Integrand size = 30, antiderivative size = 600 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
[Out]
Time = 0.31 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1126, 294, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 210
Rule 217
Rule 294
Rule 327
Rule 335
Rule 631
Rule 642
Rule 1126
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{19/2}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (17 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^4} \, dx}{16 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (221 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{192 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (663 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{512 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3315 d^8 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{2048 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 a d^{10} \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{2048 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 a d^9 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \sqrt {a} d^8 \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 )}{2048 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \sqrt {a} d^8 \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 )}{2048 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3315 \sqrt [4]{a} d^{19/2} \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 )}{4096 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3315 \sqrt [4]{a} d^{19/2} \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 )}{4096 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \sqrt {a} d^{10} \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 )}{4096 b^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \sqrt {a} d^{10} \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 )}{4096 b^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \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 )}{4096 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \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 )}{4096 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3315 \sqrt [4]{a} d^{19/2} \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 )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3315 \sqrt [4]{a} d^{19/2} \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 )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 d^9 \sqrt {d x} \left (a+b x^2\right )}{1024 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \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 )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \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 )}{2048 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3315 \sqrt [4]{a} d^{19/2} \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 )}{4096 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3315 \sqrt [4]{a} d^{19/2} \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 )}{4096 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.36 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {d^9 \sqrt {d x} \left (4 \sqrt [4]{b} \sqrt {x} \left (9945 a^4+37791 a^3 b x^2+52819 a^2 b^2 x^4+31501 a b^3 x^6+6144 b^4 x^8\right )-9945 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^4 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-9945 \sqrt {2} \sqrt [4]{a} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{12288 b^{21/4} \sqrt {x} \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \]
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Time = 0.10 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.46
method | result | size |
risch | \(\frac {2 x \,d^{10} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b^{5} \sqrt {d x}\, \left (b \,x^{2}+a \right )}-\frac {2 a \,d^{11} \left (\frac {-\frac {1267 a^{3} d^{6} \sqrt {d x}}{2048}-\frac {4405 a^{2} d^{4} b \left (d x \right )^{\frac {5}{2}}}{2048}-\frac {15955 a \,d^{2} b^{2} \left (d x \right )^{\frac {9}{2}}}{6144}-\frac {6925 b^{3} \left (d x \right )^{\frac {13}{2}}}{6144}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{4}}+\frac {3315 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16384 a \,d^{2}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b^{5} \left (b \,x^{2}+a \right )}\) | \(276\) |
default | \(\text {Expression too large to display}\) | \(1217\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 468, normalized size of antiderivative = 0.78 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {9945 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (3315 \, \sqrt {d x} d^{9} + 3315 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) + 9945 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (i \, b^{9} x^{8} + 4 i \, a b^{8} x^{6} + 6 i \, a^{2} b^{7} x^{4} + 4 i \, a^{3} b^{6} x^{2} + i \, a^{4} b^{5}\right )} \log \left (3315 \, \sqrt {d x} d^{9} + 3315 i \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) + 9945 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (-i \, b^{9} x^{8} - 4 i \, a b^{8} x^{6} - 6 i \, a^{2} b^{7} x^{4} - 4 i \, a^{3} b^{6} x^{2} - i \, a^{4} b^{5}\right )} \log \left (3315 \, \sqrt {d x} d^{9} - 3315 i \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) - 9945 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (3315 \, \sqrt {d x} d^{9} - 3315 \, \left (-\frac {a d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) - 4 \, {\left (6144 \, b^{4} d^{9} x^{8} + 31501 \, a b^{3} d^{9} x^{6} + 52819 \, a^{2} b^{2} d^{9} x^{4} + 37791 \, a^{3} b d^{9} x^{2} + 9945 \, a^{4} d^{9}\right )} \sqrt {d x}}{12288 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} \]
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Timed out. \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int { \frac {\left (d x\right )^{\frac {19}{2}}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.64 \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {1}{24576} \, d^{9} {\left (\frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {19890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {9945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {49152 \, \sqrt {d x}}{b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {8 \, {\left (6925 \, \sqrt {d x} a b^{3} d^{8} x^{6} + 15955 \, \sqrt {d x} a^{2} b^{2} d^{8} x^{4} + 13215 \, \sqrt {d x} a^{3} b d^{8} x^{2} + 3801 \, \sqrt {d x} a^{4} d^{8}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{5} \mathrm {sgn}\left (b x^{2} + a\right )}\right )} \]
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Timed out. \[ \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{19/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \]
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