Integrand size = 35, antiderivative size = 69 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {1}{2} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a+2 \text {$\#$1}^4}\&\right ] \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(587\) vs. \(2(69)=138\).
Time = 0.83 (sec) , antiderivative size = 587, normalized size of antiderivative = 8.51, number of steps used = 21, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2081, 1283, 1442, 399, 246, 218, 212, 209, 385, 214, 211} \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {\sqrt {x} \left (-a \sqrt {a^2-4 b}+a^2-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a x^4+b x^2}}+\frac {\sqrt {x} \left (a \sqrt {a^2-4 b}+a^2-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2+b}}\right )}{\left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \left (-a \sqrt {a^2-4 b}+a^2-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a x^4+b x^2}}+\frac {\sqrt {x} \left (a \sqrt {a^2-4 b}+a^2-2 b\right )^{3/4} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2+b}}\right )}{\left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a x^4+b x^2}} \]
[In]
[Out]
Rule 209
Rule 211
Rule 212
Rule 214
Rule 218
Rule 246
Rule 385
Rule 399
Rule 1283
Rule 1442
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {\left (b+a x^2\right )^{3/4}}{\sqrt {x} \left (b+a x^2+x^4\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 {\left (b+a x^4\right )^{3/4}}{b+a x^4+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (4 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {\left (b+a x^4\right )^{3/4}}{a-\sqrt {a^2-4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (4 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {\left (b+a x^4\right )^{3/4}}{a+\sqrt {a^2-4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (2 \left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (a+\sqrt {a^2-4 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \left (a^2-a \sqrt {a^2-4 b}-2 b\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (a-\sqrt {a^2-4 b}+2 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (2 \left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a^2-4 b}-\left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \left (a^2-a \sqrt {a^2-4 b}-2 b\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a^2-4 b}-\left (a \left (a-\sqrt {a^2-4 b}\right )-2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (\left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}+\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\left (a^2-a \sqrt {a^2-4 b}-2 b\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\left (a^2-a \sqrt {a^2-4 b}-2 b\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}+\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (a^2-a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (a^2+a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\left (a+\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (a^2-a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (a^2+a \sqrt {a^2-4 b}-2 b\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{b+a x^2}}\right )}{\left (a+\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{b x^2+a x^4}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.51 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}^3+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^3}{-a+2 \text {$\#$1}^4}\&\right ]}{2 \sqrt [4]{x^2 \left (b+a x^2\right )}} \]
[In]
[Out]
Time = 1.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80
method | result | size |
pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4} a +b \right )}{\sum }\left (-\frac {\textit {\_R}^{3} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{-2 \textit {\_R}^{4}+a}\right )\right )}{2}\) | \(55\) |
[In]
[Out]
Timed out. \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \]
[In]
[Out]
Not integrable
Time = 7.39 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.45 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a x^{2} + b}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{2} + b + x^{4}\right )}\, dx \]
[In]
[Out]
Not integrable
Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.51 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + a x^{2} + b\right )}} \,d x } \]
[In]
[Out]
Not integrable
Time = 3.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.51 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + a x^{2} + b\right )}} \,d x } \]
[In]
[Out]
Not integrable
Time = 5.58 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.51 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a\,x^2+b}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (x^4+a\,x^2+b\right )} \,d x \]
[In]
[Out]