Integrand size = 28, antiderivative size = 122 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\frac {\sqrt [4]{b x^2+a x^4} \left (32 b+b x^2+4 a x^4\right )}{16 x}+\frac {\left (32 a b+3 b^2\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{32 a^{3/4}}+\frac {\left (-32 a b-3 b^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{32 a^{3/4}} \]
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Leaf count is larger than twice the leaf count of optimal. \(321\) vs. \(2(122)=244\).
Time = 0.30 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.63, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2077, 2045, 2057, 335, 338, 304, 209, 212, 2046, 2049} \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\frac {3 b^2 x^{3/2} \left (a x^2+b\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{32 a^{3/4} \left (a x^4+b x^2\right )^{3/4}}-\frac {3 b^2 x^{3/2} \left (a x^2+b\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{32 a^{3/4} \left (a x^4+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} b x^{3/2} \left (a x^2+b\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\left (a x^4+b x^2\right )^{3/4}}-\frac {\sqrt [4]{a} b x^{3/2} \left (a x^2+b\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\left (a x^4+b x^2\right )^{3/4}}+\frac {1}{16} b x \sqrt [4]{a x^4+b x^2}+\frac {2 b \sqrt [4]{a x^4+b x^2}}{x}+\frac {1}{4} a x^3 \sqrt [4]{a x^4+b x^2} \]
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 338
Rule 2045
Rule 2046
Rule 2049
Rule 2057
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b \sqrt [4]{b x^2+a x^4}}{x^2}+a x^2 \sqrt [4]{b x^2+a x^4}\right ) \, dx \\ & = a \int x^2 \sqrt [4]{b x^2+a x^4} \, dx-b \int \frac {\sqrt [4]{b x^2+a x^4}}{x^2} \, dx \\ & = \frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}+\frac {1}{8} (a b) \int \frac {x^4}{\left (b x^2+a x^4\right )^{3/4}} \, dx-(a b) \int \frac {x^2}{\left (b x^2+a x^4\right )^{3/4}} \, dx \\ & = \frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{16} b x \sqrt [4]{b x^2+a x^4}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}-\frac {1}{32} \left (3 b^2\right ) \int \frac {x^2}{\left (b x^2+a x^4\right )^{3/4}} \, dx-\frac {\left (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}{\left (b x^2+a x^4\right )^{3/4}} \\ & = \frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{16} b x \sqrt [4]{b x^2+a x^4}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}-\frac {\left (2 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 )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {\left (3 b^2 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}{32 \left (b x^2+a x^4\right )^{3/4}} \\ & = \frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{16} b x \sqrt [4]{b x^2+a x^4}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}-\frac {\left (2 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 )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {\left (3 b^2 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 )}{16 \left (b x^2+a x^4\right )^{3/4}} \\ & = \frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{16} b x \sqrt [4]{b x^2+a x^4}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}-\frac {\left (\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 )}{\left (b x^2+a x^4\right )^{3/4}}+\frac {\left (\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 )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {\left (3 b^2 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 )}{16 \left (b x^2+a x^4\right )^{3/4}} \\ & = \frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{16} b x \sqrt [4]{b x^2+a x^4}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}+\frac {\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 )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {\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 )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {\left (3 b^2 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 )}{32 \sqrt {a} \left (b x^2+a x^4\right )^{3/4}}+\frac {\left (3 b^2 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 )}{32 \sqrt {a} \left (b x^2+a x^4\right )^{3/4}} \\ & = \frac {2 b \sqrt [4]{b x^2+a x^4}}{x}+\frac {1}{16} b x \sqrt [4]{b x^2+a x^4}+\frac {1}{4} a x^3 \sqrt [4]{b x^2+a x^4}+\frac {\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 )}{\left (b x^2+a x^4\right )^{3/4}}+\frac {3 b^2 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 )}{32 a^{3/4} \left (b x^2+a x^4\right )^{3/4}}-\frac {\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 )}{\left (b x^2+a x^4\right )^{3/4}}-\frac {3 b^2 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 )}{32 a^{3/4} \left (b x^2+a x^4\right )^{3/4}} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.18 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\frac {x \left (b+a x^2\right )^{3/4} \left (2 a^{3/4} \sqrt [4]{b+a x^2} \left (4 a x^4+b \left (32+x^2\right )\right )+b (32 a+3 b) \sqrt {x} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )-b (32 a+3 b) \sqrt {x} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )\right )}{32 a^{3/4} \left (x^2 \left (b+a x^2\right )\right )^{3/4}} \]
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Time = 0.36 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.06
method | result | size |
pseudoelliptic | \(-\frac {\frac {b x \left (\frac {3 b}{32}+a \right ) \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right )}{2}+b x \left (\frac {3 b}{32}+a \right ) \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )-\frac {\left (b \left (x^{2}+32\right ) a^{\frac {3}{4}}+4 a^{\frac {7}{4}} x^{4}\right ) \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{16}}{a^{\frac {3}{4}} x}\) | \(129\) |
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Timed out. \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\int \frac {\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} - b\right )}{x^{2}}\, dx \]
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\[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\int { \frac {{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}{x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (102) = 204\).
Time = 0.29 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.27 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=\frac {\frac {8 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{4}} b^{3} + 3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} a b^{3}\right )} x^{4}}{b^{2}} + 256 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} b^{2} + \frac {2 \, \sqrt {2} {\left (32 \, a b^{2} + 3 \, b^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} {\left (32 \, a b^{2} + 3 \, b^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (32 \, a b^{2} + 3 \, b^{3}\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{\left (-a\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (32 \, a b^{2} + 3 \, b^{3}\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{\left (-a\right )^{\frac {3}{4}}}}{128 \, b} \]
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Timed out. \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}}{x^2} \, dx=-\int \frac {\left (b-a\,x^4\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}}{x^2} \,d x \]
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