Integrand size = 28, antiderivative size = 335 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}+\frac {d \sqrt {d x}}{48 a b \left (a+b x^2\right )^2}+\frac {7 d \sqrt {d x}}{192 a^2 b \left (a+b x^2\right )}-\frac {7 d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}+\frac {7 d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}-\frac {7 d^{3/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{11/4} b^{5/4}}+\frac {7 d^{3/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{11/4} b^{5/4}} \]
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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, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {7 d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}+\frac {7 d^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}-\frac {7 d^{3/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{11/4} b^{5/4}}+\frac {7 d^{3/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} a^{11/4} b^{5/4}}+\frac {7 d \sqrt {d x}}{192 a^2 b \left (a+b x^2\right )}+\frac {d \sqrt {d x}}{48 a b \left (a+b x^2\right )^2}-\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3} \]
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Rule 28
Rule 210
Rule 217
Rule 294
Rule 296
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = b^4 \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^4} \, dx \\ & = -\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}+\frac {1}{12} \left (b^2 d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx \\ & = -\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}+\frac {d \sqrt {d x}}{48 a b \left (a+b x^2\right )^2}+\frac {\left (7 b d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{96 a} \\ & = -\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}+\frac {d \sqrt {d x}}{48 a b \left (a+b x^2\right )^2}+\frac {7 d \sqrt {d x}}{192 a^2 b \left (a+b x^2\right )}+\frac {\left (7 d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{128 a^2} \\ & = -\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}+\frac {d \sqrt {d x}}{48 a b \left (a+b x^2\right )^2}+\frac {7 d \sqrt {d x}}{192 a^2 b \left (a+b x^2\right )}+\frac {(7 d) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 a^2} \\ & = -\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}+\frac {d \sqrt {d x}}{48 a b \left (a+b x^2\right )^2}+\frac {7 d \sqrt {d x}}{192 a^2 b \left (a+b x^2\right )}+\frac {7 \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^{5/2}}+\frac {7 \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^{5/2}} \\ & = -\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}+\frac {d \sqrt {d x}}{48 a b \left (a+b x^2\right )^2}+\frac {7 d \sqrt {d x}}{192 a^2 b \left (a+b x^2\right )}-\frac {\left (7 d^{3/2}\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^{11/4} b^{5/4}}-\frac {\left (7 d^{3/2}\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^{11/4} b^{5/4}}+\frac {\left (7 d^2\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^{5/2} b^{3/2}}+\frac {\left (7 d^2\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^{5/2} b^{3/2}} \\ & = -\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}+\frac {d \sqrt {d x}}{48 a b \left (a+b x^2\right )^2}+\frac {7 d \sqrt {d x}}{192 a^2 b \left (a+b x^2\right )}-\frac {7 d^{3/2} \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^{11/4} b^{5/4}}+\frac {7 d^{3/2} \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^{11/4} b^{5/4}}+\frac {\left (7 d^{3/2}\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^{11/4} b^{5/4}}-\frac {\left (7 d^{3/2}\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^{11/4} b^{5/4}} \\ & = -\frac {d \sqrt {d x}}{6 b \left (a+b x^2\right )^3}+\frac {d \sqrt {d x}}{48 a b \left (a+b x^2\right )^2}+\frac {7 d \sqrt {d x}}{192 a^2 b \left (a+b x^2\right )}-\frac {7 d^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}+\frac {7 d^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{11/4} b^{5/4}}-\frac {7 d^{3/2} \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^{11/4} b^{5/4}}+\frac {7 d^{3/2} \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^{11/4} b^{5/4}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.48 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {(d x)^{3/2} \left (\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-21 a^2+18 a b x^2+7 b^2 x^4\right )}{\left (a+b x^2\right )^3}-21 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+21 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{768 a^{11/4} b^{5/4} x^{3/2}} \]
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Time = 0.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.61
method | result | size |
derivativedivides | \(2 d^{7} \left (\frac {\frac {7 b \left (d x \right )^{\frac {9}{2}}}{384 a^{2} d^{4}}+\frac {3 \left (d x \right )^{\frac {5}{2}}}{64 a \,d^{2}}-\frac {7 \sqrt {d x}}{128 b}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {7 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a^{3} d^{6} b}\right )\) | \(206\) |
default | \(2 d^{7} \left (\frac {\frac {7 b \left (d x \right )^{\frac {9}{2}}}{384 a^{2} d^{4}}+\frac {3 \left (d x \right )^{\frac {5}{2}}}{64 a \,d^{2}}-\frac {7 \sqrt {d x}}{128 b}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {7 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a^{3} d^{6} b}\right )\) | \(206\) |
pseudoelliptic | \(\frac {7 d \left (\frac {\sqrt {2}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{3} \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{2}+\sqrt {2}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+\sqrt {2}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-4 a \sqrt {d x}\, \left (-\frac {1}{3} b^{2} x^{4}-\frac {6}{7} a b \,x^{2}+a^{2}\right )\right )}{256 a^{3} b \left (b \,x^{2}+a \right )^{3}}\) | \(254\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.21 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {21 \, {\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \left (-\frac {d^{6}}{a^{11} b^{5}}\right )^{\frac {1}{4}} \log \left (7 \, a^{3} b \left (-\frac {d^{6}}{a^{11} b^{5}}\right )^{\frac {1}{4}} + 7 \, \sqrt {d x} d\right ) - 21 \, {\left (-i \, a^{2} b^{4} x^{6} - 3 i \, a^{3} b^{3} x^{4} - 3 i \, a^{4} b^{2} x^{2} - i \, a^{5} b\right )} \left (-\frac {d^{6}}{a^{11} b^{5}}\right )^{\frac {1}{4}} \log \left (7 i \, a^{3} b \left (-\frac {d^{6}}{a^{11} b^{5}}\right )^{\frac {1}{4}} + 7 \, \sqrt {d x} d\right ) - 21 \, {\left (i \, a^{2} b^{4} x^{6} + 3 i \, a^{3} b^{3} x^{4} + 3 i \, a^{4} b^{2} x^{2} + i \, a^{5} b\right )} \left (-\frac {d^{6}}{a^{11} b^{5}}\right )^{\frac {1}{4}} \log \left (-7 i \, a^{3} b \left (-\frac {d^{6}}{a^{11} b^{5}}\right )^{\frac {1}{4}} + 7 \, \sqrt {d x} d\right ) - 21 \, {\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )} \left (-\frac {d^{6}}{a^{11} b^{5}}\right )^{\frac {1}{4}} \log \left (-7 \, a^{3} b \left (-\frac {d^{6}}{a^{11} b^{5}}\right )^{\frac {1}{4}} + 7 \, \sqrt {d x} d\right ) + 4 \, {\left (7 \, b^{2} d x^{4} + 18 \, a b d x^{2} - 21 \, a^{2} d\right )} \sqrt {d x}}{768 \, {\left (a^{2} b^{4} x^{6} + 3 \, a^{3} b^{3} x^{4} + 3 \, a^{4} b^{2} x^{2} + a^{5} b\right )}} \]
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\[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\int \frac {\left (d x\right )^{\frac {3}{2}}}{\left (a + b x^{2}\right )^{4}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.99 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {8 \, {\left (7 \, \left (d x\right )^{\frac {9}{2}} b^{2} d^{4} + 18 \, \left (d x\right )^{\frac {5}{2}} a b d^{6} - 21 \, \sqrt {d x} a^{2} d^{8}\right )}}{a^{2} b^{4} d^{6} x^{6} + 3 \, a^{3} b^{3} d^{6} x^{4} + 3 \, a^{4} b^{2} d^{6} x^{2} + a^{5} b d^{6}} + \frac {21 \, {\left (\frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a^{2} b}}{1536 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.90 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {1}{1536} \, d {\left (\frac {42 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{2}} + \frac {42 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{2}} + \frac {21 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{2}} - \frac {21 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{2}} + \frac {8 \, {\left (7 \, \sqrt {d x} b^{2} d^{6} x^{4} + 18 \, \sqrt {d x} a b d^{6} x^{2} - 21 \, \sqrt {d x} a^{2} d^{6}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{2} b}\right )} \]
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Time = 13.17 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.44 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {3\,d^5\,{\left (d\,x\right )}^{5/2}}{32\,a}-\frac {7\,d^7\,\sqrt {d\,x}}{64\,b}+\frac {7\,b\,d^3\,{\left (d\,x\right )}^{9/2}}{192\,a^2}}{a^3\,d^6+3\,a^2\,b\,d^6\,x^2+3\,a\,b^2\,d^6\,x^4+b^3\,d^6\,x^6}-\frac {7\,d^{3/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{11/4}\,b^{5/4}}-\frac {7\,d^{3/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{11/4}\,b^{5/4}} \]
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