Integrand size = 33, antiderivative size = 120 \[ \int \frac {b+a x^8}{x^6 \left (-b+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx=\frac {\left (b+a x^4\right )^{5/4}}{5 b^2 x^5}+\frac {\left (a^{5/4}+\sqrt [4]{a} b\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} b^2}-\frac {\left (a^{5/4}+\sqrt [4]{a} b\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} b^2} \]
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Time = 0.46 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {6857, 277, 270, 508, 304, 209, 212} \[ \int \frac {b+a x^8}{x^6 \left (-b+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx=\frac {\sqrt [4]{a} (a+b) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} b^2}-\frac {\sqrt [4]{a} (a+b) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} b^2}+\frac {a \sqrt [4]{a x^4+b}}{5 b^2 x}+\frac {\sqrt [4]{a x^4+b}}{5 b x^5} \]
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Rule 209
Rule 212
Rule 270
Rule 277
Rule 304
Rule 508
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{x^6 \left (b+a x^4\right )^{3/4}}-\frac {a}{b x^2 \left (b+a x^4\right )^{3/4}}-\frac {a (a+b) x^2}{b \left (b-a x^4\right ) \left (b+a x^4\right )^{3/4}}\right ) \, dx \\ & = -\frac {a \int \frac {1}{x^2 \left (b+a x^4\right )^{3/4}} \, dx}{b}-\frac {(a (a+b)) \int \frac {x^2}{\left (b-a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx}{b}-\int \frac {1}{x^6 \left (b+a x^4\right )^{3/4}} \, dx \\ & = \frac {\sqrt [4]{b+a x^4}}{5 b x^5}+\frac {a \sqrt [4]{b+a x^4}}{b^2 x}+\frac {(4 a) \int \frac {1}{x^2 \left (b+a x^4\right )^{3/4}} \, dx}{5 b}-\frac {(a (a+b)) \text {Subst}\left (\int \frac {x^2}{b-2 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{b} \\ & = \frac {\sqrt [4]{b+a x^4}}{5 b x^5}+\frac {a \sqrt [4]{b+a x^4}}{5 b^2 x}-\frac {\left (\sqrt {a} (a+b)\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2} b^2}+\frac {\left (\sqrt {a} (a+b)\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2} b^2} \\ & = \frac {\sqrt [4]{b+a x^4}}{5 b x^5}+\frac {a \sqrt [4]{b+a x^4}}{5 b^2 x}+\frac {\sqrt [4]{a} (a+b) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} b^2}-\frac {\sqrt [4]{a} (a+b) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} b^2} \\ \end{align*}
Time = 1.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {b+a x^8}{x^6 \left (-b+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx=\frac {\left (b+a x^4\right )^{5/4}}{5 b^2 x^5}+\frac {\sqrt [4]{a} (a+b) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} b^2}-\frac {\sqrt [4]{a} (a+b) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} b^2} \]
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Time = 0.46 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.98
method | result | size |
pseudoelliptic | \(\frac {-5 x^{5} 2^{\frac {1}{4}} \left (a^{\frac {5}{4}}+a^{\frac {1}{4}} b \right ) \ln \left (\frac {x 2^{\frac {1}{4}} a^{\frac {1}{4}}+\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{-x 2^{\frac {1}{4}} a^{\frac {1}{4}}+\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right )-10 x^{5} 2^{\frac {1}{4}} \left (a^{\frac {5}{4}}+a^{\frac {1}{4}} b \right ) \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right )+8 \left (a \,x^{4}+b \right )^{\frac {5}{4}}}{40 b^{2} x^{5}}\) | \(117\) |
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Timed out. \[ \int \frac {b+a x^8}{x^6 \left (-b+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {b+a x^8}{x^6 \left (-b+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx=\int \frac {a x^{8} + b}{x^{6} \left (a x^{4} - b\right ) \left (a x^{4} + b\right )^{\frac {3}{4}}}\, dx \]
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\[ \int \frac {b+a x^8}{x^6 \left (-b+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx=\int { \frac {a x^{8} + b}{{\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - b\right )} x^{6}} \,d x } \]
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\[ \int \frac {b+a x^8}{x^6 \left (-b+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx=\int { \frac {a x^{8} + b}{{\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - b\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {b+a x^8}{x^6 \left (-b+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx=-\int \frac {a\,x^8+b}{x^6\,{\left (a\,x^4+b\right )}^{3/4}\,\left (b-a\,x^4\right )} \,d x \]
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