Integrand size = 30, antiderivative size = 557 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
[Out]
Time = 0.28 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1126, 294, 296, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 210
Rule 217
Rule 294
Rule 296
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)^{3/2}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \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 \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (11 b d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx}{192 a \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{1536 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^2 \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 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d \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 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 \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 a^{7/2} b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 \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 a^{7/2} b \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 d^{3/2} \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} a^{15/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 d^{3/2} \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} a^{15/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^2 \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 a^{7/2} b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^2 \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 a^{7/2} b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \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} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \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} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^{3/2} \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} a^{15/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 d^{3/2} \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} a^{15/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = \frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \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} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \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} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \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} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \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} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.36 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {(d x)^{3/2} \left (a+b x^2\right ) \left (4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-231 a^3+351 a^2 b x^2+275 a b^2 x^4+77 b^3 x^6\right )-231 \sqrt {2} \left (a+b x^2\right )^4 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+231 \sqrt {2} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{12288 a^{15/4} b^{5/4} x^{3/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1150\) vs. \(2(361)=722\).
Time = 0.05 (sec) , antiderivative size = 1151, normalized size of antiderivative = 2.07
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 473, normalized size of antiderivative = 0.85 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {231 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} \log \left (77 \, a^{4} b \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d\right ) - 231 \, {\left (-i \, a^{3} b^{5} x^{8} - 4 i \, a^{4} b^{4} x^{6} - 6 i \, a^{5} b^{3} x^{4} - 4 i \, a^{6} b^{2} x^{2} - i \, a^{7} b\right )} \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} \log \left (77 i \, a^{4} b \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d\right ) - 231 \, {\left (i \, a^{3} b^{5} x^{8} + 4 i \, a^{4} b^{4} x^{6} + 6 i \, a^{5} b^{3} x^{4} + 4 i \, a^{6} b^{2} x^{2} + i \, a^{7} b\right )} \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} \log \left (-77 i \, a^{4} b \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d\right ) - 231 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} \log \left (-77 \, a^{4} b \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d\right ) + 4 \, {\left (77 \, b^{3} d x^{6} + 275 \, a b^{2} d x^{4} + 351 \, a^{2} b d x^{2} - 231 \, a^{3} d\right )} \sqrt {d x}}{12288 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )}} \]
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\[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {\left (d x\right )^{\frac {3}{2}}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.33 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.05 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {385 \, b^{3} d^{\frac {3}{2}} x^{\frac {13}{2}} + 495 \, a b^{2} d^{\frac {3}{2}} x^{\frac {9}{2}} + 315 \, a^{2} b d^{\frac {3}{2}} x^{\frac {5}{2}} + 77 \, a^{3} d^{\frac {3}{2}} \sqrt {x}}{1024 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )}} + \frac {{\left (77 \, b^{4} d^{\frac {3}{2}} x^{5} + 66 \, a b^{3} d^{\frac {3}{2}} x^{3} + 21 \, a^{2} b^{2} d^{\frac {3}{2}} x\right )} x^{\frac {11}{2}} + 2 \, {\left (99 \, a b^{3} d^{\frac {3}{2}} x^{5} + 102 \, a^{2} b^{2} d^{\frac {3}{2}} x^{3} + 35 \, a^{3} b d^{\frac {3}{2}} x\right )} x^{\frac {7}{2}} + {\left (153 \, a^{2} b^{2} d^{\frac {3}{2}} x^{5} + 202 \, a^{3} b d^{\frac {3}{2}} x^{3} + 81 \, a^{4} d^{\frac {3}{2}} x\right )} x^{\frac {3}{2}}}{192 \, {\left (a^{6} b^{3} x^{6} + 3 \, a^{7} b^{2} x^{4} + 3 \, a^{8} b x^{2} + a^{9} + {\left (a^{3} b^{6} x^{6} + 3 \, a^{4} b^{5} x^{4} + 3 \, a^{5} b^{4} x^{2} + a^{6} b^{3}\right )} x^{6} + 3 \, {\left (a^{4} b^{5} x^{6} + 3 \, a^{5} b^{4} x^{4} + 3 \, a^{6} b^{3} x^{2} + a^{7} b^{2}\right )} x^{4} + 3 \, {\left (a^{5} b^{4} x^{6} + 3 \, a^{6} b^{3} x^{4} + 3 \, a^{7} b^{2} x^{2} + a^{8} b\right )} x^{2}\right )}} + \frac {77 \, d {\left (\frac {2 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} \sqrt {d} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} \sqrt {d} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} \sqrt {d} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{8192 \, a^{3} b} \]
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Time = 0.31 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.67 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {1}{24576} \, d {\left (\frac {462 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {462 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{4} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {231 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{4} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {231 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{4} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {8 \, {\left (77 \, \sqrt {d x} b^{3} d^{8} x^{6} + 275 \, \sqrt {d x} a b^{2} d^{8} x^{4} + 351 \, \sqrt {d x} a^{2} b d^{8} x^{2} - 231 \, \sqrt {d x} a^{3} d^{8}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{3} b \mathrm {sgn}\left (b x^{2} + a\right )}\right )} \]
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Timed out. \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{3/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \]
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