Integrand size = 20, antiderivative size = 341 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^3} \, dx=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {3 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}-\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{3/4}} \]
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Time = 0.21 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1869, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^3} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} e+21 \sqrt {b} c\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt {a} e+21 \sqrt {b} c\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {3 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{3/4}}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2} \]
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Rule 210
Rule 211
Rule 281
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 1869
Rule 1890
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}-\frac {\int \frac {-7 c-6 d x-5 e x^2}{\left (a+b x^4\right )^2} \, dx}{8 a} \\ & = \frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {\int \frac {21 c+12 d x+5 e x^2}{a+b x^4} \, dx}{32 a^2} \\ & = \frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {\int \left (\frac {12 d x}{a+b x^4}+\frac {21 c+5 e x^2}{a+b x^4}\right ) \, dx}{32 a^2} \\ & = \frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {\int \frac {21 c+5 e x^2}{a+b x^4} \, dx}{32 a^2}+\frac {(3 d) \int \frac {x}{a+b x^4} \, dx}{8 a^2} \\ & = \frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \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 (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{64 a^2 b}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}+5 e\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{64 a^2 b} \\ & = \frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \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 (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\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^{9/4} b^{3/4}}-\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\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^{9/4} b^{3/4}}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}+5 e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^2 b}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}+5 e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^2 b} \\ & = \frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \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 (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{3/4}}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\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^{3/4}}-\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\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^{3/4}} \\ & = \frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \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 \sqrt {b} c+5 \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}-\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{3/4}}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{3/4}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^3} \, dx=\frac {\frac {32 a^2 x (c+x (d+e x))}{\left (a+b x^4\right )^2}+\frac {8 a x (7 c+x (6 d+5 e x))}{a+b x^4}-\frac {2 \sqrt [4]{a} \left (21 \sqrt {2} \sqrt {b} c+24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt {2} \sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {2 \sqrt [4]{a} \left (21 \sqrt {2} \sqrt {b} c-24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt {2} \sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {\sqrt {2} \left (-21 \sqrt [4]{a} \sqrt {b} c+5 a^{3/4} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}+\frac {\sqrt {2} \left (21 \sqrt [4]{a} \sqrt {b} c-5 a^{3/4} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}}{256 a^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.49 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.33
method | result | size |
risch | \(\frac {\frac {5 b e \,x^{7}}{32 a^{2}}+\frac {3 b d \,x^{6}}{16 a^{2}}+\frac {7 b c \,x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}+\frac {11 c x}{32 a}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (5 \textit {\_R}^{2} e +12 \textit {\_R} d +21 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} b}\) | \(111\) |
default | \(c \left (\frac {x}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {\frac {7 x}{32 a \left (b \,x^{4}+a \right )}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a^{2}}}{a}\right )+d \left (\frac {x^{2}}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {\frac {3 x^{2}}{16 a \left (b \,x^{4}+a \right )}+\frac {3 \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{16 a \sqrt {a b}}}{a}\right )+e \left (\frac {x^{3}}{8 a \left (b \,x^{4}+a \right )^{2}}+\frac {\frac {5 x^{3}}{32 a \left (b \,x^{4}+a \right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a}\right )\) | \(354\) |
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Result contains complex when optimal does not.
Time = 7.02 (sec) , antiderivative size = 124787, normalized size of antiderivative = 365.94 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^3} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^3} \, dx=\frac {5 \, b e x^{7} + 6 \, b d x^{6} + 7 \, b c x^{5} + 9 \, a e x^{3} + 10 \, a d x^{2} + 11 \, a c x}{32 \, {\left (a^{2} b^{2} x^{8} + 2 \, a^{3} b x^{4} + a^{4}\right )}} + \frac {\frac {\sqrt {2} {\left (21 \, \sqrt {b} c - 5 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (21 \, \sqrt {b} c - 5 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e - 24 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e + 24 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{256 \, a^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^3} \, dx=\frac {5 \, b e x^{7} + 6 \, b d x^{6} + 7 \, b c x^{5} + 9 \, a e x^{3} + 10 \, a d x^{2} + 11 \, a c x}{32 \, {\left (b x^{4} + a\right )}^{2} a^{2}} + \frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} \]
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Time = 9.63 (sec) , antiderivative size = 826, normalized size of antiderivative = 2.42 \[ \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^3} \, dx=\frac {\frac {5\,d\,x^2}{16\,a}+\frac {9\,e\,x^3}{32\,a}+\frac {11\,c\,x}{32\,a}+\frac {7\,b\,c\,x^5}{32\,a^2}+\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 (-\frac {b\,\left (125\,a\,e^3-3024\,b\,c\,d^2+2205\,b\,c^2\,e-1728\,b\,d^3\,x+{\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )}^2\,a^5\,b^2\,c\,344064-\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )\,a^3\,b\,e^2\,x\,3200+2520\,b\,c\,d\,e\,x+\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )\,a^2\,b^2\,c^2\,x\,56448-{\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )}^2\,a^5\,b^2\,d\,x\,196608+\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )\,a^3\,b\,d\,e\,15360\right )}{a^6\,32768}\right )\,\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )\right ) \]
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