56 (sec) , antiderivative size = 259898, normalized size of antiderivative = 623.26 \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\frac {c^{2} d \log \left (d x + c\right )}{b c^{4} + a d^{4}} - \frac {\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} c^{2} d + \sqrt {a} b^{\frac {3}{2}} c^{3} + a b c d^{2}\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} c^{2} d - \sqrt {a} b^{\frac {3}{2}} c^{3} - a b c d^{2}\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} c^{3} - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} c d^{2} - 2 \, a^{\frac {3}{2}} b d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} c^{3} - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} c d^{2} + 2 \, a^{\frac {3}{2}} b d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}}}{8 \, {\left (b c^{4} + a d^{4}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 406, normalized size of antiderivative = 0.97 \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\frac {c^{2} d^{2} \log \left ({\left | d x + c \right |}\right )}{b c^{4} d + a d^{5}} - \frac {c^{2} d \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, {\left (b c^{4} + a d^{4}\right )}} - \frac {{\left (\sqrt {2} a b^{2} d - \left (a b^{3}\right )^{\frac {3}{4}} c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a b^{3} c^{2} + \sqrt {2} \sqrt {a b} a b^{2} d^{2} - 2 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c d\right )}} + \frac {{\left (\sqrt {2} a b^{2} d + \left (a b^{3}\right )^{\frac {3}{4}} c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a b^{3} c^{2} + \sqrt {2} \sqrt {a b} a b^{2} d^{2} + 2 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c d\right )}} - \frac {{\left (\left (a b^{3}\right )^{\frac {1}{4}} a b c d^{2} + \left (a b^{3}\right )^{\frac {3}{4}} c^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{4 \, {\left (\sqrt {2} a b^{3} c^{4} + \sqrt {2} a^{2} b^{2} d^{4}\right )}} + \frac {{\left (\left (a b^{3}\right )^{\frac {1}{4}} a b c d^{2} + \left (a b^{3}\right )^{\frac {3}{4}} c^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{4 \, {\left (\sqrt {2} a b^{3} c^{4} + \sqrt {2} a^{2} b^{2} d^{4}\right )}} \]
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Time = 10.07 (sec) , antiderivative size = 823, normalized size of antiderivative = 1.97 \[ \int \frac {x^2}{(c+d x) \left (a+b x^4\right )} \, dx=\left (\sum _{k=1}^4\ln \left (a\,b^2\,d\,\left (c\,d+d^2\,x-\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )\,b\,c^3+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^2\,b^2\,c^4\,x\,4+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^2\,a\,b\,d^4\,x\,36-{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^4\,a\,b^3\,c^5\,d\,128-\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )\,b\,c^2\,d\,x\,5+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^3\,a\,b^2\,c^3\,d^2\,96+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^4\,a^2\,b^2\,c\,d^5\,384+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^4\,a^2\,b^2\,d^6\,x\,320+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^2\,a\,b\,c\,d^3\,32+{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^3\,a\,b^2\,c^2\,d^3\,x\,160-{\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )}^4\,a\,b^3\,c^4\,d^2\,x\,192\right )\right )\,\mathrm {root}\left (256\,a^2\,b^2\,d^4\,z^4+256\,a\,b^3\,c^4\,z^4+256\,a\,b^2\,c^2\,d\,z^3+32\,a\,b\,d^2\,z^2+1,z,k\right )\right )+\frac {c^2\,d\,\ln \left (c+d\,x\right )}{b\,c^4+a\,d^4} \]
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