Integrand size = 80, antiderivative size = 255 \[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\frac {x \left (a b^4+\left (4 a b^3+b^4\right ) x^4+\left (6 a b^2+4 b^3\right ) x^8+\left (4 a b+6 b^2\right ) x^{12}+(a+4 b) x^{16}+x^{20}\right )^{3/4}}{4 \left (b+x^4\right )^3}+\frac {1}{8} (a-4 b) \arctan \left (\frac {\sqrt [4]{a b^4+\left (4 a b^3+b^4\right ) x^4+\left (6 a b^2+4 b^3\right ) x^8+\left (4 a b+6 b^2\right ) x^{12}+(a+4 b) x^{16}+x^{20}}}{x \left (b+x^4\right )}\right )+\frac {1}{8} (-a+4 b) \text {arctanh}\left (\frac {\sqrt [4]{a b^4+\left (4 a b^3+b^4\right ) x^4+\left (6 a b^2+4 b^3\right ) x^8+\left (4 a b+6 b^2\right ) x^{12}+(a+4 b) x^{16}+x^{20}}}{x \left (b+x^4\right )}\right ) \]
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Time = 0.36 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.54, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6, 6820, 1986, 396, 246, 218, 212, 209} \[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=-\frac {(a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right ) \arctan \left (\frac {x}{\sqrt [4]{a+x^4}}\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {(a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right ) \text {arctanh}\left (\frac {x}{\sqrt [4]{a+x^4}}\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}+\frac {x \left (a+x^4\right ) \left (b+x^4\right )}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \]
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Rule 6
Rule 209
Rule 212
Rule 218
Rule 246
Rule 396
Rule 1986
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+\left (4 a b^3+b^4\right ) x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx \\ & = \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+\left (4 a b^3+b^4\right ) x^4+\left (6 a b^2+4 b^3\right ) x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx \\ & = \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+\left (4 a b^3+b^4\right ) x^4+\left (6 a b^2+4 b^3\right ) x^8+\left (4 a b+6 b^2\right ) x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx \\ & = \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+\left (4 a b^3+b^4\right ) x^4+\left (6 a b^2+4 b^3\right ) x^8+\left (4 a b+6 b^2\right ) x^{12}+(a+4 b) x^{16}+x^{20}}} \, dx \\ & = \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \, dx \\ & = \frac {\left (\sqrt [4]{a+x^4} \left (b+x^4\right )\right ) \int \frac {b+x^4}{\sqrt [4]{a+x^4}} \, dx}{\sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \\ & = \frac {x \left (a+x^4\right ) \left (b+x^4\right )}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {\left ((a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right )\right ) \int \frac {1}{\sqrt [4]{a+x^4}} \, dx}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \\ & = \frac {x \left (a+x^4\right ) \left (b+x^4\right )}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {\left ((a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{a+x^4}}\right )}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \\ & = \frac {x \left (a+x^4\right ) \left (b+x^4\right )}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {\left ((a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{a+x^4}}\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {\left ((a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{a+x^4}}\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \\ & = \frac {x \left (a+x^4\right ) \left (b+x^4\right )}{4 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {(a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right ) \arctan \left (\frac {x}{\sqrt [4]{a+x^4}}\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}}-\frac {(a-4 b) \sqrt [4]{a+x^4} \left (b+x^4\right ) \text {arctanh}\left (\frac {x}{\sqrt [4]{a+x^4}}\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.36 \[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\frac {\left (b+x^4\right ) \left (2 x \left (a+x^4\right )-(a-4 b) \sqrt [4]{a+x^4} \arctan \left (\frac {x}{\sqrt [4]{a+x^4}}\right )-(a-4 b) \sqrt [4]{a+x^4} \text {arctanh}\left (\frac {x}{\sqrt [4]{a+x^4}}\right )\right )}{8 \sqrt [4]{\left (a+x^4\right ) \left (b+x^4\right )^4}} \]
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\[\int \frac {\left (x^{4}+b \right )^{2}}{\left (x^{20}+a \,x^{16}+4 x^{16} b +4 a b \,x^{12}+6 b^{2} x^{12}+6 a \,b^{2} x^{8}+4 b^{3} x^{8}+4 b^{3} x^{4} a +b^{4} x^{4}+a \,b^{4}\right )^{\frac {1}{4}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (247) = 494\).
Time = 0.26 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.95 \[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\frac {2 \, {\left ({\left (a - 4 \, b\right )} x^{12} + 3 \, {\left (a b - 4 \, b^{2}\right )} x^{8} + 3 \, {\left (a b^{2} - 4 \, b^{3}\right )} x^{4} + a b^{3} - 4 \, b^{4}\right )} \arctan \left (\frac {{\left (x^{20} + {\left (a + 4 \, b\right )} x^{16} + 2 \, {\left (2 \, a b + 3 \, b^{2}\right )} x^{12} + 2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} x^{8} + a b^{4} + {\left (4 \, a b^{3} + b^{4}\right )} x^{4}\right )}^{\frac {1}{4}}}{x^{5} + b x}\right ) - {\left ({\left (a - 4 \, b\right )} x^{12} + 3 \, {\left (a b - 4 \, b^{2}\right )} x^{8} + 3 \, {\left (a b^{2} - 4 \, b^{3}\right )} x^{4} + a b^{3} - 4 \, b^{4}\right )} \log \left (\frac {x^{5} + b x + {\left (x^{20} + {\left (a + 4 \, b\right )} x^{16} + 2 \, {\left (2 \, a b + 3 \, b^{2}\right )} x^{12} + 2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} x^{8} + a b^{4} + {\left (4 \, a b^{3} + b^{4}\right )} x^{4}\right )}^{\frac {1}{4}}}{x^{5} + b x}\right ) + {\left ({\left (a - 4 \, b\right )} x^{12} + 3 \, {\left (a b - 4 \, b^{2}\right )} x^{8} + 3 \, {\left (a b^{2} - 4 \, b^{3}\right )} x^{4} + a b^{3} - 4 \, b^{4}\right )} \log \left (-\frac {x^{5} + b x - {\left (x^{20} + {\left (a + 4 \, b\right )} x^{16} + 2 \, {\left (2 \, a b + 3 \, b^{2}\right )} x^{12} + 2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} x^{8} + a b^{4} + {\left (4 \, a b^{3} + b^{4}\right )} x^{4}\right )}^{\frac {1}{4}}}{x^{5} + b x}\right ) + 4 \, {\left (x^{20} + {\left (a + 4 \, b\right )} x^{16} + 2 \, {\left (2 \, a b + 3 \, b^{2}\right )} x^{12} + 2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} x^{8} + a b^{4} + {\left (4 \, a b^{3} + b^{4}\right )} x^{4}\right )}^{\frac {3}{4}} x}{16 \, {\left (x^{12} + 3 \, b x^{8} + 3 \, b^{2} x^{4} + b^{3}\right )}} \]
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\[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\int \frac {\left (b + x^{4}\right )^{2}}{\sqrt [4]{\left (a + x^{4}\right ) \left (b + x^{4}\right )^{4}}}\, dx \]
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\[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\int { \frac {{\left (x^{4} + b\right )}^{2}}{{\left (x^{20} + a x^{16} + 4 \, b x^{16} + 4 \, a b x^{12} + 6 \, b^{2} x^{12} + 6 \, a b^{2} x^{8} + 4 \, b^{3} x^{8} + 4 \, a b^{3} x^{4} + b^{4} x^{4} + a b^{4}\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\int { \frac {{\left (x^{4} + b\right )}^{2}}{{\left (x^{20} + a x^{16} + 4 \, b x^{16} + 4 \, a b x^{12} + 6 \, b^{2} x^{12} + 6 \, a b^{2} x^{8} + 4 \, b^{3} x^{8} + 4 \, a b^{3} x^{4} + b^{4} x^{4} + a b^{4}\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {\left (b+x^4\right )^2}{\sqrt [4]{a b^4+4 a b^3 x^4+b^4 x^4+6 a b^2 x^8+4 b^3 x^8+4 a b x^{12}+6 b^2 x^{12}+a x^{16}+4 b x^{16}+x^{20}}} \, dx=\int \frac {{\left (x^4+b\right )}^2}{{\left (b^4\,x^4+a\,b^4+4\,b^3\,x^8+4\,a\,b^3\,x^4+6\,b^2\,x^{12}+6\,a\,b^2\,x^8+4\,b\,x^{16}+4\,a\,b\,x^{12}+x^{20}+a\,x^{16}\right )}^{1/4}} \,d x \]
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