Integrand size = 22, antiderivative size = 310 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=-\frac {9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac {9 A b-5 a B}{2 a^3 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {\sqrt [4]{b} (9 A b-5 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (9 A b-5 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (9 A b-5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (9 A b-5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}} \]
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Time = 0.18 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {468, 331, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=-\frac {\sqrt [4]{b} (9 A b-5 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (9 A b-5 a B) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (9 A b-5 a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (9 A b-5 a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}+\frac {9 A b-5 a B}{2 a^3 \sqrt {x}}-\frac {9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )} \]
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Rule 210
Rule 303
Rule 331
Rule 335
Rule 468
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac {\left (\frac {9 A b}{2}-\frac {5 a B}{2}\right ) \int \frac {1}{x^{7/2} \left (a+b x^2\right )} \, dx}{2 a b} \\ & = -\frac {9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {(9 A b-5 a B) \int \frac {1}{x^{3/2} \left (a+b x^2\right )} \, dx}{4 a^2} \\ & = -\frac {9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac {9 A b-5 a B}{2 a^3 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac {(b (9 A b-5 a B)) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{4 a^3} \\ & = -\frac {9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac {9 A b-5 a B}{2 a^3 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac {(b (9 A b-5 a B)) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^3} \\ & = -\frac {9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac {9 A b-5 a B}{2 a^3 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {\left (\sqrt {b} (9 A b-5 a B)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^3}+\frac {\left (\sqrt {b} (9 A b-5 a B)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^3} \\ & = -\frac {9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac {9 A b-5 a B}{2 a^3 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac {(9 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 )}{8 a^3}+\frac {(9 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 )}{8 a^3}+\frac {\left (\sqrt [4]{b} (9 A b-5 a B)\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^{13/4}}+\frac {\left (\sqrt [4]{b} (9 A b-5 a B)\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^{13/4}} \\ & = -\frac {9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac {9 A b-5 a B}{2 a^3 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac {\sqrt [4]{b} (9 A b-5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (9 A b-5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}+\frac {\left (\sqrt [4]{b} (9 A b-5 a B)\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^{13/4}}-\frac {\left (\sqrt [4]{b} (9 A b-5 a B)\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^{13/4}} \\ & = -\frac {9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac {9 A b-5 a B}{2 a^3 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {\sqrt [4]{b} (9 A b-5 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (9 A b-5 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (9 A b-5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (9 A b-5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.60 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=\frac {-\frac {4 \sqrt [4]{a} \left (-45 A b^2 x^4+4 a^2 \left (A+5 B x^2\right )+a \left (-36 A b x^2+25 b B x^4\right )\right )}{x^{5/2} \left (a+b x^2\right )}+5 \sqrt {2} \sqrt [4]{b} (-9 A b+5 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+5 \sqrt {2} \sqrt [4]{b} (-9 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{40 a^{13/4}} \]
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Time = 2.80 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.55
method | result | size |
derivativedivides | \(-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 \left (-2 A b +B a \right )}{a^{3} \sqrt {x}}+\frac {2 b \left (\frac {\left (\frac {A b}{4}-\frac {B a}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {9 A b}{4}-\frac {5 B a}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}\) | \(170\) |
default | \(-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 \left (-2 A b +B a \right )}{a^{3} \sqrt {x}}+\frac {2 b \left (\frac {\left (\frac {A b}{4}-\frac {B a}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {9 A b}{4}-\frac {5 B a}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}\) | \(170\) |
risch | \(-\frac {2 \left (-10 A b \,x^{2}+5 B a \,x^{2}+A a \right )}{5 a^{3} x^{\frac {5}{2}}}+\frac {b \left (\frac {2 \left (\frac {A b}{4}-\frac {B a}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {9 A b}{4}-\frac {5 B a}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}\) | \(171\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 838, normalized size of antiderivative = 2.70 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=\frac {5 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {1}{4}} \log \left (a^{10} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {3}{4}} - {\left (125 \, B^{3} a^{3} b - 675 \, A B^{2} a^{2} b^{2} + 1215 \, A^{2} B a b^{3} - 729 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 5 \, {\left (i \, a^{3} b x^{5} + i \, a^{4} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {1}{4}} \log \left (i \, a^{10} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {3}{4}} - {\left (125 \, B^{3} a^{3} b - 675 \, A B^{2} a^{2} b^{2} + 1215 \, A^{2} B a b^{3} - 729 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 5 \, {\left (-i \, a^{3} b x^{5} - i \, a^{4} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {1}{4}} \log \left (-i \, a^{10} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {3}{4}} - {\left (125 \, B^{3} a^{3} b - 675 \, A B^{2} a^{2} b^{2} + 1215 \, A^{2} B a b^{3} - 729 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 5 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {1}{4}} \log \left (-a^{10} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {3}{4}} - {\left (125 \, B^{3} a^{3} b - 675 \, A B^{2} a^{2} b^{2} + 1215 \, A^{2} B a b^{3} - 729 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 4 \, {\left (5 \, {\left (5 \, B a b - 9 \, A b^{2}\right )} x^{4} + 4 \, A a^{2} + 4 \, {\left (5 \, B a^{2} - 9 \, A a b\right )} x^{2}\right )} \sqrt {x}}{40 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \]
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Timed out. \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=-\frac {5 \, {\left (5 \, B a b - 9 \, A b^{2}\right )} x^{4} + 4 \, A a^{2} + 4 \, {\left (5 \, B a^{2} - 9 \, A a b\right )} x^{2}}{10 \, {\left (a^{3} b x^{\frac {9}{2}} + a^{4} x^{\frac {5}{2}}\right )}} - \frac {{\left (5 \, B a b - 9 \, A b^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=-\frac {B a b x^{\frac {3}{2}} - A b^{2} x^{\frac {3}{2}}}{2 \, {\left (b x^{2} + a\right )} a^{3}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{4} b^{2}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{4} b^{2}} - \frac {2 \, {\left (5 \, B a x^{2} - 10 \, A b x^{2} + A a\right )}}{5 \, a^{3} x^{\frac {5}{2}}} \]
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Time = 5.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.39 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=\frac {\frac {2\,x^2\,\left (9\,A\,b-5\,B\,a\right )}{5\,a^2}-\frac {2\,A}{5\,a}+\frac {b\,x^4\,\left (9\,A\,b-5\,B\,a\right )}{2\,a^3}}{a\,x^{5/2}+b\,x^{9/2}}+\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )\,\left (9\,A\,b-5\,B\,a\right )}{4\,a^{13/4}}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )\,\left (9\,A\,b-5\,B\,a\right )}{4\,a^{13/4}} \]
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