Integrand size = 22, antiderivative size = 235 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\frac {2 B \sqrt {x}}{b}-\frac {(A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{5/4}}+\frac {(A b-a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{5/4}}-\frac {(A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{5/4}}+\frac {(A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{5/4}} \]
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Time = 0.13 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {470, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=-\frac {(A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{5/4}}+\frac {(A b-a B) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{3/4} b^{5/4}}-\frac {(A b-a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{5/4}}+\frac {(A b-a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{5/4}}+\frac {2 B \sqrt {x}}{b} \]
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Rule 210
Rule 217
Rule 335
Rule 470
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {2 B \sqrt {x}}{b}-\frac {\left (2 \left (-\frac {A b}{2}+\frac {a B}{2}\right )\right ) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{b} \\ & = \frac {2 B \sqrt {x}}{b}-\frac {\left (4 \left (-\frac {A b}{2}+\frac {a B}{2}\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b} \\ & = \frac {2 B \sqrt {x}}{b}+\frac {(A b-a B) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a} b}+\frac {(A b-a B) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a} b} \\ & = \frac {2 B \sqrt {x}}{b}+\frac {(A b-a B) \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 )}{2 \sqrt {a} b^{3/2}}+\frac {(A b-a B) \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 )}{2 \sqrt {a} b^{3/2}}-\frac {(A b-a B) \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 )}{2 \sqrt {2} a^{3/4} b^{5/4}}-\frac {(A b-a B) \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 )}{2 \sqrt {2} a^{3/4} b^{5/4}} \\ & = \frac {2 B \sqrt {x}}{b}-\frac {(A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{5/4}}+\frac {(A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{5/4}}+\frac {(A b-a B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{5/4}}-\frac {(A b-a B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{5/4}} \\ & = \frac {2 B \sqrt {x}}{b}-\frac {(A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{5/4}}+\frac {(A b-a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{5/4}}-\frac {(A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{5/4}}+\frac {(A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{5/4}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.57 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\frac {2 B \sqrt {x}}{b}+\frac {(-A b+a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {2} a^{3/4} b^{5/4}}-\frac {(-A b+a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{3/4} b^{5/4}} \]
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Time = 2.67 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.54
method | result | size |
derivativedivides | \(\frac {2 B \sqrt {x}}{b}+\frac {\left (A b -B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b a}\) | \(127\) |
default | \(\frac {2 B \sqrt {x}}{b}+\frac {\left (A b -B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b a}\) | \(127\) |
risch | \(\frac {2 B \sqrt {x}}{b}+\frac {\left (A b -B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b a}\) | \(127\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.43 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\frac {b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (a b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} \sqrt {x}\right ) + i \, b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (i \, a b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} \sqrt {x}\right ) - i \, b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (-i \, a b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} \sqrt {x}\right ) - b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (-a b \left (-\frac {B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{5}}\right )^{\frac {1}{4}} - {\left (B a - A b\right )} \sqrt {x}\right ) + 4 \, B \sqrt {x}}{2 \, b} \]
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Time = 1.71 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{3 x^{\frac {3}{2}}} + 2 B \sqrt {x}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} + 2 B \sqrt {x}}{b} & \text {for}\: a = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {5}{2}}}{5}}{a} & \text {for}\: b = 0 \\- \frac {A \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {A \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {A \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a} + \frac {2 B \sqrt {x}}{b} + \frac {B \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {B \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {B \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\frac {2 \, B \sqrt {x}}{b} - \frac {\frac {2 \, \sqrt {2} {\left (B a - A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (B a - A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (B a - A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B a - A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{4 \, b} \]
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Time = 0.28 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\frac {2 \, B \sqrt {x}}{b} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a b^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a b^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} B a - \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{4 \, a b^{2}} \]
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Time = 5.29 (sec) , antiderivative size = 739, normalized size of antiderivative = 3.14 \[ \int \frac {A+B x^2}{\sqrt {x} \left (a+b x^2\right )} \, dx=\frac {2\,B\,\sqrt {x}}{b}-\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )-\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}+\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )+\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}}{\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )-\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}-\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )+\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}}\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{3/4}\,b^{5/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )-\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}+\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )+\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}}{\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )-\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}-\frac {\left (A\,b-B\,a\right )\,\left (\sqrt {x}\,\left (16\,A^2\,b^3-32\,A\,B\,a\,b^2+16\,B^2\,a^2\,b\right )+\frac {\left (32\,B\,a^2\,b^2-32\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{5/4}}}\right )\,\left (A\,b-B\,a\right )}{{\left (-a\right )}^{3/4}\,b^{5/4}} \]
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