Integrand size = 26, antiderivative size = 81 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=-\frac {\text {RootSum}\left [a^4+a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(1017\) vs. \(2(81)=162\).
Time = 2.42 (sec) , antiderivative size = 1017, normalized size of antiderivative = 12.56, number of steps used = 22, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2081, 6847, 6857, 1443, 385, 218, 212, 209, 214, 211} \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\frac {\sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}} \]
[In]
[Out]
Rule 209
Rule 211
Rule 212
Rule 214
Rule 218
Rule 385
Rule 1443
Rule 2081
Rule 6847
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{-b+a x^2} \left (b+a x^8\right )} \, dx}{\sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {b} \sqrt [4]{-b+a x^4} \left (\sqrt {b}-\sqrt {-a} x^8\right )}+\frac {1}{2 \sqrt {b} \sqrt [4]{-b+a x^4} \left (\sqrt {b}+\sqrt {-a} x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (\sqrt {b}-\sqrt {-a} x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (\sqrt {b}+\sqrt {-a} x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b} \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (\sqrt {-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-\sqrt {-a}} \sqrt [4]{b}-\sqrt {-a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-\sqrt {-a}} \sqrt [4]{b}+\sqrt {-a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-a} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [4]{-a} \sqrt [4]{b}-\sqrt {-a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-a} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [4]{-a} \sqrt [4]{b}+\sqrt {-a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (\sqrt {-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\sqrt {-a}} \sqrt [4]{b}-\left (\sqrt {-\sqrt {-a}} a \sqrt [4]{b}-\sqrt {-a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\sqrt {-a}} \sqrt [4]{b}-\left (\sqrt {-\sqrt {-a}} a \sqrt [4]{b}+\sqrt {-a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-a} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a} \sqrt [4]{b}-\left (\sqrt [4]{-a} a \sqrt [4]{b}-\sqrt {-a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-a} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a} \sqrt [4]{b}-\left (\sqrt [4]{-a} a \sqrt [4]{b}+\sqrt {-a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a-\sqrt [4]{-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a-\sqrt [4]{-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a+\sqrt [4]{-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a+\sqrt [4]{-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-\sqrt {-a}}-\sqrt {\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-\sqrt {-a}}+\sqrt {\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-\sqrt {-a}}-\sqrt {\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-\sqrt {-a}}+\sqrt {\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {RootSum}\left [a^4+a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b \sqrt [4]{-b x^2+a x^4}} \]
[In]
[Out]
Time = 1.51 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-4 a \,\textit {\_Z}^{12}+6 a^{2} \textit {\_Z}^{8}-4 a^{3} \textit {\_Z}^{4}+a^{4}+a \,b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}}{8 b}\) | \(73\) |
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\text {Timed out} \]
[In]
[Out]
Not integrable
Time = 7.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.27 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{8} + b\right )}\, dx \]
[In]
[Out]
Not integrable
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\int { \frac {1}{{\left (a x^{8} + b\right )} {\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Not integrable
Time = 5.68 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\int \frac {1}{\left (a\,x^8+b\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}} \,d x \]
[In]
[Out]