Integrand size = 38, antiderivative size = 106 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{x^4 \left (b+a x^4\right )} \, dx=-\frac {2 \left (-b+a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{5 b x^3}+\frac {1}{2} a \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(236\) vs. \(2(106)=212\).
Time = 0.98 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.23, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {2081, 6857, 270, 1284, 1535, 277, 1543, 525, 524} \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{x^4 \left (b+a x^4\right )} \, dx=\frac {2 a x \sqrt [4]{a x^4-b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {-a} x^2}{\sqrt {b}},\frac {a x^2}{b}\right )}{3 b \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {2 a x \sqrt [4]{a x^4-b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {-a} x^2}{\sqrt {b}},\frac {a x^2}{b}\right )}{3 b \sqrt [4]{1-\frac {a x^2}{b}}}-\frac {4 a \sqrt [4]{a x^4-b x^2}}{5 b x}-\frac {2 \left (b-a x^2\right ) \sqrt [4]{a x^4-b x^2}}{5 b x^3}+\frac {4 \sqrt [4]{a x^4-b x^2}}{5 x^3} \]
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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} \left (-b+a x^4\right )}{x^{7/2} \left (b+a x^4\right )} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}} \\ & = \frac {\sqrt [4]{-b x^2+a x^4} \int \left (\frac {\sqrt [4]{-b+a x^2}}{x^{7/2}}-\frac {2 b \sqrt [4]{-b+a x^2}}{x^{7/2} \left (b+a x^4\right )}\right ) \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}} \\ & = \frac {\sqrt [4]{-b x^2+a x^4} \int \frac {\sqrt [4]{-b+a x^2}}{x^{7/2}} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (2 b \sqrt [4]{-b x^2+a x^4}\right ) \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 {2 \left (b-a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{5 b x^3}-\frac {\left (4 b \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 {2 \left (b-a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{5 b x^3}-\frac {\left (4 \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 )}{\sqrt {x} \sqrt [4]{-b+a x^2}}+\frac {\left (4 b \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 {4 \sqrt [4]{-b x^2+a x^4}}{5 x^3}-\frac {2 \left (b-a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{5 b x^3}-\frac {\left (4 \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 )}{\sqrt {x} \sqrt [4]{-b+a x^2}}+\frac {\left (16 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 \sqrt {x} \sqrt [4]{-b+a x^2}} \\ & = \frac {4 \sqrt [4]{-b x^2+a x^4}}{5 x^3}+\frac {16 a \sqrt [4]{-b x^2+a x^4}}{5 b x}-\frac {2 \left (b-a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{5 b x^3}-\frac {\left (4 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 )}{\sqrt {x} \sqrt [4]{-b+a x^2}}+\frac {\left (4 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 )}{\sqrt {x} \sqrt [4]{-b+a x^2}} \\ & = \frac {4 \sqrt [4]{-b x^2+a x^4}}{5 x^3}-\frac {4 a \sqrt [4]{-b x^2+a x^4}}{5 b x}-\frac {2 \left (b-a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{5 b x^3}+\frac {\left (4 a \sqrt [4]{-b x^2+a x^4}\right ) \text {Subst}\left (\int \left (-\frac {a x^2 \sqrt [4]{-b+a x^4}}{2 \sqrt {-a} \sqrt {b} \left (\sqrt {-a} \sqrt {b}-a x^4\right )}-\frac {a x^2 \sqrt [4]{-b+a x^4}}{2 \sqrt {-a} \sqrt {b} \left (\sqrt {-a} \sqrt {b}+a x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-b+a x^2}} \\ & = \frac {4 \sqrt [4]{-b x^2+a x^4}}{5 x^3}-\frac {4 a \sqrt [4]{-b x^2+a x^4}}{5 b x}-\frac {2 \left (b-a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{5 b x^3}+\frac {\left (2 \sqrt {-a} a \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 )}{\sqrt {b} \sqrt {x} \sqrt [4]{-b+a x^2}}+\frac {\left (2 \sqrt {-a} a \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 )}{\sqrt {b} \sqrt {x} \sqrt [4]{-b+a x^2}} \\ & = \frac {4 \sqrt [4]{-b x^2+a x^4}}{5 x^3}-\frac {4 a \sqrt [4]{-b x^2+a x^4}}{5 b x}-\frac {2 \left (b-a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{5 b x^3}+\frac {\left (2 \sqrt {-a} a \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 )}{\sqrt {b} \sqrt {x} \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {\left (2 \sqrt {-a} a \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 )}{\sqrt {b} \sqrt {x} \sqrt [4]{1-\frac {a x^2}{b}}} \\ & = \frac {4 \sqrt [4]{-b x^2+a x^4}}{5 x^3}-\frac {4 a \sqrt [4]{-b x^2+a x^4}}{5 b x}-\frac {2 \left (b-a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{5 b x^3}+\frac {2 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 \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {2 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 \sqrt [4]{1-\frac {a x^2}{b}}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.26 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{x^4 \left (b+a x^4\right )} \, dx=\frac {\sqrt [4]{-b x^2+a x^4} \left (-4 \left (-b+a x^2\right )^{5/4}+5 a b x^{5/2} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ]\right )}{10 b x^3 \sqrt [4]{-b+a x^2}} \]
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Time = 0.00 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {5 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}+a b \right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{4}-a}\right ) b \,x^{3}-4 \left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}} \left (a \,x^{2}-b \right )}{10 x^{3} b}\) | \(99\) |
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Timed out. \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{x^4 \left (b+a x^4\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{x^4 \left (b+a x^4\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.36 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{x^4 \left (b+a x^4\right )} \, dx=\int { \frac {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}{{\left (a x^{4} + b\right )} x^{4}} \,d x } \]
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Not integrable
Time = 0.94 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.36 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{x^4 \left (b+a x^4\right )} \, dx=\int { \frac {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}{{\left (a x^{4} + b\right )} x^{4}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.36 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{x^4 \left (b+a x^4\right )} \, dx=\int -\frac {\left (b-a\,x^4\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}}{x^4\,\left (a\,x^4+b\right )} \,d x \]
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