Integrand size = 22, antiderivative size = 298 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}-\frac {(3 A b+5 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{9/4}}-\frac {(3 A b+5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}} \]
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Time = 0.16 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {468, 294, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {(5 a B+3 A b) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(5 a B+3 A b) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{7/4} b^{9/4}}-\frac {(5 a B+3 A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(5 a B+3 A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}-\frac {\sqrt {x} (5 a B+3 A b)}{16 a b^2 \left (a+b x^2\right )}+\frac {x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]
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Rule 210
Rule 217
Rule 294
Rule 335
Rule 468
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}+\frac {\left (\frac {3 A b}{2}+\frac {5 a B}{2}\right ) \int \frac {x^{3/2}}{\left (a+b x^2\right )^2} \, dx}{4 a b} \\ & = \frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 A b+5 a B) \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{32 a b^2} \\ & = \frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 A b+5 a B) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a b^2} \\ & = \frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 A b+5 a B) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{3/2} b^2}+\frac {(3 A b+5 a B) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^{3/2} b^2} \\ & = \frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 A b+5 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 )}{64 a^{3/2} b^{5/2}}+\frac {(3 A b+5 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 )}{64 a^{3/2} b^{5/2}}-\frac {(3 A b+5 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 )}{64 \sqrt {2} a^{7/4} b^{9/4}}-\frac {(3 A b+5 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 )}{64 \sqrt {2} a^{7/4} b^{9/4}} \\ & = \frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}-\frac {(3 A b+5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 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 )}{32 \sqrt {2} a^{7/4} b^{9/4}}-\frac {(3 A b+5 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 )}{32 \sqrt {2} a^{7/4} b^{9/4}} \\ & = \frac {(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(3 A b+5 a B) \sqrt {x}}{16 a b^2 \left (a+b x^2\right )}-\frac {(3 A b+5 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} b^{9/4}}-\frac {(3 A b+5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}}+\frac {(3 A b+5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{7/4} b^{9/4}} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.58 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (5 a^2 B-A b^2 x^2+3 a b \left (A+3 B x^2\right )\right )}{\left (a+b x^2\right )^2}-\sqrt {2} (3 A b+5 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt {2} (3 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{64 a^{7/4} b^{9/4}} \]
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Time = 2.77 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.56
method | result | size |
derivativedivides | \(\frac {\frac {\left (A b -9 B a \right ) x^{\frac {5}{2}}}{16 a b}-\frac {\left (3 A b +5 B a \right ) \sqrt {x}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (3 A b +5 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b^{2} a^{2}}\) | \(167\) |
default | \(\frac {\frac {\left (A b -9 B a \right ) x^{\frac {5}{2}}}{16 a b}-\frac {\left (3 A b +5 B a \right ) \sqrt {x}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (3 A b +5 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b^{2} a^{2}}\) | \(167\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 763, normalized size of antiderivative = 2.56 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (a^{2} b^{2} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + {\left (5 \, B a + 3 \, A b\right )} \sqrt {x}\right ) - {\left (-i \, a b^{4} x^{4} - 2 i \, a^{2} b^{3} x^{2} - i \, a^{3} b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (i \, a^{2} b^{2} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + {\left (5 \, B a + 3 \, A b\right )} \sqrt {x}\right ) - {\left (i \, a b^{4} x^{4} + 2 i \, a^{2} b^{3} x^{2} + i \, a^{3} b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (-i \, a^{2} b^{2} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + {\left (5 \, B a + 3 \, A b\right )} \sqrt {x}\right ) - {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} \log \left (-a^{2} b^{2} \left (-\frac {625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac {1}{4}} + {\left (5 \, B a + 3 \, A b\right )} \sqrt {x}\right ) - 4 \, {\left (5 \, B a^{2} + 3 \, A a b + {\left (9 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt {x}}{64 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1445 vs. \(2 (287) = 574\).
Time = 149.65 (sec) , antiderivative size = 1445, normalized size of antiderivative = 4.85 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \]
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Time = 0.29 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.94 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (9 \, B a b - A b^{2}\right )} x^{\frac {5}{2}} + {\left (5 \, B a^{2} + 3 \, A a b\right )} \sqrt {x}}{16 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (5 \, B a + 3 \, A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (5 \, B a + 3 \, A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (5 \, B a + 3 \, A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, B a + 3 \, A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{128 \, a b^{2}} \]
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Time = 0.33 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{3}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{3}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{3}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{3}} - \frac {9 \, B a b x^{\frac {5}{2}} - A b^{2} x^{\frac {5}{2}} + 5 \, B a^{2} \sqrt {x} + 3 \, A a b \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} a b^{2}} \]
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Time = 0.34 (sec) , antiderivative size = 799, normalized size of antiderivative = 2.68 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {\frac {\sqrt {x}\,\left (3\,A\,b+5\,B\,a\right )}{16\,b^2}-\frac {x^{5/2}\,\left (A\,b-9\,B\,a\right )}{16\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}-\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}+\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}+\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}}{\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}-\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}-\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}+\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}}\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{32\,{\left (-a\right )}^{7/4}\,b^{9/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}-\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}+\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}+\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}}{\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}-\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}-\frac {\left (3\,A\,b+5\,B\,a\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,b^2+30\,A\,B\,a\,b+25\,B^2\,a^2\right )}{64\,a^2\,b}+\frac {\left (3\,A\,b^2+5\,B\,a\,b\right )\,\left (3\,A\,b+5\,B\,a\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-a\right )}^{7/4}\,b^{9/4}}}\right )\,\left (3\,A\,b+5\,B\,a\right )}{32\,{\left (-a\right )}^{7/4}\,b^{9/4}} \]
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