Integrand size = 28, antiderivative size = 300 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {7}{6 a^2 d (d x)^{3/2}}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}+\frac {7 b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{11/4} d^{5/2}}-\frac {7 b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{11/4} d^{5/2}}+\frac {7 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{11/4} d^{5/2}}-\frac {7 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{11/4} d^{5/2}} \]
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Time = 0.19 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 296, 331, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\frac {7 b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{11/4} d^{5/2}}-\frac {7 b^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} a^{11/4} d^{5/2}}+\frac {7 b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{11/4} d^{5/2}}-\frac {7 b^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{11/4} d^{5/2}}-\frac {7}{6 a^2 d (d x)^{3/2}}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )} \]
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Rule 28
Rule 210
Rule 217
Rule 296
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^2} \, dx \\ & = \frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}+\frac {(7 b) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )} \, dx}{4 a} \\ & = -\frac {7}{6 a^2 d (d x)^{3/2}}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (7 b^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{4 a^2 d^2} \\ & = -\frac {7}{6 a^2 d (d x)^{3/2}}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (7 b^2\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{2 a^2 d^3} \\ & = -\frac {7}{6 a^2 d (d x)^{3/2}}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (7 b^2\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 )}{4 a^{5/2} d^4}-\frac {\left (7 b^2\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 )}{4 a^{5/2} d^4} \\ & = -\frac {7}{6 a^2 d (d x)^{3/2}}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}+\frac {\left (7 b^{3/4}\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 )}{8 \sqrt {2} a^{11/4} d^{5/2}}+\frac {\left (7 b^{3/4}\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 )}{8 \sqrt {2} a^{11/4} d^{5/2}}-\frac {\left (7 \sqrt {b}\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 )}{8 a^{5/2} d^2}-\frac {\left (7 \sqrt {b}\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 )}{8 a^{5/2} d^2} \\ & = -\frac {7}{6 a^2 d (d x)^{3/2}}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}+\frac {7 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{11/4} d^{5/2}}-\frac {7 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{11/4} d^{5/2}}-\frac {\left (7 b^{3/4}\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 )}{4 \sqrt {2} a^{11/4} d^{5/2}}+\frac {\left (7 b^{3/4}\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 )}{4 \sqrt {2} a^{11/4} d^{5/2}} \\ & = -\frac {7}{6 a^2 d (d x)^{3/2}}+\frac {1}{2 a d (d x)^{3/2} \left (a+b x^2\right )}+\frac {7 b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{11/4} d^{5/2}}-\frac {7 b^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{11/4} d^{5/2}}+\frac {7 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{11/4} d^{5/2}}-\frac {7 b^{3/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{11/4} d^{5/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.55 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\frac {x \left (-4 a^{3/4} \left (4 a+7 b x^2\right )+21 \sqrt {2} b^{3/4} x^{3/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-21 \sqrt {2} b^{3/4} x^{3/2} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{24 a^{11/4} (d x)^{5/2} \left (a+b x^2\right )} \]
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Time = 0.21 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {2}{3 a^{2} x \sqrt {d x}\, d^{2}}-\frac {2 b \left (\frac {\sqrt {d x}}{4 b \,d^{2} x^{2}+4 a \,d^{2}}+\frac {7 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,d^{2}}\right )}{a^{2} d}\) | \(192\) |
derivativedivides | \(2 d^{3} \left (-\frac {1}{3 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\sqrt {d x}}{4 b \,d^{2} x^{2}+4 a \,d^{2}}+\frac {7 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,d^{2}}\right )}{a^{2} d^{4}}\right )\) | \(194\) |
default | \(2 d^{3} \left (-\frac {1}{3 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\sqrt {d x}}{4 b \,d^{2} x^{2}+4 a \,d^{2}}+\frac {7 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \,d^{2}}\right )}{a^{2} d^{4}}\right )\) | \(194\) |
pseudoelliptic | \(d^{3} \left (-\frac {2}{3 a^{2} d^{4} \left (d x \right )^{\frac {3}{2}}}-\frac {\sqrt {d x}\, b}{2 d^{6} \left (b \,x^{2}+a \right ) a^{2}}-\frac {7 \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right ) \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b}{16 a^{3} d^{6}}-\frac {7 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b}{8 a^{3} d^{6}}-\frac {7 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b}{8 a^{3} d^{6}}\right )\) | \(245\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {21 \, {\left (a^{2} b d^{3} x^{4} + a^{3} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{11} d^{10}}\right )^{\frac {1}{4}} \log \left (7 \, a^{3} d^{3} \left (-\frac {b^{3}}{a^{11} d^{10}}\right )^{\frac {1}{4}} + 7 \, \sqrt {d x} b\right ) + 21 \, {\left (i \, a^{2} b d^{3} x^{4} + i \, a^{3} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{11} d^{10}}\right )^{\frac {1}{4}} \log \left (7 i \, a^{3} d^{3} \left (-\frac {b^{3}}{a^{11} d^{10}}\right )^{\frac {1}{4}} + 7 \, \sqrt {d x} b\right ) + 21 \, {\left (-i \, a^{2} b d^{3} x^{4} - i \, a^{3} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{11} d^{10}}\right )^{\frac {1}{4}} \log \left (-7 i \, a^{3} d^{3} \left (-\frac {b^{3}}{a^{11} d^{10}}\right )^{\frac {1}{4}} + 7 \, \sqrt {d x} b\right ) - 21 \, {\left (a^{2} b d^{3} x^{4} + a^{3} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{11} d^{10}}\right )^{\frac {1}{4}} \log \left (-7 \, a^{3} d^{3} \left (-\frac {b^{3}}{a^{11} d^{10}}\right )^{\frac {1}{4}} + 7 \, \sqrt {d x} b\right ) + 4 \, {\left (7 \, b x^{2} + 4 \, a\right )} \sqrt {d x}}{24 \, {\left (a^{2} b d^{3} x^{4} + a^{3} d^{3} x^{2}\right )}} \]
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\[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\int \frac {1}{\left (d x\right )^{\frac {5}{2}} \left (a + b x^{2}\right )^{2}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {\frac {8 \, {\left (7 \, b d^{2} x^{2} + 4 \, a d^{2}\right )}}{\left (d x\right )^{\frac {7}{2}} a^{2} b + \left (d x\right )^{\frac {3}{2}} a^{3} d^{2}} + \frac {21 \, {\left (\frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a} d} + \frac {2 \, \sqrt {2} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a} d}\right )}}{a^{2}}}{48 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {\sqrt {d x} b}{2 \, {\left (b d^{2} x^{2} + a d^{2}\right )} a^{2} d} - \frac {7 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} d^{3}} - \frac {7 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{3} d^{3}} - \frac {7 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{16 \, a^{3} d^{3}} + \frac {7 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{16 \, a^{3} d^{3}} - \frac {2}{3 \, \sqrt {d x} a^{2} d^{2} x} \]
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Time = 13.86 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.34 \[ \int \frac {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\frac {7\,{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{4\,a^{11/4}\,d^{5/2}}-\frac {\frac {2\,d}{3\,a}+\frac {7\,b\,d\,x^2}{6\,a^2}}{b\,{\left (d\,x\right )}^{7/2}+a\,d^2\,{\left (d\,x\right )}^{3/2}}+\frac {7\,{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{4\,a^{11/4}\,d^{5/2}} \]
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