Integrand size = 28, antiderivative size = 388 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}-\frac {117 d^{15/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 d^{15/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}-\frac {117 d^{15/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 d^{15/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{7/4} b^{17/4}} \]
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Time = 0.28 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 16, 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} \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {117 d^{15/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 d^{15/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}-\frac {117 d^{15/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 d^{15/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{7/4} b^{17/4}}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5} \]
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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^6 \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^6} \, dx \\ & = -\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (13 b^4 d^2\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^5} \, dx \\ & = -\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {1}{320} \left (117 b^2 d^4\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^4} \, dx \\ & = -\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}+\frac {1}{256} \left (39 d^6\right ) \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^3} \, dx \\ & = -\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {\left (39 d^8\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{2048 b^2} \\ & = -\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}+\frac {\left (117 d^8\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a b^3} \\ & = -\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}+\frac {\left (117 d^7\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a b^3} \\ & = -\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}+\frac {\left (117 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 )}{8192 a^{3/2} b^3}+\frac {\left (117 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 )}{8192 a^{3/2} b^3} \\ & = -\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}-\frac {\left (117 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 )}{16384 \sqrt {2} a^{7/4} b^{17/4}}-\frac {\left (117 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 )}{16384 \sqrt {2} a^{7/4} b^{17/4}}+\frac {\left (117 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 )}{16384 a^{3/2} b^{9/2}}+\frac {\left (117 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 )}{16384 a^{3/2} b^{9/2}} \\ & = -\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}-\frac {117 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 )}{16384 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 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 )}{16384 \sqrt {2} a^{7/4} b^{17/4}}+\frac {\left (117 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 )}{8192 \sqrt {2} a^{7/4} b^{17/4}}-\frac {\left (117 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 )}{8192 \sqrt {2} a^{7/4} b^{17/4}} \\ & = -\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}-\frac {117 d^{15/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 d^{15/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}-\frac {117 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 )}{16384 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 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 )}{16384 \sqrt {2} a^{7/4} b^{17/4}} \\ \end{align*}
Time = 0.75 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.48 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {d^7 \sqrt {d x} \left (-\frac {4 a^{3/4} \sqrt [4]{b} \left (585 a^4+2808 a^3 b x^2+5330 a^2 b^2 x^4+4960 a b^3 x^6-195 b^4 x^8\right )}{\left (a+b x^2\right )^5}-\frac {585 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt {x}}+\frac {585 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {x}}\right )}{81920 a^{7/4} b^{17/4}} \]
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Time = 19.94 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.61
method | result | size |
derivativedivides | \(2 d^{11} \left (\frac {-\frac {117 d^{6} a^{3} \sqrt {d x}}{8192 b^{4}}-\frac {351 d^{4} a^{2} \left (d x \right )^{\frac {5}{2}}}{5120 b^{3}}-\frac {533 d^{2} a \left (d x \right )^{\frac {9}{2}}}{4096 b^{2}}-\frac {31 \left (d x \right )^{\frac {13}{2}}}{256 b}+\frac {39 \left (d x \right )^{\frac {17}{2}}}{8192 a \,d^{2}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {117 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{2} d^{4} b^{4}}\right )\) | \(238\) |
default | \(2 d^{11} \left (\frac {-\frac {117 d^{6} a^{3} \sqrt {d x}}{8192 b^{4}}-\frac {351 d^{4} a^{2} \left (d x \right )^{\frac {5}{2}}}{5120 b^{3}}-\frac {533 d^{2} a \left (d x \right )^{\frac {9}{2}}}{4096 b^{2}}-\frac {31 \left (d x \right )^{\frac {13}{2}}}{256 b}+\frac {39 \left (d x \right )^{\frac {17}{2}}}{8192 a \,d^{2}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {117 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{2} d^{4} b^{4}}\right )\) | \(238\) |
pseudoelliptic | \(\frac {\left (\left (1560 a \,x^{8} b^{4}-39680 a^{2} x^{6} b^{3}-42640 a^{3} x^{4} b^{2}-22464 x^{2} a^{4} b -4680 a^{5}\right ) \sqrt {d x}+585 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right ) d^{7}}{163840 b^{4} a^{2} \left (b \,x^{2}+a \right )^{5}}\) | \(240\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.46 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {585 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} \log \left (117 \, \sqrt {d x} d^{7} + 117 \, \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} a^{2} b^{4}\right ) - 585 \, {\left (-i \, a b^{9} x^{10} - 5 i \, a^{2} b^{8} x^{8} - 10 i \, a^{3} b^{7} x^{6} - 10 i \, a^{4} b^{6} x^{4} - 5 i \, a^{5} b^{5} x^{2} - i \, a^{6} b^{4}\right )} \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} \log \left (117 \, \sqrt {d x} d^{7} + 117 i \, \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} a^{2} b^{4}\right ) - 585 \, {\left (i \, a b^{9} x^{10} + 5 i \, a^{2} b^{8} x^{8} + 10 i \, a^{3} b^{7} x^{6} + 10 i \, a^{4} b^{6} x^{4} + 5 i \, a^{5} b^{5} x^{2} + i \, a^{6} b^{4}\right )} \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} \log \left (117 \, \sqrt {d x} d^{7} - 117 i \, \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} a^{2} b^{4}\right ) - 585 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} \log \left (117 \, \sqrt {d x} d^{7} - 117 \, \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} a^{2} b^{4}\right ) + 4 \, {\left (195 \, b^{4} d^{7} x^{8} - 4960 \, a b^{3} d^{7} x^{6} - 5330 \, a^{2} b^{2} d^{7} x^{4} - 2808 \, a^{3} b d^{7} x^{2} - 585 \, a^{4} d^{7}\right )} \sqrt {d x}}{81920 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )}} \]
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Timed out. \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.01 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (195 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{10} - 4960 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{12} - 5330 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{14} - 2808 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{16} - 585 \, \sqrt {d x} a^{4} d^{18}\right )}}{a b^{9} d^{10} x^{10} + 5 \, a^{2} b^{8} d^{10} x^{8} + 10 \, a^{3} b^{7} d^{10} x^{6} + 10 \, a^{4} b^{6} d^{10} x^{4} + 5 \, a^{5} b^{5} d^{10} x^{2} + a^{6} b^{4} d^{10}} + \frac {585 \, {\left (\frac {\sqrt {2} d^{10} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{10} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{9} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{9} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a b^{4}}}{163840 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.88 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1}{163840} \, d^{7} {\left (\frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{5}} + \frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{5}} + \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{2} b^{5}} - \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{2} b^{5}} + \frac {8 \, {\left (195 \, \sqrt {d x} b^{4} d^{10} x^{8} - 4960 \, \sqrt {d x} a b^{3} d^{10} x^{6} - 5330 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{4} - 2808 \, \sqrt {d x} a^{3} b d^{10} x^{2} - 585 \, \sqrt {d x} a^{4} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a b^{4}}\right )} \]
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Time = 0.13 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.54 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {117\,d^{15/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{7/4}\,b^{17/4}}-\frac {\frac {31\,d^{11}\,{\left (d\,x\right )}^{13/2}}{128\,b}-\frac {39\,d^9\,{\left (d\,x\right )}^{17/2}}{4096\,a}+\frac {351\,a^2\,d^{15}\,{\left (d\,x\right )}^{5/2}}{2560\,b^3}+\frac {117\,a^3\,d^{17}\,\sqrt {d\,x}}{4096\,b^4}+\frac {533\,a\,d^{13}\,{\left (d\,x\right )}^{9/2}}{2048\,b^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}+\frac {117\,d^{15/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{7/4}\,b^{17/4}} \]
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