Integrand size = 17, antiderivative size = 273 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^3} \, dx=-\frac {(b c-a d) x}{8 c d \left (c+d x^4\right )^2}+\frac {(b c+7 a d) x}{32 c^2 d \left (c+d x^4\right )}-\frac {3 (b c+7 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{5/4}}+\frac {3 (b c+7 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{5/4}}-\frac {3 (b c+7 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{5/4}}+\frac {3 (b c+7 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{5/4}} \]
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Time = 0.14 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {393, 205, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^3} \, dx=-\frac {3 (7 a d+b c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{5/4}}+\frac {3 (7 a d+b c) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{64 \sqrt {2} c^{11/4} d^{5/4}}-\frac {3 (7 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{5/4}}+\frac {3 (7 a d+b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{5/4}}+\frac {x (7 a d+b c)}{32 c^2 d \left (c+d x^4\right )}-\frac {x (b c-a d)}{8 c d \left (c+d x^4\right )^2} \]
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Rule 205
Rule 210
Rule 217
Rule 393
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {(b c-a d) x}{8 c d \left (c+d x^4\right )^2}+\frac {(b c+7 a d) \int \frac {1}{\left (c+d x^4\right )^2} \, dx}{8 c d} \\ & = -\frac {(b c-a d) x}{8 c d \left (c+d x^4\right )^2}+\frac {(b c+7 a d) x}{32 c^2 d \left (c+d x^4\right )}+\frac {(3 (b c+7 a d)) \int \frac {1}{c+d x^4} \, dx}{32 c^2 d} \\ & = -\frac {(b c-a d) x}{8 c d \left (c+d x^4\right )^2}+\frac {(b c+7 a d) x}{32 c^2 d \left (c+d x^4\right )}+\frac {(3 (b c+7 a d)) \int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx}{64 c^{5/2} d}+\frac {(3 (b c+7 a d)) \int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx}{64 c^{5/2} d} \\ & = -\frac {(b c-a d) x}{8 c d \left (c+d x^4\right )^2}+\frac {(b c+7 a d) x}{32 c^2 d \left (c+d x^4\right )}+\frac {(3 (b c+7 a d)) \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{128 c^{5/2} d^{3/2}}+\frac {(3 (b c+7 a d)) \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{128 c^{5/2} d^{3/2}}-\frac {(3 (b c+7 a d)) \int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{128 \sqrt {2} c^{11/4} d^{5/4}}-\frac {(3 (b c+7 a d)) \int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{128 \sqrt {2} c^{11/4} d^{5/4}} \\ & = -\frac {(b c-a d) x}{8 c d \left (c+d x^4\right )^2}+\frac {(b c+7 a d) x}{32 c^2 d \left (c+d x^4\right )}-\frac {3 (b c+7 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{5/4}}+\frac {3 (b c+7 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{5/4}}+\frac {(3 (b c+7 a d)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{5/4}}-\frac {(3 (b c+7 a d)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{5/4}} \\ & = -\frac {(b c-a d) x}{8 c d \left (c+d x^4\right )^2}+\frac {(b c+7 a d) x}{32 c^2 d \left (c+d x^4\right )}-\frac {3 (b c+7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{5/4}}+\frac {3 (b c+7 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{5/4}}-\frac {3 (b c+7 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{5/4}}+\frac {3 (b c+7 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{5/4}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^3} \, dx=\frac {-\frac {32 c^{7/4} \sqrt [4]{d} (b c-a d) x}{\left (c+d x^4\right )^2}+\frac {8 c^{3/4} \sqrt [4]{d} (b c+7 a d) x}{c+d x^4}-6 \sqrt {2} (b c+7 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+6 \sqrt {2} (b c+7 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )-3 \sqrt {2} (b c+7 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )+3 \sqrt {2} (b c+7 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{256 c^{11/4} d^{5/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.93 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.31
method | result | size |
risch | \(\frac {\frac {\left (7 a d +b c \right ) x^{5}}{32 c^{2}}+\frac {\left (11 a d -3 b c \right ) x}{32 c d}}{\left (d \,x^{4}+c \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+c \right )}{\sum }\frac {\left (7 a d +b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{128 c^{2} d^{2}}\) | \(84\) |
default | \(\frac {\frac {\left (7 a d +b c \right ) x^{5}}{32 c^{2}}+\frac {\left (11 a d -3 b c \right ) x}{32 c d}}{\left (d \,x^{4}+c \right )^{2}}+\frac {3 \left (7 a d +b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3} d}\) | \(159\) |
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 739, normalized size of antiderivative = 2.71 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^3} \, dx=\frac {4 \, {\left (b c d + 7 \, a d^{2}\right )} x^{5} + 3 \, {\left (c^{2} d^{3} x^{8} + 2 \, c^{3} d^{2} x^{4} + c^{4} d\right )} \left (-\frac {b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac {1}{4}} \log \left (3 \, c^{3} d \left (-\frac {b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac {1}{4}} + 3 \, {\left (b c + 7 \, a d\right )} x\right ) - 3 \, {\left (-i \, c^{2} d^{3} x^{8} - 2 i \, c^{3} d^{2} x^{4} - i \, c^{4} d\right )} \left (-\frac {b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac {1}{4}} \log \left (3 i \, c^{3} d \left (-\frac {b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac {1}{4}} + 3 \, {\left (b c + 7 \, a d\right )} x\right ) - 3 \, {\left (i \, c^{2} d^{3} x^{8} + 2 i \, c^{3} d^{2} x^{4} + i \, c^{4} d\right )} \left (-\frac {b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac {1}{4}} \log \left (-3 i \, c^{3} d \left (-\frac {b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac {1}{4}} + 3 \, {\left (b c + 7 \, a d\right )} x\right ) - 3 \, {\left (c^{2} d^{3} x^{8} + 2 \, c^{3} d^{2} x^{4} + c^{4} d\right )} \left (-\frac {b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac {1}{4}} \log \left (-3 \, c^{3} d \left (-\frac {b^{4} c^{4} + 28 \, a b^{3} c^{3} d + 294 \, a^{2} b^{2} c^{2} d^{2} + 1372 \, a^{3} b c d^{3} + 2401 \, a^{4} d^{4}}{c^{11} d^{5}}\right )^{\frac {1}{4}} + 3 \, {\left (b c + 7 \, a d\right )} x\right ) - 4 \, {\left (3 \, b c^{2} - 11 \, a c d\right )} x}{128 \, {\left (c^{2} d^{3} x^{8} + 2 \, c^{3} d^{2} x^{4} + c^{4} d\right )}} \]
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Time = 0.52 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.55 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^3} \, dx=\frac {x^{5} \cdot \left (7 a d^{2} + b c d\right ) + x \left (11 a c d - 3 b c^{2}\right )}{32 c^{4} d + 64 c^{3} d^{2} x^{4} + 32 c^{2} d^{3} x^{8}} + \operatorname {RootSum} {\left (268435456 t^{4} c^{11} d^{5} + 194481 a^{4} d^{4} + 111132 a^{3} b c d^{3} + 23814 a^{2} b^{2} c^{2} d^{2} + 2268 a b^{3} c^{3} d + 81 b^{4} c^{4}, \left ( t \mapsto t \log {\left (\frac {128 t c^{3} d}{21 a d + 3 b c} + x \right )} \right )\right )} \]
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Time = 0.29 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.99 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^3} \, dx=\frac {{\left (b c d + 7 \, a d^{2}\right )} x^{5} - {\left (3 \, b c^{2} - 11 \, a c d\right )} x}{32 \, {\left (c^{2} d^{3} x^{8} + 2 \, c^{3} d^{2} x^{4} + c^{4} d\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (b c + 7 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b c + 7 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b c + 7 \, a d\right )} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b c + 7 \, a d\right )} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}\right )}}{256 \, c^{2} d} \]
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Time = 0.29 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.05 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^3} \, dx=\frac {3 \, \sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b c + 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{128 \, c^{3} d^{2}} + \frac {3 \, \sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b c + 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{128 \, c^{3} d^{2}} + \frac {3 \, \sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b c + 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{256 \, c^{3} d^{2}} - \frac {3 \, \sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b c + 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{256 \, c^{3} d^{2}} + \frac {b c d x^{5} + 7 \, a d^{2} x^{5} - 3 \, b c^{2} x + 11 \, a c d x}{32 \, {\left (d x^{4} + c\right )}^{2} c^{2} d} \]
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Time = 5.93 (sec) , antiderivative size = 762, normalized size of antiderivative = 2.79 \[ \int \frac {a+b x^4}{\left (c+d x^4\right )^3} \, dx=\frac {\frac {x^5\,\left (7\,a\,d+b\,c\right )}{32\,c^2}+\frac {x\,\left (11\,a\,d-3\,b\,c\right )}{32\,c\,d}}{c^2+2\,c\,d\,x^4+d^2\,x^8}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {9\,x\,\left (49\,a^2\,d^3+14\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{256\,c^4}-\frac {9\,\left (7\,a\,d+b\,c\right )\,\left (7\,a\,d^3+b\,c\,d^2\right )}{256\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (7\,a\,d+b\,c\right )\,3{}\mathrm {i}}{128\,{\left (-c\right )}^{11/4}\,d^{5/4}}+\frac {\left (\frac {9\,x\,\left (49\,a^2\,d^3+14\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{256\,c^4}+\frac {9\,\left (7\,a\,d+b\,c\right )\,\left (7\,a\,d^3+b\,c\,d^2\right )}{256\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (7\,a\,d+b\,c\right )\,3{}\mathrm {i}}{128\,{\left (-c\right )}^{11/4}\,d^{5/4}}}{\frac {3\,\left (\frac {9\,x\,\left (49\,a^2\,d^3+14\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{256\,c^4}-\frac {9\,\left (7\,a\,d+b\,c\right )\,\left (7\,a\,d^3+b\,c\,d^2\right )}{256\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (7\,a\,d+b\,c\right )}{128\,{\left (-c\right )}^{11/4}\,d^{5/4}}-\frac {3\,\left (\frac {9\,x\,\left (49\,a^2\,d^3+14\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{256\,c^4}+\frac {9\,\left (7\,a\,d+b\,c\right )\,\left (7\,a\,d^3+b\,c\,d^2\right )}{256\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (7\,a\,d+b\,c\right )}{128\,{\left (-c\right )}^{11/4}\,d^{5/4}}}\right )\,\left (7\,a\,d+b\,c\right )\,3{}\mathrm {i}}{64\,{\left (-c\right )}^{11/4}\,d^{5/4}}-\frac {3\,\mathrm {atan}\left (\frac {\frac {3\,\left (\frac {9\,x\,\left (49\,a^2\,d^3+14\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{256\,c^4}-\frac {\left (7\,a\,d+b\,c\right )\,\left (7\,a\,d^3+b\,c\,d^2\right )\,9{}\mathrm {i}}{256\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (7\,a\,d+b\,c\right )}{128\,{\left (-c\right )}^{11/4}\,d^{5/4}}+\frac {3\,\left (\frac {9\,x\,\left (49\,a^2\,d^3+14\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{256\,c^4}+\frac {\left (7\,a\,d+b\,c\right )\,\left (7\,a\,d^3+b\,c\,d^2\right )\,9{}\mathrm {i}}{256\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (7\,a\,d+b\,c\right )}{128\,{\left (-c\right )}^{11/4}\,d^{5/4}}}{\frac {\left (\frac {9\,x\,\left (49\,a^2\,d^3+14\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{256\,c^4}-\frac {\left (7\,a\,d+b\,c\right )\,\left (7\,a\,d^3+b\,c\,d^2\right )\,9{}\mathrm {i}}{256\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (7\,a\,d+b\,c\right )\,3{}\mathrm {i}}{128\,{\left (-c\right )}^{11/4}\,d^{5/4}}-\frac {\left (\frac {9\,x\,\left (49\,a^2\,d^3+14\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{256\,c^4}+\frac {\left (7\,a\,d+b\,c\right )\,\left (7\,a\,d^3+b\,c\,d^2\right )\,9{}\mathrm {i}}{256\,{\left (-c\right )}^{15/4}\,d^{5/4}}\right )\,\left (7\,a\,d+b\,c\right )\,3{}\mathrm {i}}{128\,{\left (-c\right )}^{11/4}\,d^{5/4}}}\right )\,\left (7\,a\,d+b\,c\right )}{64\,{\left (-c\right )}^{11/4}\,d^{5/4}} \]
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