Integrand size = 30, antiderivative size = 460 \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {5 (d x)^{3/2}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 \sqrt {d} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 \sqrt {d} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.22 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1126, 296, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {5 (d x)^{3/2}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 \sqrt {d} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 \sqrt {d} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 210
Rule 296
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1126
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 b \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{8 a \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {5 (d x)^{3/2}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{32 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {5 (d x)^{3/2}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {5 (d x)^{3/2}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (5 \left (a b+b^2 x^2\right )\right ) \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 )}{32 a^2 \sqrt {b} d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 \left (a b+b^2 x^2\right )\right ) \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 )}{32 a^2 \sqrt {b} d \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {5 (d x)^{3/2}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 \sqrt {d} \left (a b+b^2 x^2\right )\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 )}{64 \sqrt {2} a^{9/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 \sqrt {d} \left (a b+b^2 x^2\right )\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 )}{64 \sqrt {2} a^{9/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 d \left (a b+b^2 x^2\right )\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 )}{64 a^2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 d \left (a b+b^2 x^2\right )\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 )}{64 a^2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {5 (d x)^{3/2}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 \sqrt {d} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 \sqrt {d} \left (a b+b^2 x^2\right )\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 )}{32 \sqrt {2} a^{9/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (5 \sqrt {d} \left (a b+b^2 x^2\right )\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 )}{32 \sqrt {2} a^{9/4} b^{7/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {5 (d x)^{3/2}}{16 a^2 d \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(d x)^{3/2}}{4 a d \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 \sqrt {d} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \sqrt {d} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 \sqrt {d} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{9/4} b^{3/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {\sqrt {d x} \left (4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (9 a+5 b x^2\right )-5 \sqrt {2} \left (a+b x^2\right )^2 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-5 \sqrt {2} \left (a+b x^2\right )^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{64 a^{9/4} b^{3/4} \sqrt {x} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(616\) vs. \(2(294)=588\).
Time = 0.04 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.34
method | result | size |
default | \(\frac {\left (5 \sqrt {2}\, \ln \left (-\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\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 ) b^{2} d^{2} x^{4}+10 \sqrt {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 ) b^{2} d^{2} x^{4}+10 \sqrt {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 ) b^{2} d^{2} x^{4}+40 \left (d x \right )^{\frac {3}{2}} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{2} x^{2}+10 \sqrt {2}\, \ln \left (-\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\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 ) a b \,d^{2} x^{2}+20 \sqrt {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 ) a b \,d^{2} x^{2}+20 \sqrt {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 ) a b \,d^{2} x^{2}+72 \left (d x \right )^{\frac {3}{2}} a b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}+5 \sqrt {2}\, \ln \left (-\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\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 ) a^{2} d^{2}+10 \sqrt {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 ) a^{2} d^{2}+10 \sqrt {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 ) a^{2} d^{2}\right ) \left (b \,x^{2}+a \right )}{128 d \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b \,a^{2} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}\) | \(617\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {5 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \left (-\frac {d^{2}}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (125 \, a^{7} b^{2} \left (-\frac {d^{2}}{a^{9} b^{3}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} d\right ) - 5 \, {\left (i \, a^{2} b^{2} x^{4} + 2 i \, a^{3} b x^{2} + i \, a^{4}\right )} \left (-\frac {d^{2}}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (125 i \, a^{7} b^{2} \left (-\frac {d^{2}}{a^{9} b^{3}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} d\right ) - 5 \, {\left (-i \, a^{2} b^{2} x^{4} - 2 i \, a^{3} b x^{2} - i \, a^{4}\right )} \left (-\frac {d^{2}}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (-125 i \, a^{7} b^{2} \left (-\frac {d^{2}}{a^{9} b^{3}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} d\right ) - 5 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \left (-\frac {d^{2}}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (-125 \, a^{7} b^{2} \left (-\frac {d^{2}}{a^{9} b^{3}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} d\right ) + 4 \, {\left (5 \, b x^{3} + 9 \, a x\right )} \sqrt {d x}}{64 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} \]
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\[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {d x}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {\sqrt {d} x^{\frac {3}{2}}}{2 \, {\left (a^{2} b x^{2} + a^{3} + {\left (a b^{2} x^{2} + a^{2} b\right )} x^{2}\right )}} + \frac {5 \, b \sqrt {d} x^{\frac {7}{2}} + a \sqrt {d} x^{\frac {3}{2}}}{16 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} + \frac {5 \, \sqrt {d} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, a^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {\frac {8 \, {\left (5 \, \sqrt {d x} b d^{5} x^{3} + 9 \, \sqrt {d x} a d^{5} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{3} \mathrm {sgn}\left (b x^{2} + a\right )}}{128 \, d} \]
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Timed out. \[ \int \frac {\sqrt {d x}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {d\,x}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \]
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