Integrand size = 30, antiderivative size = 458 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {5 d^3 \sqrt {d x}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^{7/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 d^{7/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Time = 0.21 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1126, 294, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {5 d^3 \sqrt {d x}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^{7/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 d^{7/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^{7/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rule 210
Rule 217
Rule 294
Rule 335
Rule 631
Rule 642
Rule 1126
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{8 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {5 d^3 \sqrt {d x}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{32 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {5 d^3 \sqrt {d x}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 d^3 \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 )}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {5 d^3 \sqrt {d x}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 d^2 \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 )}{32 \sqrt {a} b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 d^2 \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 )}{32 \sqrt {a} b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {5 d^3 \sqrt {d x}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (5 d^{7/2} \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 )}{64 \sqrt {2} a^{3/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (5 d^{7/2} \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 )}{64 \sqrt {2} a^{3/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 d^4 \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 )}{64 \sqrt {a} b^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 d^4 \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 )}{64 \sqrt {a} b^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {5 d^3 \sqrt {d x}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^{7/2} \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 )}{64 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 d^{7/2} \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 )}{64 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (5 d^{7/2} \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 )}{32 \sqrt {2} a^{3/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (5 d^{7/2} \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 )}{32 \sqrt {2} a^{3/4} b^{13/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {5 d^3 \sqrt {d x}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^{7/2} \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 )}{32 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 d^{7/2} \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 )}{32 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 d^{7/2} \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 )}{64 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 d^{7/2} \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 )}{64 \sqrt {2} a^{3/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.40 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {d^3 \sqrt {d x} \left (4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (5 a+9 b x^2\right )+5 \sqrt {2} \left (a+b x^2\right )^2 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-5 \sqrt {2} \left (a+b x^2\right )^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{64 a^{3/4} b^{9/4} \sqrt {x} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(674\) vs. \(2(292)=584\).
Time = 0.04 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.47
method | result | size |
default | \(-\frac {\left (-5 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \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}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\sqrt {\frac {a \,d^{2}}{b}}}\right ) b^{2} d^{2} x^{4}-10 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {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 ) b^{2} d^{2} x^{4}-10 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {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 ) b^{2} d^{2} x^{4}-10 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \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}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\sqrt {\frac {a \,d^{2}}{b}}}\right ) a b \,d^{2} x^{2}-20 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {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 ) a b \,d^{2} x^{2}-20 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {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 ) a b \,d^{2} x^{2}+72 \left (d x \right )^{\frac {5}{2}} a b -5 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \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}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\sqrt {\frac {a \,d^{2}}{b}}}\right ) a^{2} d^{2}-10 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {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 ) a^{2} d^{2}-10 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {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 ) a^{2} d^{2}+40 \sqrt {d x}\, a^{2} d^{2}\right ) d \left (b \,x^{2}+a \right )}{128 a \,b^{2} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}\) | \(675\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.73 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {5 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac {d^{14}}{a^{3} b^{9}}\right )^{\frac {1}{4}} \log \left (5 \, \sqrt {d x} d^{3} + 5 \, \left (-\frac {d^{14}}{a^{3} b^{9}}\right )^{\frac {1}{4}} a b^{2}\right ) - 5 \, {\left (-i \, b^{4} x^{4} - 2 i \, a b^{3} x^{2} - i \, a^{2} b^{2}\right )} \left (-\frac {d^{14}}{a^{3} b^{9}}\right )^{\frac {1}{4}} \log \left (5 \, \sqrt {d x} d^{3} + 5 i \, \left (-\frac {d^{14}}{a^{3} b^{9}}\right )^{\frac {1}{4}} a b^{2}\right ) - 5 \, {\left (i \, b^{4} x^{4} + 2 i \, a b^{3} x^{2} + i \, a^{2} b^{2}\right )} \left (-\frac {d^{14}}{a^{3} b^{9}}\right )^{\frac {1}{4}} \log \left (5 \, \sqrt {d x} d^{3} - 5 i \, \left (-\frac {d^{14}}{a^{3} b^{9}}\right )^{\frac {1}{4}} a b^{2}\right ) - 5 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac {d^{14}}{a^{3} b^{9}}\right )^{\frac {1}{4}} \log \left (5 \, \sqrt {d x} d^{3} - 5 \, \left (-\frac {d^{14}}{a^{3} b^{9}}\right )^{\frac {1}{4}} a b^{2}\right ) - 4 \, {\left (9 \, b d^{3} x^{2} + 5 \, a d^{3}\right )} \sqrt {d x}}{64 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
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\[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {\left (d x\right )^{\frac {7}{2}}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.61 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {d^{\frac {7}{2}} x^{\frac {5}{2}}}{2 \, {\left (a b^{2} x^{2} + a^{2} b + {\left (b^{3} x^{2} + a b^{2}\right )} x^{2}\right )}} + \frac {5 \, d^{3} {\left (\frac {2 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} \sqrt {d} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} \sqrt {d} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} \sqrt {d} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{128 \, b^{2}} - \frac {b d^{\frac {7}{2}} x^{\frac {5}{2}} + 5 \, a d^{\frac {7}{2}} \sqrt {x}}{16 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.72 \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {1}{128} \, d^{3} {\left (\frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {8 \, {\left (9 \, \sqrt {d x} b d^{4} x^{2} + 5 \, \sqrt {d x} a d^{4}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )}\right )} \]
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Timed out. \[ \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {{\left (d\,x\right )}^{7/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \]
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