Integrand size = 30, antiderivative size = 106 \[ \int \frac {\sqrt [4]{b x^2+a x^4}}{x^4 \left (-b+a x^4\right )} \, dx=\frac {2 \left (b+a x^2\right ) \sqrt [4]{b x^2+a x^4}}{5 b^2 x^3}+\frac {a \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ]}{4 b} \]
[Out]
Time = 0.79 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.86, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2081, 1284, 1535, 277, 270, 6857, 1543, 525, 524} \[ \int \frac {\sqrt [4]{b x^2+a x^4}}{x^4 \left (-b+a x^4\right )} \, dx=-\frac {a x \sqrt [4]{a x^4+b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 b^2 \sqrt [4]{\frac {a x^2}{b}+1}}-\frac {a x \sqrt [4]{a x^4+b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 b^2 \sqrt [4]{\frac {a x^2}{b}+1}}+\frac {2 a \sqrt [4]{a x^4+b x^2}}{5 b^2 x}+\frac {2 \sqrt [4]{a x^4+b x^2}}{5 b x^3} \]
[In]
[Out]
Rule 270
Rule 277
Rule 524
Rule 525
Rule 1284
Rule 1535
Rule 1543
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{b x^2+a x^4} \int \frac {\sqrt [4]{b+a x^2}}{x^{7/2} \left (-b+a x^4\right )} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{b+a x^4}}{x^6 \left (-b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}} \\ & = -\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{x^6 \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {-a b-a b x^4}{x^2 \left (b+a x^4\right )^{3/4} \left (-b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {2 \sqrt [4]{b x^2+a x^4}}{5 b x^3}-\frac {\left (2 \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \left (\frac {a}{x^2 \left (b+a x^4\right )^{3/4}}-\frac {a x^2 \sqrt [4]{b+a x^4}}{-b+a x^8}\right ) \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (8 a \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{5 b \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {2 \sqrt [4]{b x^2+a x^4}}{5 b x^3}-\frac {8 a \sqrt [4]{b x^2+a x^4}}{5 b^2 x}-\frac {\left (2 a \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 a \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{-b+a x^8} \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {2 \sqrt [4]{b x^2+a x^4}}{5 b x^3}+\frac {2 a \sqrt [4]{b x^2+a x^4}}{5 b^2 x}+\frac {\left (2 a \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {a} x^2 \sqrt [4]{b+a x^4}}{2 \sqrt {b} \left (\sqrt {a} \sqrt {b}-a x^4\right )}-\frac {\sqrt {a} x^2 \sqrt [4]{b+a x^4}}{2 \sqrt {b} \left (\sqrt {a} \sqrt {b}+a x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {2 \sqrt [4]{b x^2+a x^4}}{5 b x^3}+\frac {2 a \sqrt [4]{b x^2+a x^4}}{5 b^2 x}-\frac {\left (a^{3/2} \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {a} \sqrt {b}-a x^4} \, dx,x,\sqrt {x}\right )}{b^{3/2} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (a^{3/2} \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {a} \sqrt {b}+a x^4} \, dx,x,\sqrt {x}\right )}{b^{3/2} \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {2 \sqrt [4]{b x^2+a x^4}}{5 b x^3}+\frac {2 a \sqrt [4]{b x^2+a x^4}}{5 b^2 x}-\frac {\left (a^{3/2} \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{\sqrt {a} \sqrt {b}-a x^4} \, dx,x,\sqrt {x}\right )}{b^{3/2} \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {\left (a^{3/2} \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{\sqrt {a} \sqrt {b}+a x^4} \, dx,x,\sqrt {x}\right )}{b^{3/2} \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}} \\ & = \frac {2 \sqrt [4]{b x^2+a x^4}}{5 b x^3}+\frac {2 a \sqrt [4]{b x^2+a x^4}}{5 b^2 x}-\frac {a x \sqrt [4]{b x^2+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 b^2 \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {a x \sqrt [4]{b x^2+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 b^2 \sqrt [4]{1+\frac {a x^2}{b}}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt [4]{b x^2+a x^4}}{x^4 \left (-b+a x^4\right )} \, dx=\frac {\sqrt [4]{x^2 \left (b+a x^2\right )} \left (8 \left (b+a x^2\right )^{5/4}+5 a b x^{5/2} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ]\right )}{20 b^2 x^3 \sqrt [4]{b+a x^2}} \]
[In]
[Out]
Time = 0.48 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {5 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}-a b \right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{4}-a}\right ) b \,x^{3}+8 \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}} \left (a \,x^{2}+b \right )}{20 x^{3} b^{2}}\) | \(94\) |
[In]
[Out]
Timed out. \[ \int \frac {\sqrt [4]{b x^2+a x^4}}{x^4 \left (-b+a x^4\right )} \, dx=\text {Timed out} \]
[In]
[Out]
Not integrable
Time = 7.38 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.23 \[ \int \frac {\sqrt [4]{b x^2+a x^4}}{x^4 \left (-b+a x^4\right )} \, dx=\int \frac {\sqrt [4]{x^{2} \left (a x^{2} + b\right )}}{x^{4} \left (a x^{4} - b\right )}\, dx \]
[In]
[Out]
Not integrable
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt [4]{b x^2+a x^4}}{x^4 \left (-b+a x^4\right )} \, dx=\int { \frac {{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}}{{\left (a x^{4} - b\right )} x^{4}} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.84 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt [4]{b x^2+a x^4}}{x^4 \left (-b+a x^4\right )} \, dx=\int { \frac {{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}}{{\left (a x^{4} - b\right )} x^{4}} \,d x } \]
[In]
[Out]
Not integrable
Time = 6.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt [4]{b x^2+a x^4}}{x^4 \left (-b+a x^4\right )} \, dx=-\int \frac {{\left (a\,x^4+b\,x^2\right )}^{1/4}}{x^4\,\left (b-a\,x^4\right )} \,d x \]
[In]
[Out]