Integrand size = 28, antiderivative size = 335 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}-\frac {5 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d^{5/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{9/4} b^{7/4}}-\frac {5 d^{5/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{9/4} b^{7/4}} \]
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Time = 0.22 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 294, 296, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {5 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d^{5/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{9/4} b^{7/4}}-\frac {5 d^{5/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}-\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3} \]
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Rule 28
Rule 210
Rule 294
Rule 296
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = b^4 \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^4} \, dx \\ & = -\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {1}{4} \left (b^2 d^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^3} \, dx \\ & = -\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {\left (5 b d^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{32 a} \\ & = -\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}+\frac {\left (5 d^2\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{128 a^2} \\ & = -\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}+\frac {(5 d) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 a^2} \\ & = -\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}-\frac {(5 d) \text {Subst}\left (\int \frac {\sqrt {a} d-\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{128 a^2 \sqrt {b}}+\frac {(5 d) \text {Subst}\left (\int \frac {\sqrt {a} d+\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{128 a^2 \sqrt {b}} \\ & = -\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}+\frac {\left (5 d^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{256 \sqrt {2} a^{9/4} b^{7/4}}+\frac {\left (5 d^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{256 \sqrt {2} a^{9/4} b^{7/4}}+\frac {\left (5 d^3\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{256 a^2 b^2}+\frac {\left (5 d^3\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{256 a^2 b^2} \\ & = -\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}+\frac {5 d^{5/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{9/4} b^{7/4}}-\frac {5 d^{5/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{9/4} b^{7/4}}+\frac {\left (5 d^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}-\frac {\left (5 d^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{9/4} b^{7/4}} \\ & = -\frac {d (d x)^{3/2}}{6 b \left (a+b x^2\right )^3}+\frac {d (d x)^{3/2}}{16 a b \left (a+b x^2\right )^2}+\frac {5 d (d x)^{3/2}}{64 a^2 b \left (a+b x^2\right )}-\frac {5 d^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {5 d^{5/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{9/4} b^{7/4}}-\frac {5 d^{5/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{9/4} b^{7/4}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.48 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {(d x)^{5/2} \left (\frac {4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (-5 a^2+42 a b x^2+15 b^2 x^4\right )}{\left (a+b x^2\right )^3}-15 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-15 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{768 a^{9/4} b^{7/4} x^{5/2}} \]
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Time = 2.24 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.61
method | result | size |
derivativedivides | \(2 d^{7} \left (\frac {\frac {5 b \left (d x \right )^{\frac {11}{2}}}{128 a^{2} d^{4}}+\frac {7 \left (d x \right )^{\frac {7}{2}}}{64 a \,d^{2}}-\frac {5 \left (d x \right )^{\frac {3}{2}}}{384 b}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a^{2} d^{4} b^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(206\) |
default | \(2 d^{7} \left (\frac {\frac {5 b \left (d x \right )^{\frac {11}{2}}}{128 a^{2} d^{4}}+\frac {7 \left (d x \right )^{\frac {7}{2}}}{64 a \,d^{2}}-\frac {5 \left (d x \right )^{\frac {3}{2}}}{384 b}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a^{2} d^{4} b^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(206\) |
pseudoelliptic | \(-\frac {5 d^{2} \left (8 x b \sqrt {d x}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (-3 b^{2} x^{4}-\frac {42}{5} a b \,x^{2}+a^{2}\right )-3 \sqrt {2}\, d \left (b \,x^{2}+a \right )^{3} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )\right )\right )}{1536 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{2} a^{2} \left (b \,x^{2}+a \right )^{3}}\) | \(227\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.28 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {15 \, {\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {1}{4}} \log \left (125 \, a^{7} b^{5} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} d^{7}\right ) - 15 \, {\left (i \, a^{2} b^{4} x^{6} + 3 i \, a^{3} b^{3} x^{4} + 3 i \, a^{4} b^{2} x^{2} + i \, a^{5} b\right )} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {1}{4}} \log \left (125 i \, a^{7} b^{5} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} d^{7}\right ) - 15 \, {\left (-i \, a^{2} b^{4} x^{6} - 3 i \, a^{3} b^{3} x^{4} - 3 i \, a^{4} b^{2} x^{2} - i \, a^{5} b\right )} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {1}{4}} \log \left (-125 i \, a^{7} b^{5} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} d^{7}\right ) - 15 \, {\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {1}{4}} \log \left (-125 \, a^{7} b^{5} \left (-\frac {d^{10}}{a^{9} b^{7}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} d^{7}\right ) + 4 \, {\left (15 \, b^{2} d^{2} x^{5} + 42 \, a b d^{2} x^{3} - 5 \, a^{2} d^{2} x\right )} \sqrt {d x}}{768 \, {\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )}} \]
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\[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\int \frac {\left (d x\right )^{\frac {5}{2}}}{\left (a + b x^{2}\right )^{4}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.96 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {15 \, d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{2} b} + \frac {8 \, {\left (15 \, \left (d x\right )^{\frac {11}{2}} b^{2} d^{4} + 42 \, \left (d x\right )^{\frac {7}{2}} a b d^{6} - 5 \, \left (d x\right )^{\frac {3}{2}} a^{2} d^{8}\right )}}{a^{2} b^{4} d^{6} x^{6} + 3 \, a^{3} b^{3} d^{6} x^{4} + 3 \, a^{4} b^{2} d^{6} x^{2} + a^{5} b d^{6}}}{1536 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.95 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {1}{1536} \, d^{2} {\left (\frac {8 \, {\left (15 \, \sqrt {d x} b^{2} d^{6} x^{5} + 42 \, \sqrt {d x} a b d^{6} x^{3} - 5 \, \sqrt {d x} a^{2} d^{6} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{2} b} + \frac {30 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{4} d} + \frac {30 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{4} d} - \frac {15 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{4} d} + \frac {15 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{4} d}\right )} \]
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Time = 13.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.44 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {7\,d^5\,{\left (d\,x\right )}^{7/2}}{32\,a}-\frac {5\,d^7\,{\left (d\,x\right )}^{3/2}}{192\,b}+\frac {5\,b\,d^3\,{\left (d\,x\right )}^{11/2}}{64\,a^2}}{a^3\,d^6+3\,a^2\,b\,d^6\,x^2+3\,a\,b^2\,d^6\,x^4+b^3\,d^6\,x^6}+\frac {5\,d^{5/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{9/4}\,b^{7/4}}-\frac {5\,d^{5/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{9/4}\,b^{7/4}} \]
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