Integrand size = 28, antiderivative size = 402 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac {13923 \sqrt [4]{a} d^{23/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{25/4}}+\frac {13923 \sqrt [4]{a} d^{23/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} b^{25/4}} \]
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Time = 0.35 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 17, 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} \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {13923 \sqrt [4]{a} d^{23/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} b^{25/4}}+\frac {13923 \sqrt [4]{a} d^{23/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} b^{25/4}}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}+\frac {13923 d^{11} \sqrt {d x}}{4096 b^6} \]
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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^6 \int \frac {(d x)^{23/2}}{\left (a b+b^2 x^2\right )^6} \, dx \\ & = -\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (21 b^4 d^2\right ) \int \frac {(d x)^{19/2}}{\left (a b+b^2 x^2\right )^5} \, dx \\ & = -\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {1}{320} \left (357 b^2 d^4\right ) \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^4} \, dx \\ & = -\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}+\frac {\left (1547 d^6\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280} \\ & = -\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}+\frac {\left (13923 d^8\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{10240 b^2} \\ & = -\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac {\left (13923 d^{10}\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{8192 b^4} \\ & = \frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac {\left (13923 a d^{12}\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 b^5} \\ & = \frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac {\left (13923 a d^{11}\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 b^5} \\ & = \frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac {\left (13923 \sqrt {a} d^{10}\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 b^5}-\frac {\left (13923 \sqrt {a} d^{10}\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 b^5} \\ & = \frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac {\left (13923 \sqrt [4]{a} d^{23/2}\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} b^{25/4}}+\frac {\left (13923 \sqrt [4]{a} d^{23/2}\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} b^{25/4}}-\frac {\left (13923 \sqrt {a} d^{12}\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 b^{13/2}}-\frac {\left (13923 \sqrt {a} d^{12}\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 b^{13/2}} \\ & = \frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac {13923 \sqrt [4]{a} d^{23/2} \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} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \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} b^{25/4}}-\frac {\left (13923 \sqrt [4]{a} d^{23/2}\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} b^{25/4}}+\frac {\left (13923 \sqrt [4]{a} d^{23/2}\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} b^{25/4}} \\ & = \frac {13923 d^{11} \sqrt {d x}}{4096 b^6}-\frac {d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac {21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac {13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} b^{25/4}}+\frac {13923 \sqrt [4]{a} d^{23/2} \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} b^{25/4}}-\frac {13923 \sqrt [4]{a} d^{23/2} \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} b^{25/4}} \\ \end{align*}
Time = 0.92 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.54 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {d^{11} \sqrt {d x} \left (4 \sqrt [4]{b} \sqrt {x} \left (69615 a^5+334152 a^4 b x^2+634270 a^3 b^2 x^4+590240 a^2 b^3 x^6+263515 a b^4 x^8+40960 b^5 x^{10}\right )-69615 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^5 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-69615 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 b^{25/4} \sqrt {x} \left (a+b x^2\right )^5} \]
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Time = 20.32 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(-\frac {d^{11} \left (\left (-327680 x^{10} b^{5}-2108120 a \,x^{8} b^{4}-4721920 a^{2} x^{6} b^{3}-5074160 a^{3} x^{4} b^{2}-2673216 x^{2} a^{4} b -556920 a^{5}\right ) \sqrt {d x}+69615 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right )}{163840 b^{6} \left (b \,x^{2}+a \right )^{5}}\) | \(245\) |
derivativedivides | \(2 d^{11} \left (\frac {\sqrt {d x}}{b^{6}}-\frac {a \,d^{2} \left (\frac {-\frac {5731 a^{4} d^{8} \sqrt {d x}}{8192}-\frac {16169 a^{3} b \,d^{6} \left (d x \right )^{\frac {5}{2}}}{5120}-\frac {22467 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{4096}-\frac {1129 a \,b^{3} d^{2} \left (d x \right )^{\frac {13}{2}}}{256}-\frac {11743 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {13923 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{b^{6}}\right )\) | \(252\) |
default | \(2 d^{11} \left (\frac {\sqrt {d x}}{b^{6}}-\frac {a \,d^{2} \left (\frac {-\frac {5731 a^{4} d^{8} \sqrt {d x}}{8192}-\frac {16169 a^{3} b \,d^{6} \left (d x \right )^{\frac {5}{2}}}{5120}-\frac {22467 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{4096}-\frac {1129 a \,b^{3} d^{2} \left (d x \right )^{\frac {13}{2}}}{256}-\frac {11743 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {13923 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{b^{6}}\right )\) | \(252\) |
risch | \(\frac {2 x \,d^{12}}{b^{6} \sqrt {d x}}-\frac {2 a \,d^{13} \left (\frac {-\frac {5731 a^{4} d^{8} \sqrt {d x}}{8192}-\frac {16169 a^{3} b \,d^{6} \left (d x \right )^{\frac {5}{2}}}{5120}-\frac {22467 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {9}{2}}}{4096}-\frac {1129 a \,b^{3} d^{2} \left (d x \right )^{\frac {13}{2}}}{256}-\frac {11743 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {13923 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2}}\right )}{b^{6}}\) | \(252\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.34 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {69615 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \log \left (13923 \, \sqrt {d x} d^{11} + 13923 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) + 69615 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (i \, b^{11} x^{10} + 5 i \, a b^{10} x^{8} + 10 i \, a^{2} b^{9} x^{6} + 10 i \, a^{3} b^{8} x^{4} + 5 i \, a^{4} b^{7} x^{2} + i \, a^{5} b^{6}\right )} \log \left (13923 \, \sqrt {d x} d^{11} + 13923 i \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) + 69615 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (-i \, b^{11} x^{10} - 5 i \, a b^{10} x^{8} - 10 i \, a^{2} b^{9} x^{6} - 10 i \, a^{3} b^{8} x^{4} - 5 i \, a^{4} b^{7} x^{2} - i \, a^{5} b^{6}\right )} \log \left (13923 \, \sqrt {d x} d^{11} - 13923 i \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 69615 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \log \left (13923 \, \sqrt {d x} d^{11} - 13923 \, \left (-\frac {a d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 4 \, {\left (40960 \, b^{5} d^{11} x^{10} + 263515 \, a b^{4} d^{11} x^{8} + 590240 \, a^{2} b^{3} d^{11} x^{6} + 634270 \, a^{3} b^{2} d^{11} x^{4} + 334152 \, a^{4} b d^{11} x^{2} + 69615 \, a^{5} d^{11}\right )} \sqrt {d x}}{81920 \, {\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} \]
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Timed out. \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {327680 \, \sqrt {d x} d^{12}}{b^{6}} + \frac {8 \, {\left (58715 \, \left (d x\right )^{\frac {17}{2}} a b^{4} d^{14} + 180640 \, \left (d x\right )^{\frac {13}{2}} a^{2} b^{3} d^{16} + 224670 \, \left (d x\right )^{\frac {9}{2}} a^{3} b^{2} d^{18} + 129352 \, \left (d x\right )^{\frac {5}{2}} a^{4} b d^{20} + 28655 \, \sqrt {d x} a^{5} d^{22}\right )}}{b^{11} d^{10} x^{10} + 5 \, a b^{10} d^{10} x^{8} + 10 \, a^{2} b^{9} d^{10} x^{6} + 10 \, a^{3} b^{8} d^{10} x^{4} + 5 \, a^{4} b^{7} d^{10} x^{2} + a^{5} b^{6} d^{10}} - \frac {69615 \, {\left (\frac {\sqrt {2} d^{14} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{14} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{13} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{13} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )} a}{b^{6}}}{163840 \, d} \]
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Time = 0.28 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.85 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {1}{163840} \, d^{11} {\left (\frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{7}} + \frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{7}} + \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{7}} - \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{7}} - \frac {327680 \, \sqrt {d x}}{b^{6}} - \frac {8 \, {\left (58715 \, \sqrt {d x} a b^{4} d^{10} x^{8} + 180640 \, \sqrt {d x} a^{2} b^{3} d^{10} x^{6} + 224670 \, \sqrt {d x} a^{3} b^{2} d^{10} x^{4} + 129352 \, \sqrt {d x} a^{4} b d^{10} x^{2} + 28655 \, \sqrt {d x} a^{5} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{6}}\right )} \]
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Time = 13.61 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.57 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {5731\,a^5\,d^{21}\,\sqrt {d\,x}}{4096}+\frac {22467\,a^3\,b^2\,d^{17}\,{\left (d\,x\right )}^{9/2}}{2048}+\frac {1129\,a^2\,b^3\,d^{15}\,{\left (d\,x\right )}^{13/2}}{128}+\frac {16169\,a^4\,b\,d^{19}\,{\left (d\,x\right )}^{5/2}}{2560}+\frac {11743\,a\,b^4\,d^{13}\,{\left (d\,x\right )}^{17/2}}{4096}}{a^5\,b^6\,d^{10}+5\,a^4\,b^7\,d^{10}\,x^2+10\,a^3\,b^8\,d^{10}\,x^4+10\,a^2\,b^9\,d^{10}\,x^6+5\,a\,b^{10}\,d^{10}\,x^8+b^{11}\,d^{10}\,x^{10}}+\frac {2\,d^{11}\,\sqrt {d\,x}}{b^6}-\frac {13923\,{\left (-a\right )}^{1/4}\,d^{23/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,b^{25/4}}+\frac {{\left (-a\right )}^{1/4}\,d^{23/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,13923{}\mathrm {i}}{8192\,b^{25/4}} \]
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