Integrand size = 28, antiderivative size = 98 \[ \int \frac {\left (2 b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\frac {\left (-8 b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{2 x}-\frac {9}{4} \sqrt [4]{a} b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )+\frac {9}{4} \sqrt [4]{a} b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.73, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2063, 2029, 2057, 335, 338, 304, 209, 212} \[ \int \frac {\left (2 b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=-\frac {9 \sqrt [4]{a} b x^{3/2} \left (a x^2+b\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 \left (a x^4+b x^2\right )^{3/4}}+\frac {9 \sqrt [4]{a} b x^{3/2} \left (a x^2+b\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 \left (a x^4+b x^2\right )^{3/4}}+\frac {9}{2} a x \sqrt [4]{a x^4+b x^2}-\frac {4 \left (a x^4+b x^2\right )^{5/4}}{x^3} \]
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 338
Rule 2029
Rule 2057
Rule 2063
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}+(9 a) \int \sqrt [4]{b x^2+a x^4} \, dx \\ & = \frac {9}{2} a x \sqrt [4]{b x^2+a x^4}-\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}+\frac {1}{4} (9 a b) \int \frac {x^2}{\left (b x^2+a x^4\right )^{3/4}} \, dx \\ & = \frac {9}{2} a x \sqrt [4]{b x^2+a x^4}-\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}+\frac {\left (9 a b x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \int \frac {\sqrt {x}}{\left (b+a x^2\right )^{3/4}} \, dx}{4 \left (b x^2+a x^4\right )^{3/4}} \\ & = \frac {9}{2} a x \sqrt [4]{b x^2+a x^4}-\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}+\frac {\left (9 a b x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \left (b x^2+a x^4\right )^{3/4}} \\ & = \frac {9}{2} a x \sqrt [4]{b x^2+a x^4}-\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}+\frac {\left (9 a b x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \left (b x^2+a x^4\right )^{3/4}} \\ & = \frac {9}{2} a x \sqrt [4]{b x^2+a x^4}-\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}+\frac {\left (9 \sqrt {a} b x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 \left (b x^2+a x^4\right )^{3/4}}-\frac {\left (9 \sqrt {a} b x^{3/2} \left (b+a x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 \left (b x^2+a x^4\right )^{3/4}} \\ & = \frac {9}{2} a x \sqrt [4]{b x^2+a x^4}-\frac {4 \left (b x^2+a x^4\right )^{5/4}}{x^3}-\frac {9 \sqrt [4]{a} b x^{3/2} \left (b+a x^2\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 \left (b x^2+a x^4\right )^{3/4}}+\frac {9 \sqrt [4]{a} b x^{3/2} \left (b+a x^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 \left (b x^2+a x^4\right )^{3/4}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.29 \[ \int \frac {\left (2 b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\frac {x \left (b+a x^2\right )^{3/4} \left (2 \left (-8 b+a x^2\right ) \sqrt [4]{b+a x^2}-9 \sqrt [4]{a} b \sqrt {x} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+9 \sqrt [4]{a} b \sqrt {x} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )\right )}{4 \left (x^2 \left (b+a x^2\right )\right )^{3/4}} \]
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Time = 1.01 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.11
method | result | size |
pseudoelliptic | \(\frac {9 b x \left (\ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )\right ) a^{\frac {1}{4}}+4 \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}} \left (a \,x^{2}-8 b \right )}{8 x}\) | \(109\) |
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Timed out. \[ \int \frac {\left (2 b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (2 b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\int \frac {\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{2} + 2 b\right )}{x^{2}}\, dx \]
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\[ \int \frac {\left (2 b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\int { \frac {{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{2} + 2 \, b\right )}}{x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (78) = 156\).
Time = 0.29 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.26 \[ \int \frac {\left (2 b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\frac {8 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} a b x^{2} + 18 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + 18 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + 9 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right ) - 9 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right ) - 64 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} b^{2}}{16 \, b} \]
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Time = 6.57 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.91 \[ \int \frac {\left (2 b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\frac {2\,a\,x\,{\left (a\,x^4+b\,x^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ -\frac {a\,x^2}{b}\right )}{3\,{\left (\frac {a\,x^2}{b}+1\right )}^{1/4}}-\frac {4\,b\,{\left (a\,x^4+b\,x^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ -\frac {a\,x^2}{b}\right )}{x\,{\left (\frac {a\,x^2}{b}+1\right )}^{1/4}} \]
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