Integrand size = 15, antiderivative size = 329 \[ \int \frac {x^{5/2}}{\left (a+c x^4\right )^3} \, dx=\frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}-\frac {9 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}-\frac {9 \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}+\frac {9 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}}-\frac {9 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}} \]
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Time = 0.22 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {296, 335, 307, 303, 1176, 631, 210, 1179, 642, 304, 211, 214} \[ \int \frac {x^{5/2}}{\left (a+c x^4\right )^3} \, dx=\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}-\frac {9 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}-\frac {9 \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}+\frac {9 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}}-\frac {9 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {x^{7/2}}{8 a \left (a+c x^4\right )^2} \]
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Rule 210
Rule 211
Rule 214
Rule 296
Rule 303
Rule 304
Rule 307
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 \int \frac {x^{5/2}}{\left (a+c x^4\right )^2} \, dx}{16 a} \\ & = \frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}+\frac {9 \int \frac {x^{5/2}}{a+c x^4} \, dx}{128 a^2} \\ & = \frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}+\frac {9 \text {Subst}\left (\int \frac {x^6}{a+c x^8} \, dx,x,\sqrt {x}\right )}{64 a^2} \\ & = \frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}-\frac {9 \text {Subst}\left (\int \frac {x^2}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 a^2 \sqrt {c}}+\frac {9 \text {Subst}\left (\int \frac {x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 a^2 \sqrt {c}} \\ & = \frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}-\frac {9 \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 a^2 c^{3/4}}+\frac {9 \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 a^2 c^{3/4}}-\frac {9 \text {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{256 a^2 c^{3/4}}+\frac {9 \text {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{256 a^2 c^{3/4}} \\ & = \frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}+\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}+\frac {9 \text {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{512 a^2 c}+\frac {9 \text {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{512 a^2 c}+\frac {9 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}+2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}-2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}} \\ & = \frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}+\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}+\frac {9 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}}-\frac {9 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}}-\frac {9 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}} \\ & = \frac {x^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {9 x^{7/2}}{64 a^2 \left (a+c x^4\right )}-\frac {9 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{17/8} c^{7/8}}+\frac {9 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{17/8} c^{7/8}}+\frac {9 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}}-\frac {9 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{17/8} c^{7/8}} \\ \end{align*}
Time = 0.96 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.87 \[ \int \frac {x^{5/2}}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {8 \sqrt [8]{a} x^{7/2} \left (17 a+9 c x^4\right )}{\left (a+c x^4\right )^2}-\frac {9 \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{c^{7/8}}-\frac {9 \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{c^{7/8}}-\frac {9 \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{c^{7/8}}-\frac {9 \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{c^{7/8}}}{512 a^{17/8}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.41 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.19
method | result | size |
derivativedivides | \(\frac {\frac {17 x^{\frac {7}{2}}}{64 a}+\frac {9 c \,x^{\frac {15}{2}}}{64 a^{2}}}{\left (x^{4} c +a \right )^{2}}+\frac {9 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}\right )}{512 a^{2} c}\) | \(62\) |
default | \(\frac {\frac {17 x^{\frac {7}{2}}}{64 a}+\frac {9 c \,x^{\frac {15}{2}}}{64 a^{2}}}{\left (x^{4} c +a \right )^{2}}+\frac {9 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}\right )}{512 a^{2} c}\) | \(62\) |
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.67 \[ \int \frac {x^{5/2}}{\left (a+c x^4\right )^3} \, dx=-\frac {9 \, \sqrt {2} {\left (\left (i - 1\right ) \, a^{2} c^{2} x^{8} + \left (2 i - 2\right ) \, a^{3} c x^{4} + \left (i - 1\right ) \, a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{15} c^{6} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 9 \, \sqrt {2} {\left (-\left (i + 1\right ) \, a^{2} c^{2} x^{8} - \left (2 i + 2\right ) \, a^{3} c x^{4} - \left (i + 1\right ) \, a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{15} c^{6} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 9 \, \sqrt {2} {\left (\left (i + 1\right ) \, a^{2} c^{2} x^{8} + \left (2 i + 2\right ) \, a^{3} c x^{4} + \left (i + 1\right ) \, a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{15} c^{6} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 9 \, \sqrt {2} {\left (-\left (i - 1\right ) \, a^{2} c^{2} x^{8} - \left (2 i - 2\right ) \, a^{3} c x^{4} - \left (i - 1\right ) \, a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{15} c^{6} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) - 18 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \log \left (a^{15} c^{6} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 18 \, {\left (i \, a^{2} c^{2} x^{8} + 2 i \, a^{3} c x^{4} + i \, a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \log \left (i \, a^{15} c^{6} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 18 \, {\left (-i \, a^{2} c^{2} x^{8} - 2 i \, a^{3} c x^{4} - i \, a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \log \left (-i \, a^{15} c^{6} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + 18 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {1}{8}} \log \left (-a^{15} c^{6} \left (-\frac {1}{a^{17} c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) - 16 \, {\left (9 \, c x^{7} + 17 \, a x^{3}\right )} \sqrt {x}}{1024 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} \]
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Timed out. \[ \int \frac {x^{5/2}}{\left (a+c x^4\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {x^{5/2}}{\left (a+c x^4\right )^3} \, dx=\int { \frac {x^{\frac {5}{2}}}{{\left (c x^{4} + a\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (224) = 448\).
Time = 0.46 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.43 \[ \int \frac {x^{5/2}}{\left (a+c x^4\right )^3} \, dx=\frac {9 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{256 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {9 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{256 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {9 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{256 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {9 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{256 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} - \frac {9 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{512 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {9 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{512 \, a^{3} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {9 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{512 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {9 \, \left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{512 \, a^{3} \sqrt {2 \, \sqrt {2} + 4}} + \frac {9 \, c x^{\frac {15}{2}} + 17 \, a x^{\frac {7}{2}}}{64 \, {\left (c x^{4} + a\right )}^{2} a^{2}} \]
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Time = 5.58 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.48 \[ \int \frac {x^{5/2}}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {17\,x^{7/2}}{64\,a}+\frac {9\,c\,x^{15/2}}{64\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\frac {9\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{256\,{\left (-a\right )}^{17/8}\,c^{7/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,9{}\mathrm {i}}{256\,{\left (-a\right )}^{17/8}\,c^{7/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {9}{512}-\frac {9}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{17/8}\,c^{7/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {9}{512}+\frac {9}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{17/8}\,c^{7/8}} \]
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