Integrand size = 24, antiderivative size = 368 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )^2} \, dx=-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (5 b c+7 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (5 b c+7 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}} \]
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Time = 0.30 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 479, 584, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2 (7 a d+5 b c)}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (7 a d+5 b c)}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (7 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (7 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 x^{3/2} (3 b c-7 a d)}{6 a b^2}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b \sqrt {x} \left (a+b x^2\right )} \]
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Rule 210
Rule 303
Rule 477
Rule 479
Rule 584
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right ) \\ & = \frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (-c (5 b c-a d)+d (3 b c-7 a d) x^4\right )}{x^2 \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right )}{2 a b} \\ & = \frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {c^2 (-5 b c+a d)}{a x^2}+\frac {d^2 (3 b c-7 a d) x^2}{b}+\frac {(-b c+a d)^2 (5 b c+7 a d) x^2}{a b \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b} \\ & = -\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2 b^2} \\ & = -\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^2 b^{5/2}}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^2 b^{5/2}} \\ & = -\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^2 b^3}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^2 b^3}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \text {Subst}\left (\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,x,\sqrt {x}\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \text {Subst}\left (\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,x,\sqrt {x}\right )}{8 \sqrt {2} a^{9/4} b^{11/4}} \\ & = -\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}+\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}} \\ & = -\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (5 b c+7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (5 b c+7 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.62 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{a} b^{3/4} \left (-15 b^3 c^3 x^2+7 a^3 d^3 x^2+3 a b^2 c^2 \left (-4 c+3 d x^2\right )+a^2 b d^2 x^2 \left (-9 c+4 d x^2\right )\right )}{\sqrt {x} \left (a+b x^2\right )}+3 \sqrt {2} (b c-a d)^2 (5 b c+7 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+3 \sqrt {2} (b c-a d)^2 (5 b c+7 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{24 a^{9/4} b^{11/4}} \]
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Time = 2.79 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.54
method | result | size |
risch | \(\frac {-2 b^{2} c^{3}+\frac {2 a^{2} d^{3} x^{2}}{3}}{a^{2} \sqrt {x}\, b^{2}}-\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {7 a d}{4}+\frac {5 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{b^{2} a^{2}}\) | \(200\) |
derivativedivides | \(\frac {2 d^{3} x^{\frac {3}{2}}}{3 b^{2}}-\frac {2 c^{3}}{a^{2} \sqrt {x}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {7}{4} a^{3} d^{3}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {5}{4} b^{3} c^{3}-\frac {9}{4} a^{2} b c \,d^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{2}}\) | \(225\) |
default | \(\frac {2 d^{3} x^{\frac {3}{2}}}{3 b^{2}}-\frac {2 c^{3}}{a^{2} \sqrt {x}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {7}{4} a^{3} d^{3}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {5}{4} b^{3} c^{3}-\frac {9}{4} a^{2} b c \,d^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{2}}\) | \(225\) |
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Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 2104, normalized size of antiderivative = 5.72 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \]
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Time = 0.33 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.83 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {2 \, d^{3} x^{\frac {3}{2}}}{3 \, b^{2}} - \frac {4 \, a b^{2} c^{3} + {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}}{2 \, {\left (a^{2} b^{3} x^{\frac {5}{2}} + a^{3} b^{2} \sqrt {x}\right )}} - \frac {{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a^{2} b^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.37 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {2 \, d^{3} x^{\frac {3}{2}}}{3 \, b^{2}} - \frac {5 \, b^{3} c^{3} x^{2} - 3 \, a b^{2} c^{2} d x^{2} + 3 \, a^{2} b c d^{2} x^{2} - a^{3} d^{3} x^{2} + 4 \, a b^{2} c^{3}}{2 \, {\left (b x^{\frac {5}{2}} + a \sqrt {x}\right )} a^{2} b^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b^{5}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} b^{5}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b^{5}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{3} b^{5}} \]
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Time = 0.25 (sec) , antiderivative size = 657, normalized size of antiderivative = 1.79 \[ \int \frac {\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {\frac {x^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-5\,b^3\,c^3\right )}{2\,a^2}-\frac {2\,b^2\,c^3}{a}}{b^3\,x^{5/2}+a\,b^2\,\sqrt {x}}+\frac {2\,d^3\,x^{3/2}}{3\,b^2}-\frac {\mathrm {atan}\left (\frac {\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )\,\left (1568\,a^{13}\,b^8\,d^6-4032\,a^{12}\,b^9\,c\,d^5+1248\,a^{11}\,b^{10}\,c^2\,d^4+3968\,a^{10}\,b^{11}\,c^3\,d^3-2592\,a^9\,b^{12}\,c^4\,d^2-960\,a^8\,b^{13}\,c^5\,d+800\,a^7\,b^{14}\,c^6\right )}{4\,{\left (-a\right )}^{9/4}\,b^{11/4}\,\left (2744\,a^{14}\,b^5\,d^9-10584\,a^{13}\,b^6\,c\,d^8+10080\,a^{12}\,b^7\,c^2\,d^7+9120\,a^{11}\,b^8\,c^3\,d^6-19440\,a^{10}\,b^9\,c^4\,d^5+2736\,a^9\,b^{10}\,c^5\,d^4+10464\,a^8\,b^{11}\,c^6\,d^3-4320\,a^7\,b^{12}\,c^7\,d^2-1800\,a^6\,b^{13}\,c^8\,d+1000\,a^5\,b^{14}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )}{4\,{\left (-a\right )}^{9/4}\,b^{11/4}}-\frac {\mathrm {atan}\left (\frac {\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )\,\left (1568\,a^{13}\,b^8\,d^6-4032\,a^{12}\,b^9\,c\,d^5+1248\,a^{11}\,b^{10}\,c^2\,d^4+3968\,a^{10}\,b^{11}\,c^3\,d^3-2592\,a^9\,b^{12}\,c^4\,d^2-960\,a^8\,b^{13}\,c^5\,d+800\,a^7\,b^{14}\,c^6\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{9/4}\,b^{11/4}\,\left (2744\,a^{14}\,b^5\,d^9-10584\,a^{13}\,b^6\,c\,d^8+10080\,a^{12}\,b^7\,c^2\,d^7+9120\,a^{11}\,b^8\,c^3\,d^6-19440\,a^{10}\,b^9\,c^4\,d^5+2736\,a^9\,b^{10}\,c^5\,d^4+10464\,a^8\,b^{11}\,c^6\,d^3-4320\,a^7\,b^{12}\,c^7\,d^2-1800\,a^6\,b^{13}\,c^8\,d+1000\,a^5\,b^{14}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{9/4}\,b^{11/4}} \]
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