Integrand size = 28, antiderivative size = 387 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}-\frac {4389 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}} \]
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Time = 0.29 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 296, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {4389 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5} \]
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Rule 28
Rule 210
Rule 217
Rule 296
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = b^6 \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^6} \, dx \\ & = \frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {\left (19 b^5\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^5} \, dx}{20 a} \\ & = \frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {\left (57 b^4\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^4} \, dx}{64 a^2} \\ & = \frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {\left (209 b^3\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx}{256 a^3} \\ & = \frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {\left (1463 b^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a^4} \\ & = \frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(4389 b) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a^5} \\ & = \frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(4389 b) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^5 d} \\ & = \frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(4389 b) \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 )}{8192 a^{11/2} d^2}+\frac {(4389 b) \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 )}{8192 a^{11/2} d^2} \\ & = \frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {4389 \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 )}{16384 a^{11/2} \sqrt {b}}+\frac {4389 \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 )}{16384 a^{11/2} \sqrt {b}}-\frac {4389 \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 )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \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 )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}} \\ & = \frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}-\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \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 )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \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 )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}} \\ & = \frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}-\frac {4389 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\sqrt {x} \left (\frac {4 a^{3/4} \sqrt {x} \left (19015 a^4+50312 a^3 b x^2+59470 a^2 b^2 x^4+33440 a b^3 x^6+7315 b^4 x^8\right )}{\left (a+b x^2\right )^5}-\frac {21945 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b}}+\frac {21945 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b}}\right )}{81920 a^{23/4} \sqrt {d x}} \]
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Time = 0.40 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(\frac {\left (58520 a \,x^{8} b^{4}+267520 a^{2} x^{6} b^{3}+475760 a^{3} x^{4} b^{2}+402496 x^{2} a^{4} b +152120 a^{5}\right ) \sqrt {d x}+21945 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )}{163840 d \,a^{6} \left (b \,x^{2}+a \right )^{5}}\) | \(237\) |
derivativedivides | \(2 d^{11} \left (\frac {\frac {3803 \sqrt {d x}}{8192 a \,d^{2}}+\frac {6289 b \left (d x \right )^{\frac {5}{2}}}{5120 a^{2} d^{4}}+\frac {5947 b^{2} \left (d x \right )^{\frac {9}{2}}}{4096 a^{3} d^{6}}+\frac {209 b^{3} \left (d x \right )^{\frac {13}{2}}}{256 a^{4} d^{8}}+\frac {1463 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192 a^{5} d^{10}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {4389 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{6} d^{12}}\right )\) | \(241\) |
default | \(2 d^{11} \left (\frac {\frac {3803 \sqrt {d x}}{8192 a \,d^{2}}+\frac {6289 b \left (d x \right )^{\frac {5}{2}}}{5120 a^{2} d^{4}}+\frac {5947 b^{2} \left (d x \right )^{\frac {9}{2}}}{4096 a^{3} d^{6}}+\frac {209 b^{3} \left (d x \right )^{\frac {13}{2}}}{256 a^{4} d^{8}}+\frac {1463 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192 a^{5} d^{10}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {4389 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{6} d^{12}}\right )\) | \(241\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {21945 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \log \left (a^{6} d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 21945 \, {\left (-i \, a^{5} b^{5} d x^{10} - 5 i \, a^{6} b^{4} d x^{8} - 10 i \, a^{7} b^{3} d x^{6} - 10 i \, a^{8} b^{2} d x^{4} - 5 i \, a^{9} b d x^{2} - i \, a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \log \left (i \, a^{6} d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 21945 \, {\left (i \, a^{5} b^{5} d x^{10} + 5 i \, a^{6} b^{4} d x^{8} + 10 i \, a^{7} b^{3} d x^{6} + 10 i \, a^{8} b^{2} d x^{4} + 5 i \, a^{9} b d x^{2} + i \, a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \log \left (-i \, a^{6} d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 21945 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \log \left (-a^{6} d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) + 4 \, {\left (7315 \, b^{4} x^{8} + 33440 \, a b^{3} x^{6} + 59470 \, a^{2} b^{2} x^{4} + 50312 \, a^{3} b x^{2} + 19015 \, a^{4}\right )} \sqrt {d x}}{81920 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )}} \]
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\[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {1}{\sqrt {d x} \left (a + b x^{2}\right )^{6}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (7315 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{2} + 33440 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{4} + 59470 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{6} + 50312 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{8} + 19015 \, \sqrt {d x} a^{4} d^{10}\right )}}{a^{5} b^{5} d^{10} x^{10} + 5 \, a^{6} b^{4} d^{10} x^{8} + 10 \, a^{7} b^{3} d^{10} x^{6} + 10 \, a^{8} b^{2} d^{10} x^{4} + 5 \, a^{9} b d^{10} x^{2} + a^{10} d^{10}} + \frac {21945 \, {\left (\frac {\sqrt {2} d^{2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a^{5}}}{163840 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {4389 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{16384 \, a^{6} b d} + \frac {4389 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{16384 \, a^{6} b d} + \frac {4389 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{32768 \, a^{6} b d} - \frac {4389 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{32768 \, a^{6} b d} + \frac {7315 \, \sqrt {d x} b^{4} d^{9} x^{8} + 33440 \, \sqrt {d x} a b^{3} d^{9} x^{6} + 59470 \, \sqrt {d x} a^{2} b^{2} d^{9} x^{4} + 50312 \, \sqrt {d x} a^{3} b d^{9} x^{2} + 19015 \, \sqrt {d x} a^{4} d^{9}}{20480 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{5}} \]
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Time = 14.19 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.54 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {3803\,d^9\,\sqrt {d\,x}}{4096\,a}+\frac {5947\,b^2\,d^5\,{\left (d\,x\right )}^{9/2}}{2048\,a^3}+\frac {209\,b^3\,d^3\,{\left (d\,x\right )}^{13/2}}{128\,a^4}+\frac {6289\,b\,d^7\,{\left (d\,x\right )}^{5/2}}{2560\,a^2}+\frac {1463\,b^4\,d\,{\left (d\,x\right )}^{17/2}}{4096\,a^5}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}+\frac {4389\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{23/4}\,b^{1/4}\,\sqrt {d}}+\frac {4389\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{23/4}\,b^{1/4}\,\sqrt {d}} \]
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