Integrand size = 28, antiderivative size = 350 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac {195 \sqrt [4]{a} d^{15/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{17/4}}+\frac {195 \sqrt [4]{a} d^{15/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} b^{17/4}} \]
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Time = 0.24 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 294, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {195 \sqrt [4]{a} d^{15/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} b^{17/4}}+\frac {195 \sqrt [4]{a} d^{15/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} b^{17/4}}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}+\frac {195 d^7 \sqrt {d x}}{64 b^4} \]
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Rule 28
Rule 210
Rule 217
Rule 294
Rule 327
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = b^4 \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^4} \, dx \\ & = -\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}+\frac {1}{12} \left (13 b^2 d^2\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^3} \, dx \\ & = -\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac {1}{32} \left (39 d^4\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^2} \, dx \\ & = -\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac {\left (195 d^6\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{128 b^2} \\ & = \frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac {\left (195 a d^8\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{128 b^3} \\ & = \frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac {\left (195 a d^7\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 b^3} \\ & = \frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}-\frac {\left (195 \sqrt {a} d^6\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 )}{128 b^3}-\frac {\left (195 \sqrt {a} d^6\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 )}{128 b^3} \\ & = \frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac {\left (195 \sqrt [4]{a} d^{15/2}\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} b^{17/4}}+\frac {\left (195 \sqrt [4]{a} d^{15/2}\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} b^{17/4}}-\frac {\left (195 \sqrt {a} d^8\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 b^{9/2}}-\frac {\left (195 \sqrt {a} d^8\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 b^{9/2}} \\ & = \frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac {195 \sqrt [4]{a} d^{15/2} \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} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \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} b^{17/4}}-\frac {\left (195 \sqrt [4]{a} d^{15/2}\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} b^{17/4}}+\frac {\left (195 \sqrt [4]{a} d^{15/2}\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} b^{17/4}} \\ & = \frac {195 d^7 \sqrt {d x}}{64 b^4}-\frac {d (d x)^{13/2}}{6 b \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{64 b^3 \left (a+b x^2\right )}+\frac {195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{17/4}}+\frac {195 \sqrt [4]{a} d^{15/2} \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} b^{17/4}}-\frac {195 \sqrt [4]{a} d^{15/2} \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} b^{17/4}} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.50 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {d^7 \sqrt {d x} \left (\frac {4 \sqrt [4]{b} \sqrt {x} \left (585 a^3+1638 a^2 b x^2+1469 a b^2 x^4+384 b^3 x^6\right )}{\left (a+b x^2\right )^3}+585 \sqrt {2} \sqrt [4]{a} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-585 \sqrt {2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{768 b^{17/4} \sqrt {x}} \]
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Time = 2.35 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(2 d^{7} \left (\frac {\sqrt {d x}}{b^{4}}-\frac {a \,d^{2} \left (\frac {-\frac {317 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}-\frac {81 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}-\frac {67 a^{2} d^{4} \sqrt {d x}}{128}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {195 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{b^{4}}\right )\) | \(220\) |
default | \(2 d^{7} \left (\frac {\sqrt {d x}}{b^{4}}-\frac {a \,d^{2} \left (\frac {-\frac {317 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}-\frac {81 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}-\frac {67 a^{2} d^{4} \sqrt {d x}}{128}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {195 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{b^{4}}\right )\) | \(220\) |
risch | \(\frac {2 x \,d^{8}}{b^{4} \sqrt {d x}}-\frac {2 a \,d^{9} \left (\frac {-\frac {317 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}-\frac {81 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}-\frac {67 a^{2} d^{4} \sqrt {d x}}{128}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {195 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{b^{4}}\) | \(220\) |
pseudoelliptic | \(-\frac {d^{7} \left (\left (-3072 b^{3} x^{6}-11752 b^{2} x^{4} a -13104 a^{2} b \,x^{2}-4680 a^{3}\right ) \sqrt {d x}+585 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{3} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right )}{1536 \left (b \,x^{2}+a \right )^{3} b^{4}}\) | \(223\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.14 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {585 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \log \left (195 \, \sqrt {d x} d^{7} + 195 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) + 585 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (i \, b^{7} x^{6} + 3 i \, a b^{6} x^{4} + 3 i \, a^{2} b^{5} x^{2} + i \, a^{3} b^{4}\right )} \log \left (195 \, \sqrt {d x} d^{7} + 195 i \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) + 585 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (-i \, b^{7} x^{6} - 3 i \, a b^{6} x^{4} - 3 i \, a^{2} b^{5} x^{2} - i \, a^{3} b^{4}\right )} \log \left (195 \, \sqrt {d x} d^{7} - 195 i \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 585 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )} \log \left (195 \, \sqrt {d x} d^{7} - 195 \, \left (-\frac {a d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 4 \, {\left (384 \, b^{3} d^{7} x^{6} + 1469 \, a b^{2} d^{7} x^{4} + 1638 \, a^{2} b d^{7} x^{2} + 585 \, a^{3} d^{7}\right )} \sqrt {d x}}{768 \, {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}} \]
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\[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\int \frac {\left (d x\right )^{\frac {15}{2}}}{\left (a + b x^{2}\right )^{4}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.98 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {3072 \, \sqrt {d x} d^{8}}{b^{4}} + \frac {8 \, {\left (317 \, \left (d x\right )^{\frac {9}{2}} a b^{2} d^{10} + 486 \, \left (d x\right )^{\frac {5}{2}} a^{2} b d^{12} + 201 \, \sqrt {d x} a^{3} d^{14}\right )}}{b^{7} d^{6} x^{6} + 3 \, a b^{6} d^{6} x^{4} + 3 \, a^{2} b^{5} d^{6} x^{2} + a^{3} b^{4} d^{6}} - \frac {585 \, {\left (\frac {\sqrt {2} d^{10} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{10} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{9} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{9} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )} a}{b^{4}}}{1536 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.86 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {1}{1536} \, d^{7} {\left (\frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{5}} + \frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{5}} + \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{5}} - \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{5}} - \frac {3072 \, \sqrt {d x}}{b^{4}} - \frac {8 \, {\left (317 \, \sqrt {d x} a b^{2} d^{6} x^{4} + 486 \, \sqrt {d x} a^{2} b d^{6} x^{2} + 201 \, \sqrt {d x} a^{3} d^{6}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{4}}\right )} \]
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Time = 13.99 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.49 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {67\,a^3\,d^{13}\,\sqrt {d\,x}}{64}+\frac {81\,a^2\,b\,d^{11}\,{\left (d\,x\right )}^{5/2}}{32}+\frac {317\,a\,b^2\,d^9\,{\left (d\,x\right )}^{9/2}}{192}}{a^3\,b^4\,d^6+3\,a^2\,b^5\,d^6\,x^2+3\,a\,b^6\,d^6\,x^4+b^7\,d^6\,x^6}+\frac {2\,d^7\,\sqrt {d\,x}}{b^4}-\frac {195\,{\left (-a\right )}^{1/4}\,d^{15/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,b^{17/4}}+\frac {{\left (-a\right )}^{1/4}\,d^{15/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,195{}\mathrm {i}}{128\,b^{17/4}} \]
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