Integrand size = 26, antiderivative size = 237 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )} \, dx=-\frac {2 A}{3 b x^{3/2}}-\frac {(b B-A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{7/4} \sqrt [4]{c}}+\frac {(b B-A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{7/4} \sqrt [4]{c}}-\frac {(b B-A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} b^{7/4} \sqrt [4]{c}}+\frac {(b B-A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} b^{7/4} \sqrt [4]{c}} \]
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Time = 0.13 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1598, 464, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )} \, dx=-\frac {(b B-A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{7/4} \sqrt [4]{c}}+\frac {(b B-A c) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} b^{7/4} \sqrt [4]{c}}-\frac {(b B-A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} b^{7/4} \sqrt [4]{c}}+\frac {(b B-A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} b^{7/4} \sqrt [4]{c}}-\frac {2 A}{3 b x^{3/2}} \]
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Rule 210
Rule 217
Rule 335
Rule 464
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {A+B x^2}{x^{5/2} \left (b+c x^2\right )} \, dx \\ & = -\frac {2 A}{3 b x^{3/2}}-\frac {\left (2 \left (-\frac {3 b B}{2}+\frac {3 A c}{2}\right )\right ) \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{3 b} \\ & = -\frac {2 A}{3 b x^{3/2}}-\frac {\left (4 \left (-\frac {3 b B}{2}+\frac {3 A c}{2}\right )\right ) \text {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{3 b} \\ & = -\frac {2 A}{3 b x^{3/2}}+\frac {(b B-A c) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{b^{3/2}}+\frac {(b B-A c) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{b^{3/2}} \\ & = -\frac {2 A}{3 b x^{3/2}}+\frac {(b B-A c) \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 b^{3/2} \sqrt {c}}+\frac {(b B-A c) \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 b^{3/2} \sqrt {c}}-\frac {(b B-A c) \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} b^{7/4} \sqrt [4]{c}}-\frac {(b B-A c) \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} b^{7/4} \sqrt [4]{c}} \\ & = -\frac {2 A}{3 b x^{3/2}}-\frac {(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} b^{7/4} \sqrt [4]{c}}+\frac {(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} b^{7/4} \sqrt [4]{c}}+\frac {(b B-A c) \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} b^{7/4} \sqrt [4]{c}}-\frac {(b B-A c) \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} b^{7/4} \sqrt [4]{c}} \\ & = -\frac {2 A}{3 b x^{3/2}}-\frac {(b B-A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{7/4} \sqrt [4]{c}}+\frac {(b B-A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{7/4} \sqrt [4]{c}}-\frac {(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} b^{7/4} \sqrt [4]{c}}+\frac {(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} b^{7/4} \sqrt [4]{c}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.57 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )} \, dx=-\frac {2 A}{3 b x^{3/2}}-\frac {(b B-A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt {2} b^{7/4} \sqrt [4]{c}}+\frac {(b B-A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{\sqrt {2} b^{7/4} \sqrt [4]{c}} \]
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Time = 1.78 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.52
method | result | size |
derivativedivides | \(\frac {\left (-A c +B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{2}}-\frac {2 A}{3 b \,x^{\frac {3}{2}}}\) | \(124\) |
default | \(\frac {\left (-A c +B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{2}}-\frac {2 A}{3 b \,x^{\frac {3}{2}}}\) | \(124\) |
risch | \(-\frac {2 A}{3 b \,x^{\frac {3}{2}}}-\frac {\left (A c -B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{2}}\) | \(124\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.49 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )} \, dx=-\frac {3 \, b x^{2} \left (-\frac {B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac {1}{4}} \log \left (b^{2} \left (-\frac {B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac {1}{4}} - {\left (B b - A c\right )} \sqrt {x}\right ) + 3 i \, b x^{2} \left (-\frac {B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac {1}{4}} \log \left (i \, b^{2} \left (-\frac {B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac {1}{4}} - {\left (B b - A c\right )} \sqrt {x}\right ) - 3 i \, b x^{2} \left (-\frac {B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac {1}{4}} \log \left (-i \, b^{2} \left (-\frac {B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac {1}{4}} - {\left (B b - A c\right )} \sqrt {x}\right ) - 3 \, b x^{2} \left (-\frac {B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac {1}{4}} \log \left (-b^{2} \left (-\frac {B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{7} c}\right )^{\frac {1}{4}} - {\left (B b - A c\right )} \sqrt {x}\right ) + 4 \, A \sqrt {x}}{6 \, b x^{2}} \]
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Time = 13.82 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{c} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} + 2 B \sqrt {x}}{b} & \text {for}\: c = 0 \\- \frac {2 A}{3 b x^{\frac {3}{2}}} + \frac {A c \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {b}{c}} \right )}}{2 b^{2}} - \frac {A c \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {b}{c}} \right )}}{2 b^{2}} - \frac {A c \sqrt [4]{- \frac {b}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {b}{c}}} \right )}}{b^{2}} - \frac {B \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {b}{c}} \right )}}{2 b} + \frac {B \sqrt [4]{- \frac {b}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {b}{c}} \right )}}{2 b} + \frac {B \sqrt [4]{- \frac {b}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {b}{c}}} \right )}}{b} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )} \, dx=\frac {\frac {2 \, \sqrt {2} {\left (B b - A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} {\left (B b - A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} {\left (B b - A c\right )} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B b - A c\right )} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}}}{4 \, b} - \frac {2 \, A}{3 \, b x^{\frac {3}{2}}} \]
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Time = 0.28 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )} \, dx=\frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b - \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, b^{2} c} + \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b - \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, b^{2} c} + \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b - \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, b^{2} c} - \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b - \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, b^{2} c} - \frac {2 \, A}{3 \, b x^{\frac {3}{2}}} \]
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Time = 9.12 (sec) , antiderivative size = 811, normalized size of antiderivative = 3.42 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )} \, dx=-\frac {2\,A}{3\,b\,x^{3/2}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,c-B\,b\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3\,c^5-32\,A\,B\,b^4\,c^4+16\,B^2\,b^5\,c^3\right )-\frac {\left (A\,c-B\,b\right )\,\left (32\,A\,b^5\,c^4-32\,B\,b^6\,c^3\right )}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}+\frac {\left (A\,c-B\,b\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3\,c^5-32\,A\,B\,b^4\,c^4+16\,B^2\,b^5\,c^3\right )+\frac {\left (A\,c-B\,b\right )\,\left (32\,A\,b^5\,c^4-32\,B\,b^6\,c^3\right )}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}}{\frac {\left (A\,c-B\,b\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3\,c^5-32\,A\,B\,b^4\,c^4+16\,B^2\,b^5\,c^3\right )-\frac {\left (A\,c-B\,b\right )\,\left (32\,A\,b^5\,c^4-32\,B\,b^6\,c^3\right )}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}\right )}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}-\frac {\left (A\,c-B\,b\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3\,c^5-32\,A\,B\,b^4\,c^4+16\,B^2\,b^5\,c^3\right )+\frac {\left (A\,c-B\,b\right )\,\left (32\,A\,b^5\,c^4-32\,B\,b^6\,c^3\right )}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}\right )}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}}\right )\,\left (A\,c-B\,b\right )\,1{}\mathrm {i}}{{\left (-b\right )}^{7/4}\,c^{1/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,c-B\,b\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3\,c^5-32\,A\,B\,b^4\,c^4+16\,B^2\,b^5\,c^3\right )-\frac {\left (A\,c-B\,b\right )\,\left (32\,A\,b^5\,c^4-32\,B\,b^6\,c^3\right )\,1{}\mathrm {i}}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}\right )}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}+\frac {\left (A\,c-B\,b\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3\,c^5-32\,A\,B\,b^4\,c^4+16\,B^2\,b^5\,c^3\right )+\frac {\left (A\,c-B\,b\right )\,\left (32\,A\,b^5\,c^4-32\,B\,b^6\,c^3\right )\,1{}\mathrm {i}}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}\right )}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}}{\frac {\left (A\,c-B\,b\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3\,c^5-32\,A\,B\,b^4\,c^4+16\,B^2\,b^5\,c^3\right )-\frac {\left (A\,c-B\,b\right )\,\left (32\,A\,b^5\,c^4-32\,B\,b^6\,c^3\right )\,1{}\mathrm {i}}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}-\frac {\left (A\,c-B\,b\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3\,c^5-32\,A\,B\,b^4\,c^4+16\,B^2\,b^5\,c^3\right )+\frac {\left (A\,c-B\,b\right )\,\left (32\,A\,b^5\,c^4-32\,B\,b^6\,c^3\right )\,1{}\mathrm {i}}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-b\right )}^{7/4}\,c^{1/4}}}\right )\,\left (A\,c-B\,b\right )}{{\left (-b\right )}^{7/4}\,c^{1/4}} \]
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