Integrand size = 30, antiderivative size = 394 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac {4 a f-x \left (7 b c+a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {3 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{5/4}} \]
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Time = 0.30 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {1872, 1868, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} \sqrt {b} e+3 a g+21 b c\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt {a} \sqrt {b} e+3 a g+21 b c\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {3 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-5 \sqrt {a} \sqrt {b} e+3 a g+21 b c\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-5 \sqrt {a} \sqrt {b} e+3 a g+21 b c\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}-\frac {4 a f-x \left (a g+7 b c+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {x \left (-a g+b c+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2} \]
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Rule 210
Rule 211
Rule 281
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 1868
Rule 1872
Rule 1890
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac {\int \frac {-7 b c-a g-6 b d x-5 b e x^2-4 b f x^3}{\left (a+b x^4\right )^2} \, dx}{8 a b} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac {4 a f-x \left (7 b c+a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {\int \frac {-3 (-7 b c-a g)+12 b d x+5 b e x^2}{a+b x^4} \, dx}{32 a^2 b} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac {4 a f-x \left (7 b c+a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {\int \left (\frac {12 b d x}{a+b x^4}+\frac {-3 (-7 b c-a g)+5 b e x^2}{a+b x^4}\right ) \, dx}{32 a^2 b} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac {4 a f-x \left (7 b c+a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {\int \frac {-3 (-7 b c-a g)+5 b e x^2}{a+b x^4} \, dx}{32 a^2 b}+\frac {(3 d) \int \frac {x}{a+b x^4} \, dx}{8 a^2} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac {4 a f-x \left (7 b c+a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {(3 d) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{16 a^2}+\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{64 a^{5/2} b^{3/2}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{64 a^{5/2} b^{3/2}} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac {4 a f-x \left (7 b c+a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\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}{128 \sqrt {2} a^{11/4} b^{5/4}}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\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}{128 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^{5/2} b^{3/2}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^{5/2} b^{3/2}} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac {4 a f-x \left (7 b c+a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}-\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{5/4}} \\ & = \frac {x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac {4 a f-x \left (7 b c+a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e+3 a g\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{5/4}}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}+\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{5/4}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\frac {\frac {8 a^{3/4} \sqrt [4]{b} x (7 b c+a g+b x (6 d+5 e x))}{a+b x^4}-\frac {32 a^{7/4} \sqrt [4]{b} (a (f+g x)-b x (c+x (d+e x)))}{\left (a+b x^4\right )^2}-2 \left (21 \sqrt {2} b c+24 \sqrt [4]{a} b^{3/4} d+5 \sqrt {2} \sqrt {a} \sqrt {b} e+3 \sqrt {2} a g\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (21 \sqrt {2} b c-24 \sqrt [4]{a} b^{3/4} d+5 \sqrt {2} \sqrt {a} \sqrt {b} e+3 \sqrt {2} a g\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt {2} \left (-21 b c+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+\sqrt {2} \left (21 b c-5 \sqrt {a} \sqrt {b} e+3 a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{256 a^{11/4} b^{5/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.53 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.37
method | result | size |
risch | \(\frac {\frac {5 b e \,x^{7}}{32 a^{2}}+\frac {3 b d \,x^{6}}{16 a^{2}}+\frac {\left (a g +7 b c \right ) x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}-\frac {\left (3 a g -11 b c \right ) x}{32 a b}-\frac {f}{8 b}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (5 \textit {\_R}^{2} e +12 \textit {\_R} d +\frac {3 a g +21 b c}{b}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} b}\) | \(144\) |
default | \(\frac {\frac {5 b e \,x^{7}}{32 a^{2}}+\frac {3 b d \,x^{6}}{16 a^{2}}+\frac {\left (a g +7 b c \right ) x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}-\frac {\left (3 a g -11 b c \right ) x}{32 a b}-\frac {f}{8 b}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\frac {\left (3 a g +21 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {6 b d \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{\sqrt {a b}}+\frac {5 e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{32 a^{2} b}\) | \(330\) |
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Result contains complex when optimal does not.
Time = 58.34 (sec) , antiderivative size = 358509, normalized size of antiderivative = 909.92 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.05 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} + 9 \, a b e x^{3} + {\left (7 \, b^{2} c + a b g\right )} x^{5} + 10 \, a b d x^{2} - 4 \, a^{2} f + {\left (11 \, a b c - 3 \, a^{2} g\right )} x}{32 \, {\left (a^{2} b^{3} x^{8} + 2 \, a^{3} b^{2} x^{4} + a^{4} b\right )}} + \frac {\frac {\sqrt {2} {\left (21 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e + 3 \, a \sqrt {b} g\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (21 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e + 3 \, a \sqrt {b} g\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + 3 \, \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g - 24 \, \sqrt {a} b^{\frac {3}{2}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + 3 \, \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g + 24 \, \sqrt {a} b^{\frac {3}{2}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{256 \, a^{2} b} \]
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Time = 0.28 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} + \frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} + a b g x^{5} + 9 \, a b e x^{3} + 10 \, a b d x^{2} + 11 \, a b c x - 3 \, a^{2} g x - 4 \, a^{2} f}{32 \, {\left (b x^{4} + a\right )}^{2} a^{2} b} \]
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Time = 9.75 (sec) , antiderivative size = 1001, normalized size of antiderivative = 2.54 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^3} \, dx=\frac {\frac {5\,d\,x^2}{16\,a}-\frac {f}{8\,b}+\frac {9\,e\,x^3}{32\,a}+\frac {x^5\,\left (7\,b\,c+a\,g\right )}{32\,a^2}+\frac {x\,\left (11\,b\,c-3\,a\,g\right )}{32\,a\,b}+\frac {3\,b\,d\,x^6}{16\,a^2}+\frac {5\,b\,e\,x^7}{32\,a^2}}{a^2+2\,a\,b\,x^4+b^2\,x^8}+\left (\sum _{k=1}^4\ln \left (-\mathrm {root}\left (268435456\,a^{11}\,b^5\,z^4+983040\,a^7\,b^3\,e\,g\,z^2+6881280\,a^6\,b^4\,c\,e\,z^2+4718592\,a^6\,b^4\,d^2\,z^2-774144\,a^4\,b^3\,c\,d\,g\,z-55296\,a^5\,b^2\,d\,g^2\,z+153600\,a^4\,b^3\,d\,e^2\,z-2709504\,a^3\,b^4\,c^2\,d\,z-8640\,a^2\,b^2\,d^2\,e\,g+6300\,a^2\,b^2\,c\,e^2\,g-60480\,a\,b^3\,c\,d^2\,e+111132\,a\,b^3\,c^3\,g+2268\,a^3\,b\,c\,g^3+23814\,a^2\,b^2\,c^2\,g^2+450\,a^3\,b\,e^2\,g^2+22050\,a\,b^3\,c^2\,e^2+625\,a^2\,b^2\,e^4+20736\,a\,b^3\,d^4+81\,a^4\,g^4+194481\,b^4\,c^4,z,k\right )\,\left (\mathrm {root}\left (268435456\,a^{11}\,b^5\,z^4+983040\,a^7\,b^3\,e\,g\,z^2+6881280\,a^6\,b^4\,c\,e\,z^2+4718592\,a^6\,b^4\,d^2\,z^2-774144\,a^4\,b^3\,c\,d\,g\,z-55296\,a^5\,b^2\,d\,g^2\,z+153600\,a^4\,b^3\,d\,e^2\,z-2709504\,a^3\,b^4\,c^2\,d\,z-8640\,a^2\,b^2\,d^2\,e\,g+6300\,a^2\,b^2\,c\,e^2\,g-60480\,a\,b^3\,c\,d^2\,e+111132\,a\,b^3\,c^3\,g+2268\,a^3\,b\,c\,g^3+23814\,a^2\,b^2\,c^2\,g^2+450\,a^3\,b\,e^2\,g^2+22050\,a\,b^3\,c^2\,e^2+625\,a^2\,b^2\,e^4+20736\,a\,b^3\,d^4+81\,a^4\,g^4+194481\,b^4\,c^4,z,k\right )\,\left (\frac {49152\,g\,a^6\,b^2+344064\,c\,a^5\,b^3}{32768\,a^6}-\frac {6\,b^3\,d\,x}{a}\right )+\frac {x\,\left (144\,a^4\,b\,g^2+2016\,a^3\,b^2\,c\,g-400\,a^3\,b^2\,e^2+7056\,a^2\,b^3\,c^2\right )}{4096\,a^6}+\frac {15\,b^2\,d\,e}{32\,a^3}\right )-\frac {45\,a^2\,e\,g^2+630\,a\,b\,c\,e\,g-432\,a\,b\,d^2\,g+125\,a\,b\,e^3+2205\,b^2\,c^2\,e-3024\,b^2\,c\,d^2}{32768\,a^6}-\frac {x\,\left (-216\,b^2\,d^3+315\,c\,e\,b^2\,d+45\,a\,e\,g\,b\,d\right )}{4096\,a^6}\right )\,\mathrm {root}\left (268435456\,a^{11}\,b^5\,z^4+983040\,a^7\,b^3\,e\,g\,z^2+6881280\,a^6\,b^4\,c\,e\,z^2+4718592\,a^6\,b^4\,d^2\,z^2-774144\,a^4\,b^3\,c\,d\,g\,z-55296\,a^5\,b^2\,d\,g^2\,z+153600\,a^4\,b^3\,d\,e^2\,z-2709504\,a^3\,b^4\,c^2\,d\,z-8640\,a^2\,b^2\,d^2\,e\,g+6300\,a^2\,b^2\,c\,e^2\,g-60480\,a\,b^3\,c\,d^2\,e+111132\,a\,b^3\,c^3\,g+2268\,a^3\,b\,c\,g^3+23814\,a^2\,b^2\,c^2\,g^2+450\,a^3\,b\,e^2\,g^2+22050\,a\,b^3\,c^2\,e^2+625\,a^2\,b^2\,e^4+20736\,a\,b^3\,d^4+81\,a^4\,g^4+194481\,b^4\,c^4,z,k\right )\right ) \]
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