Integrand size = 28, antiderivative size = 281 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}-\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{3/4} b^{5/4}}+\frac {d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{3/4} b^{5/4}}-\frac {d^{3/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{3/4} b^{5/4}}+\frac {d^{3/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{8 \sqrt {2} a^{3/4} b^{5/4}} \]
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Time = 0.17 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 294, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{3/4} b^{5/4}}+\frac {d^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} a^{3/4} b^{5/4}}-\frac {d^{3/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{3/4} b^{5/4}}+\frac {d^{3/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} a^{3/4} b^{5/4}}-\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )} \]
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Rule 28
Rule 210
Rule 217
Rule 294
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^2} \, dx \\ & = -\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}+\frac {1}{4} d^2 \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx \\ & = -\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}+\frac {1}{2} d \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right ) \\ & = -\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}+\frac {\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 \sqrt {a}}+\frac {\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 \sqrt {a}} \\ & = -\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}-\frac {d^{3/2} \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^{3/4} b^{5/4}}-\frac {d^{3/2} \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^{3/4} b^{5/4}}+\frac {d^2 \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 \sqrt {a} b^{3/2}}+\frac {d^2 \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 \sqrt {a} b^{3/2}} \\ & = -\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}-\frac {d^{3/2} \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^{3/4} b^{5/4}}+\frac {d^{3/2} \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^{3/4} b^{5/4}}+\frac {d^{3/2} \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^{3/4} b^{5/4}}-\frac {d^{3/2} \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^{3/4} b^{5/4}} \\ & = -\frac {d \sqrt {d x}}{2 b \left (a+b x^2\right )}-\frac {d^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{3/4} b^{5/4}}+\frac {d^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{3/4} b^{5/4}}-\frac {d^{3/2} \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^{3/4} b^{5/4}}+\frac {d^{3/2} \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^{3/4} b^{5/4}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.54 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {(d x)^{3/2} \left (-4 a^{3/4} \sqrt [4]{b} \sqrt {x}-\sqrt {2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt {2} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{8 a^{3/4} b^{5/4} x^{3/2} \left (a+b x^2\right )} \]
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Time = 0.17 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(2 d^{3} \left (-\frac {\sqrt {d x}}{4 b \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b a \,d^{2}}\right )\) | \(177\) |
default | \(2 d^{3} \left (-\frac {\sqrt {d x}}{4 b \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b a \,d^{2}}\right )\) | \(177\) |
pseudoelliptic | \(\frac {\left (\frac {\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right ) \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{2}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-4 \sqrt {d x}\, a \right ) d}{8 b \left (b \,x^{2}+a \right ) a}\) | \(229\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.83 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {{\left (b^{2} x^{2} + a b\right )} \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (a b \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} + \sqrt {d x} d\right ) - {\left (-i \, b^{2} x^{2} - i \, a b\right )} \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (i \, a b \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} + \sqrt {d x} d\right ) - {\left (i \, b^{2} x^{2} + i \, a b\right )} \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (-i \, a b \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} + \sqrt {d x} d\right ) - {\left (b^{2} x^{2} + a b\right )} \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} \log \left (-a b \left (-\frac {d^{6}}{a^{3} b^{5}}\right )^{\frac {1}{4}} + \sqrt {d x} d\right ) - 4 \, \sqrt {d x} d}{8 \, {\left (b^{2} x^{2} + a b\right )}} \]
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\[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=\int \frac {\left (d x\right )^{\frac {3}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
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none
Time = 0.29 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.94 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {\frac {8 \, \sqrt {d x} d^{4}}{b^{2} d^{2} x^{2} + a b d^{2}} - \frac {\frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}}{b}}{16 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.93 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {1}{16} \, d {\left (\frac {8 \, \sqrt {d x} d^{2}}{{\left (b d^{2} x^{2} + a d^{2}\right )} b} - \frac {2 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{2}} - \frac {2 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{2}} - \frac {\sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{2}} + \frac {\sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{2}}\right )} \]
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Time = 13.44 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.33 \[ \int \frac {(d x)^{3/2}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {d^{3/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,{\left (-a\right )}^{3/4}\,b^{5/4}}-\frac {d^{3/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,{\left (-a\right )}^{3/4}\,b^{5/4}}-\frac {d^3\,\sqrt {d\,x}}{2\,b\,\left (b\,d^2\,x^2+a\,d^2\right )} \]
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