Integrand size = 26, antiderivative size = 255 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=-\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{5/2}}{5 c}-\frac {\sqrt [4]{b} (b B-A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{9/4}}+\frac {\sqrt [4]{b} (b B-A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{9/4}}-\frac {\sqrt [4]{b} (b B-A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}+\frac {\sqrt [4]{b} (b B-A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}} \]
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Time = 0.15 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1598, 470, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=-\frac {\sqrt [4]{b} (b B-A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{9/4}}+\frac {\sqrt [4]{b} (b B-A c) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} c^{9/4}}-\frac {\sqrt [4]{b} (b B-A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}+\frac {\sqrt [4]{b} (b B-A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}-\frac {2 \sqrt {x} (b B-A c)}{c^2}+\frac {2 B x^{5/2}}{5 c} \]
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Rule 210
Rule 217
Rule 327
Rule 335
Rule 470
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{3/2} \left (A+B x^2\right )}{b+c x^2} \, dx \\ & = \frac {2 B x^{5/2}}{5 c}-\frac {\left (2 \left (\frac {5 b B}{2}-\frac {5 A c}{2}\right )\right ) \int \frac {x^{3/2}}{b+c x^2} \, dx}{5 c} \\ & = -\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{5/2}}{5 c}+\frac {(b (b B-A c)) \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{c^2} \\ & = -\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{5/2}}{5 c}+\frac {(2 b (b B-A c)) \text {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{c^2} \\ & = -\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{5/2}}{5 c}+\frac {\left (\sqrt {b} (b B-A c)\right ) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{c^2}+\frac {\left (\sqrt {b} (b B-A c)\right ) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{c^2} \\ & = -\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{5/2}}{5 c}+\frac {\left (\sqrt {b} (b B-A c)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{2 c^{5/2}}+\frac {\left (\sqrt {b} (b B-A c)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{2 c^{5/2}}-\frac {\left (\sqrt [4]{b} (b B-A c)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} c^{9/4}}-\frac {\left (\sqrt [4]{b} (b B-A c)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} c^{9/4}} \\ & = -\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{5/2}}{5 c}-\frac {\sqrt [4]{b} (b B-A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}+\frac {\sqrt [4]{b} (b B-A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}+\frac {\left (\sqrt [4]{b} (b B-A c)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{9/4}}-\frac {\left (\sqrt [4]{b} (b B-A c)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{9/4}} \\ & = -\frac {2 (b B-A c) \sqrt {x}}{c^2}+\frac {2 B x^{5/2}}{5 c}-\frac {\sqrt [4]{b} (b B-A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{9/4}}+\frac {\sqrt [4]{b} (b B-A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{9/4}}-\frac {\sqrt [4]{b} (b B-A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}}+\frac {\sqrt [4]{b} (b B-A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{9/4}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.59 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {4 \sqrt [4]{c} \sqrt {x} \left (-5 b B+5 A c+B c x^2\right )-5 \sqrt {2} \sqrt [4]{b} (b B-A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )+5 \sqrt {2} \sqrt [4]{b} (b B-A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{10 c^{9/4}} \]
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Time = 1.77 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.54
method | result | size |
risch | \(\frac {2 \left (B c \,x^{2}+5 A c -5 B b \right ) \sqrt {x}}{5 c^{2}}-\frac {\left (A c -B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{2}}\) | \(138\) |
derivativedivides | \(\frac {\frac {2 B c \,x^{\frac {5}{2}}}{5}+2 A c \sqrt {x}-2 b B \sqrt {x}}{c^{2}}-\frac {\left (A c -B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{2}}\) | \(141\) |
default | \(\frac {\frac {2 B c \,x^{\frac {5}{2}}}{5}+2 A c \sqrt {x}-2 b B \sqrt {x}}{c^{2}}-\frac {\left (A c -B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{2}}\) | \(141\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.34 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=-\frac {5 \, c^{2} \left (-\frac {B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac {1}{4}} \log \left (c^{2} \left (-\frac {B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac {1}{4}} - {\left (B b - A c\right )} \sqrt {x}\right ) + 5 i \, c^{2} \left (-\frac {B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac {1}{4}} \log \left (i \, c^{2} \left (-\frac {B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac {1}{4}} - {\left (B b - A c\right )} \sqrt {x}\right ) - 5 i \, c^{2} \left (-\frac {B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac {1}{4}} \log \left (-i \, c^{2} \left (-\frac {B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac {1}{4}} - {\left (B b - A c\right )} \sqrt {x}\right ) - 5 \, c^{2} \left (-\frac {B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac {1}{4}} \log \left (-c^{2} \left (-\frac {B^{4} b^{5} - 4 \, A B^{3} b^{4} c + 6 \, A^{2} B^{2} b^{3} c^{2} - 4 \, A^{3} B b^{2} c^{3} + A^{4} b c^{4}}{c^{9}}\right )^{\frac {1}{4}} - {\left (B b - A c\right )} \sqrt {x}\right ) - 4 \, {\left (B c x^{2} - 5 \, B b + 5 \, A c\right )} \sqrt {x}}{10 \, c^{2}} \]
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Time = 98.31 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.08 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\begin {cases} \tilde {\infty } \left (2 A \sqrt {x} + \frac {2 B x^{\frac {5}{2}}}{5}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {9}{2}}}{9}}{b} & \text {for}\: c = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {5}{2}}}{5}}{c} & \text {for}\: b = 0 \\\frac {2 A \sqrt {x}}{c} + \frac {A \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {b}{c}} \right )}}{2 c} - \frac {A \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {b}{c}} \right )}}{2 c} - \frac {A \sqrt [4]{- \frac {b}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {b}{c}}} \right )}}{c} - \frac {2 B b \sqrt {x}}{c^{2}} - \frac {B b \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {b}{c}} \right )}}{2 c^{2}} + \frac {B b \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {b}{c}} \right )}}{2 c^{2}} + \frac {B b \sqrt [4]{- \frac {b}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {b}{c}}} \right )}}{c^{2}} + \frac {2 B x^{\frac {5}{2}}}{5 c} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.92 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {{\left (\frac {2 \, \sqrt {2} {\left (B b - A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} {\left (B b - A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} {\left (B b - A c\right )} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B b - A c\right )} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}}\right )} b}{4 \, c^{2}} + \frac {2 \, {\left (B c x^{\frac {5}{2}} - 5 \, {\left (B b - A c\right )} \sqrt {x}\right )}}{5 \, c^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.03 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b - \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{3}} + \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b - \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{3}} + \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b - \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, c^{3}} - \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b - \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, c^{3}} + \frac {2 \, {\left (B c^{4} x^{\frac {5}{2}} - 5 \, B b c^{3} \sqrt {x} + 5 \, A c^{4} \sqrt {x}\right )}}{5 \, c^{5}} \]
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Time = 9.14 (sec) , antiderivative size = 789, normalized size of antiderivative = 3.09 \[ \int \frac {x^{7/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\sqrt {x}\,\left (\frac {2\,A}{c}-\frac {2\,B\,b}{c^2}\right )+\frac {2\,B\,x^{5/2}}{5\,c}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-b\right )}^{1/4}\,\left (A\,c-B\,b\right )\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,b^2\,c^2-2\,A\,B\,b^3\,c+B^2\,b^4\right )}{c}-\frac {{\left (-b\right )}^{1/4}\,\left (32\,A\,b^2\,c^2-32\,B\,b^3\,c\right )\,\left (A\,c-B\,b\right )}{2\,c^{9/4}}\right )\,1{}\mathrm {i}}{2\,c^{9/4}}+\frac {{\left (-b\right )}^{1/4}\,\left (A\,c-B\,b\right )\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,b^2\,c^2-2\,A\,B\,b^3\,c+B^2\,b^4\right )}{c}+\frac {{\left (-b\right )}^{1/4}\,\left (32\,A\,b^2\,c^2-32\,B\,b^3\,c\right )\,\left (A\,c-B\,b\right )}{2\,c^{9/4}}\right )\,1{}\mathrm {i}}{2\,c^{9/4}}}{\frac {{\left (-b\right )}^{1/4}\,\left (A\,c-B\,b\right )\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,b^2\,c^2-2\,A\,B\,b^3\,c+B^2\,b^4\right )}{c}-\frac {{\left (-b\right )}^{1/4}\,\left (32\,A\,b^2\,c^2-32\,B\,b^3\,c\right )\,\left (A\,c-B\,b\right )}{2\,c^{9/4}}\right )}{2\,c^{9/4}}-\frac {{\left (-b\right )}^{1/4}\,\left (A\,c-B\,b\right )\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,b^2\,c^2-2\,A\,B\,b^3\,c+B^2\,b^4\right )}{c}+\frac {{\left (-b\right )}^{1/4}\,\left (32\,A\,b^2\,c^2-32\,B\,b^3\,c\right )\,\left (A\,c-B\,b\right )}{2\,c^{9/4}}\right )}{2\,c^{9/4}}}\right )\,\left (A\,c-B\,b\right )\,1{}\mathrm {i}}{c^{9/4}}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-b\right )}^{1/4}\,\left (A\,c-B\,b\right )\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,b^2\,c^2-2\,A\,B\,b^3\,c+B^2\,b^4\right )}{c}-\frac {{\left (-b\right )}^{1/4}\,\left (32\,A\,b^2\,c^2-32\,B\,b^3\,c\right )\,\left (A\,c-B\,b\right )\,1{}\mathrm {i}}{2\,c^{9/4}}\right )}{2\,c^{9/4}}+\frac {{\left (-b\right )}^{1/4}\,\left (A\,c-B\,b\right )\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,b^2\,c^2-2\,A\,B\,b^3\,c+B^2\,b^4\right )}{c}+\frac {{\left (-b\right )}^{1/4}\,\left (32\,A\,b^2\,c^2-32\,B\,b^3\,c\right )\,\left (A\,c-B\,b\right )\,1{}\mathrm {i}}{2\,c^{9/4}}\right )}{2\,c^{9/4}}}{\frac {{\left (-b\right )}^{1/4}\,\left (A\,c-B\,b\right )\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,b^2\,c^2-2\,A\,B\,b^3\,c+B^2\,b^4\right )}{c}-\frac {{\left (-b\right )}^{1/4}\,\left (32\,A\,b^2\,c^2-32\,B\,b^3\,c\right )\,\left (A\,c-B\,b\right )\,1{}\mathrm {i}}{2\,c^{9/4}}\right )\,1{}\mathrm {i}}{2\,c^{9/4}}-\frac {{\left (-b\right )}^{1/4}\,\left (A\,c-B\,b\right )\,\left (\frac {16\,\sqrt {x}\,\left (A^2\,b^2\,c^2-2\,A\,B\,b^3\,c+B^2\,b^4\right )}{c}+\frac {{\left (-b\right )}^{1/4}\,\left (32\,A\,b^2\,c^2-32\,B\,b^3\,c\right )\,\left (A\,c-B\,b\right )\,1{}\mathrm {i}}{2\,c^{9/4}}\right )\,1{}\mathrm {i}}{2\,c^{9/4}}}\right )\,\left (A\,c-B\,b\right )}{c^{9/4}} \]
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