Integrand size = 26, antiderivative size = 179 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^2} \, dx=36 c \left (b^2-4 a c\right ) d^5 \sqrt {b d+2 c d x}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}-18 c \left (b^2-4 a c\right )^{5/4} d^{11/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-18 c \left (b^2-4 a c\right )^{5/4} d^{11/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {700, 706, 708, 335, 218, 212, 209} \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^2} \, dx=-18 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-18 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+36 c d^5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 700
Rule 706
Rule 708
Rubi steps \begin{align*} \text {integral}& = -\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\left (9 c d^2\right ) \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx \\ & = \frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\left (9 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx \\ & = 36 c \left (b^2-4 a c\right ) d^5 \sqrt {b d+2 c d x}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\left (9 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx \\ & = 36 c \left (b^2-4 a c\right ) d^5 \sqrt {b d+2 c d x}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\frac {1}{2} \left (9 \left (b^2-4 a c\right )^2 d^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )} \, dx,x,b d+2 c d x\right ) \\ & = 36 c \left (b^2-4 a c\right ) d^5 \sqrt {b d+2 c d x}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\left (9 \left (b^2-4 a c\right )^2 d^5\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right ) \\ & = 36 c \left (b^2-4 a c\right ) d^5 \sqrt {b d+2 c d x}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}-\left (18 c \left (b^2-4 a c\right )^{3/2} d^6\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d-x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )-\left (18 c \left (b^2-4 a c\right )^{3/2} d^6\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right ) \\ & = 36 c \left (b^2-4 a c\right ) d^5 \sqrt {b d+2 c d x}+\frac {36}{5} c d^3 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}-18 c \left (b^2-4 a c\right )^{5/4} d^{11/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-18 c \left (b^2-4 a c\right )^{5/4} d^{11/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.65 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^2} \, dx=\left (\frac {1}{5}+\frac {i}{5}\right ) c (d (b+2 c x))^{11/2} \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (45 b^4-360 a b^2 c+720 a^2 c^2-36 b^2 (b+2 c x)^2+144 a c (b+2 c x)^2-4 (b+2 c x)^4\right )}{c (b+2 c x)^5 (a+x (b+c x))}-\frac {45 i \left (b^2-4 a c\right )^{5/4} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{11/2}}+\frac {45 i \left (b^2-4 a c\right )^{5/4} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{11/2}}+\frac {45 i \left (b^2-4 a c\right )^{5/4} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{(b+2 c x)^{11/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(372\) vs. \(2(151)=302\).
Time = 2.95 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.08
method | result | size |
derivativedivides | \(16 c \,d^{3} \left (-8 a c \,d^{2} \sqrt {2 c d x +b d}+2 b^{2} d^{2} \sqrt {2 c d x +b d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+d^{4} \left (\frac {\left (-a^{2} c^{2}+\frac {1}{2} a \,b^{2} c -\frac {1}{16} b^{4}\right ) \sqrt {2 c d x +b d}}{a c \,d^{2}-\frac {b^{2} d^{2}}{4}+\frac {\left (2 c d x +b d \right )^{2}}{4}}+\frac {9 \left (4 a^{2} c^{2}-2 a \,b^{2} c +\frac {1}{4} b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\) | \(373\) |
default | \(16 c \,d^{3} \left (-8 a c \,d^{2} \sqrt {2 c d x +b d}+2 b^{2} d^{2} \sqrt {2 c d x +b d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+d^{4} \left (\frac {\left (-a^{2} c^{2}+\frac {1}{2} a \,b^{2} c -\frac {1}{16} b^{4}\right ) \sqrt {2 c d x +b d}}{a c \,d^{2}-\frac {b^{2} d^{2}}{4}+\frac {\left (2 c d x +b d \right )^{2}}{4}}+\frac {9 \left (4 a^{2} c^{2}-2 a \,b^{2} c +\frac {1}{4} b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\) | \(373\) |
pseudoelliptic | \(-\frac {144 \left (-\frac {\left (d \left (2 c x +b \right )\right )^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right ) c \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}}}{45}+\frac {d^{2} \left (4 a c -b^{2}\right ) \left (2 \left (\frac {8 c^{2} x^{2}}{9}+\left (\frac {8 b x}{9}+a \right ) c -\frac {b^{2}}{36}\right ) \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \sqrt {d \left (2 c x +b \right )}-c \,d^{2} \sqrt {2}\, \left (c \,x^{2}+b x +a \right ) \left (-\frac {b^{2}}{4}+a c \right ) \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )\right )\right )}{8}\right ) d^{3}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (c \,x^{2}+b x +a \right )}\) | \(387\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 816, normalized size of antiderivative = 4.56 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {45 \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\left (c x^{2} + b x + a\right )} \log \left (-9 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {2 \, c d x + b d} d^{5} + 9 \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) - 45 \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\left (-i \, c x^{2} - i \, b x - i \, a\right )} \log \left (-9 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {2 \, c d x + b d} d^{5} + 9 i \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) - 45 \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\left (i \, c x^{2} + i \, b x + i \, a\right )} \log \left (-9 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {2 \, c d x + b d} d^{5} - 9 i \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) - 45 \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\left (c x^{2} + b x + a\right )} \log \left (-9 \, {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {2 \, c d x + b d} d^{5} - 9 \, \left ({\left (b^{10} c^{4} - 20 \, a b^{8} c^{5} + 160 \, a^{2} b^{6} c^{6} - 640 \, a^{3} b^{4} c^{7} + 1280 \, a^{4} b^{2} c^{8} - 1024 \, a^{5} c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) + {\left (64 \, c^{4} d^{5} x^{4} + 128 \, b c^{3} d^{5} x^{3} + 48 \, {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} d^{5} x^{2} + 16 \, {\left (11 \, b^{3} c - 36 \, a b c^{2}\right )} d^{5} x - {\left (5 \, b^{4} - 216 \, a b^{2} c + 720 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt {2 \, c d x + b d}}{5 \, {\left (c x^{2} + b x + a\right )}} \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (151) = 302\).
Time = 0.31 (sec) , antiderivative size = 646, normalized size of antiderivative = 3.61 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^2} \, dx=32 \, \sqrt {2 \, c d x + b d} b^{2} c d^{5} - 128 \, \sqrt {2 \, c d x + b d} a c^{2} d^{5} + \frac {16}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c d^{3} - 9 \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} c d^{5} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c^{2} d^{5}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - 9 \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} c d^{5} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c^{2} d^{5}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {9}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} c d^{5} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c^{2} d^{5}\right )} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {9}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} c d^{5} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c^{2} d^{5}\right )} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {4 \, {\left (\sqrt {2 \, c d x + b d} b^{4} c d^{7} - 8 \, \sqrt {2 \, c d x + b d} a b^{2} c^{2} d^{7} + 16 \, \sqrt {2 \, c d x + b d} a^{2} c^{3} d^{7}\right )}}{b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 834, normalized size of antiderivative = 4.66 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {16\,c\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}{5}-\frac {\sqrt {b\,d+2\,c\,d\,x}\,\left (64\,a^2\,c^3\,d^7-32\,a\,b^2\,c^2\,d^7+4\,b^4\,c\,d^7\right )}{{\left (b\,d+2\,c\,d\,x\right )}^2-b^2\,d^2+4\,a\,c\,d^2}-18\,c\,d^{11/2}\,\mathrm {atan}\left (\frac {9\,c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (1327104\,a^4\,c^6\,d^{14}-1327104\,a^3\,b^2\,c^5\,d^{14}+497664\,a^2\,b^4\,c^4\,d^{14}-82944\,a\,b^6\,c^3\,d^{14}+5184\,b^8\,c^2\,d^{14}\right )-c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (-36864\,a^3\,c^4\,d^9+27648\,a^2\,b^2\,c^3\,d^9-6912\,a\,b^4\,c^2\,d^9+576\,b^6\,c\,d^9\right )\,9{}\mathrm {i}\right )+9\,c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (1327104\,a^4\,c^6\,d^{14}-1327104\,a^3\,b^2\,c^5\,d^{14}+497664\,a^2\,b^4\,c^4\,d^{14}-82944\,a\,b^6\,c^3\,d^{14}+5184\,b^8\,c^2\,d^{14}\right )+c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (-36864\,a^3\,c^4\,d^9+27648\,a^2\,b^2\,c^3\,d^9-6912\,a\,b^4\,c^2\,d^9+576\,b^6\,c\,d^9\right )\,9{}\mathrm {i}\right )}{c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (1327104\,a^4\,c^6\,d^{14}-1327104\,a^3\,b^2\,c^5\,d^{14}+497664\,a^2\,b^4\,c^4\,d^{14}-82944\,a\,b^6\,c^3\,d^{14}+5184\,b^8\,c^2\,d^{14}\right )-c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (-36864\,a^3\,c^4\,d^9+27648\,a^2\,b^2\,c^3\,d^9-6912\,a\,b^4\,c^2\,d^9+576\,b^6\,c\,d^9\right )\,9{}\mathrm {i}\right )\,9{}\mathrm {i}-c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (1327104\,a^4\,c^6\,d^{14}-1327104\,a^3\,b^2\,c^5\,d^{14}+497664\,a^2\,b^4\,c^4\,d^{14}-82944\,a\,b^6\,c^3\,d^{14}+5184\,b^8\,c^2\,d^{14}\right )+c\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (-36864\,a^3\,c^4\,d^9+27648\,a^2\,b^2\,c^3\,d^9-6912\,a\,b^4\,c^2\,d^9+576\,b^6\,c\,d^9\right )\,9{}\mathrm {i}\right )\,9{}\mathrm {i}}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}-32\,c\,d^5\,\sqrt {b\,d+2\,c\,d\,x}\,\left (4\,a\,c-b^2\right )+c\,d^{11/2}\,\mathrm {atan}\left (\frac {b^2\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}-a\,c\,\sqrt {b\,d+2\,c\,d\,x}\,4{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{5/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}\,18{}\mathrm {i} \]
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