Integrand size = 28, antiderivative size = 30 \[ \int \frac {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x+x^5}}{1+x^4}\right )}{\sqrt {a}} \]
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\[ \int \frac {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\int \frac {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {1+x^4}\right ) \int \frac {-1+3 x^4}{\sqrt {x} \sqrt {1+x^4} \left (1-a x+x^4\right )} \, dx}{\sqrt {x+x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {-1+3 x^8}{\sqrt {1+x^8} \left (1-a x^2+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \left (\frac {3}{\sqrt {1+x^8}}-\frac {4-3 a x^2}{\sqrt {1+x^8} \left (1-a x^2+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}} \\ & = -\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {4-3 a x^2}{\sqrt {1+x^8} \left (1-a x^2+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}} \\ & = -\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \left (-\frac {4}{\left (-1+a x^2-x^8\right ) \sqrt {1+x^8}}+\frac {3 a x^2}{\left (-1+a x^2-x^8\right ) \sqrt {1+x^8}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (3 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (3 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {1+x^2}{\sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}} \\ & = \frac {3 x^2 \sqrt {\frac {(1+x)^2}{x}} \sqrt {-\frac {1+x^4}{x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {\sqrt {2}-2 x+\sqrt {2} x^2}{x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{\sqrt {2+\sqrt {2}} (1+x) \sqrt {x+x^5}}-\frac {3 \sqrt {-\frac {(1-x)^2}{x}} x^2 \sqrt {-\frac {1+x^4}{x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {\sqrt {2}+2 x+\sqrt {2} x^2}{x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{\sqrt {2+\sqrt {2}} (1-x) \sqrt {x+x^5}}+\frac {\left (8 \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+a x^2-x^8\right ) \sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}-\frac {\left (6 a \sqrt {x} \sqrt {1+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+a x^2-x^8\right ) \sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}} \\ \end{align*}
Time = 10.51 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x+x^5}}{1+x^4}\right )}{\sqrt {a}} \]
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Time = 0.91 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77
method | result | size |
pseudoelliptic | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{4}+1\right )}}{x \sqrt {a}}\right )}{\sqrt {a}}\) | \(23\) |
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Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 4.23 \[ \int \frac {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\left [\frac {\log \left (\frac {x^{8} + 6 \, a x^{5} + a^{2} x^{2} + 2 \, x^{4} - 4 \, \sqrt {x^{5} + x} {\left (x^{4} + a x + 1\right )} \sqrt {a} + 6 \, a x + 1}{x^{8} - 2 \, a x^{5} + a^{2} x^{2} + 2 \, x^{4} - 2 \, a x + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {x^{5} + x} {\left (x^{4} + a x + 1\right )} \sqrt {-a}}{2 \, {\left (a x^{5} + a x\right )}}\right )}{a}\right ] \]
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Timed out. \[ \int \frac {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\text {Timed out} \]
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\[ \int \frac {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\int { \frac {3 \, x^{4} - 1}{\sqrt {x^{5} + x} {\left (x^{4} - a x + 1\right )}} \,d x } \]
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\[ \int \frac {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\int { \frac {3 \, x^{4} - 1}{\sqrt {x^{5} + x} {\left (x^{4} - a x + 1\right )}} \,d x } \]
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Time = 5.50 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\frac {\ln \left (\frac {a\,x-2\,\sqrt {a}\,\sqrt {x^5+x}+x^4+1}{x^4-a\,x+1}\right )}{\sqrt {a}} \]
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