Integrand size = 31, antiderivative size = 100 \[ \int \frac {-b+c x^4+a x^8}{x^2 \left (-b+a x^4\right )^{3/4}} \, dx=\frac {\left (-4+x^4\right ) \sqrt [4]{-b+a x^4}}{4 x}+\frac {(-3 b-4 c) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{8 a^{3/4}}+\frac {(3 b+4 c) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{8 a^{3/4}} \]
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Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1500, 462, 338, 304, 209, 212} \[ \int \frac {-b+c x^4+a x^8}{x^2 \left (-b+a x^4\right )^{3/4}} \, dx=-\frac {(3 b+4 c) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{8 a^{3/4}}+\frac {(3 b+4 c) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{8 a^{3/4}}-\frac {\sqrt [4]{a x^4-b}}{x}+\frac {1}{4} x^3 \sqrt [4]{a x^4-b} \]
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Rule 209
Rule 212
Rule 304
Rule 338
Rule 462
Rule 1500
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^3 \sqrt [4]{-b+a x^4}+\frac {\int \frac {-4 a b+a (3 b+4 c) x^4}{x^2 \left (-b+a x^4\right )^{3/4}} \, dx}{4 a} \\ & = -\frac {\sqrt [4]{-b+a x^4}}{x}+\frac {1}{4} x^3 \sqrt [4]{-b+a x^4}+\frac {1}{4} (3 b+4 c) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{-b+a x^4}}{x}+\frac {1}{4} x^3 \sqrt [4]{-b+a x^4}+\frac {1}{4} (3 b+4 c) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right ) \\ & = -\frac {\sqrt [4]{-b+a x^4}}{x}+\frac {1}{4} x^3 \sqrt [4]{-b+a x^4}+\frac {(3 b+4 c) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt {a}}-\frac {(3 b+4 c) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt {a}} \\ & = -\frac {\sqrt [4]{-b+a x^4}}{x}+\frac {1}{4} x^3 \sqrt [4]{-b+a x^4}-\frac {(3 b+4 c) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{8 a^{3/4}}+\frac {(3 b+4 c) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{8 a^{3/4}} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {-b+c x^4+a x^8}{x^2 \left (-b+a x^4\right )^{3/4}} \, dx=\frac {\left (-4+x^4\right ) \sqrt [4]{-b+a x^4}}{4 x}+\frac {(-3 b-4 c) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{8 a^{3/4}}+\frac {(3 b+4 c) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{8 a^{3/4}} \]
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Time = 0.45 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {\frac {3 x \left (b +\frac {4 c}{3}\right ) \ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right )}{4}+\frac {3 x \left (b +\frac {4 c}{3}\right ) \arctan \left (\frac {\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{2}+a^{\frac {3}{4}} \left (x^{4}-4\right ) \left (a \,x^{4}-b \right )^{\frac {1}{4}}}{4 a^{\frac {3}{4}} x}\) | \(108\) |
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Timed out. \[ \int \frac {-b+c x^4+a x^8}{x^2 \left (-b+a x^4\right )^{3/4}} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 1.89 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.68 \[ \int \frac {-b+c x^4+a x^8}{x^2 \left (-b+a x^4\right )^{3/4}} \, dx=- \frac {a x^{7} e^{\frac {i \pi }{4}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{4 b^{\frac {3}{4}} \Gamma \left (\frac {11}{4}\right )} - b \left (\begin {cases} - \frac {\sqrt [4]{a} \sqrt [4]{-1 + \frac {b}{a x^{4}}} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{4}\right )}{4 b \Gamma \left (\frac {3}{4}\right )} & \text {for}\: \left |{\frac {b}{a x^{4}}}\right | > 1 \\- \frac {\sqrt [4]{a} \sqrt [4]{1 - \frac {b}{a x^{4}}} \Gamma \left (- \frac {1}{4}\right )}{4 b \Gamma \left (\frac {3}{4}\right )} & \text {otherwise} \end {cases}\right ) + \frac {c x^{3} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {a x^{4}}{b}} \right )}}{4 b^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (80) = 160\).
Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.17 \[ \int \frac {-b+c x^4+a x^8}{x^2 \left (-b+a x^4\right )^{3/4}} \, dx=\frac {1}{4} \, c {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {3}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {3}{4}}}\right )} + \frac {1}{16} \, a {\left (\frac {3 \, {\left (\frac {2 \, b \arctan \left (\frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {3}{4}}} - \frac {b \log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {3}{4}}}\right )}}{a} + \frac {4 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}} b}{{\left (a^{2} - \frac {{\left (a x^{4} - b\right )} a}{x^{4}}\right )} x}\right )} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x} \]
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\[ \int \frac {-b+c x^4+a x^8}{x^2 \left (-b+a x^4\right )^{3/4}} \, dx=\int { \frac {a x^{8} + c x^{4} - b}{{\left (a x^{4} - b\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {-b+c x^4+a x^8}{x^2 \left (-b+a x^4\right )^{3/4}} \, dx=\int \frac {a\,x^8+c\,x^4-b}{x^2\,{\left (a\,x^4-b\right )}^{3/4}} \,d x \]
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