Integrand size = 38, antiderivative size = 334 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^4} \, dx=\frac {x \sqrt [4]{b x^2+a x^4}}{2 a}+\frac {\left (4 a^2-b\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{4 a^{7/4}}+\frac {\left (-4 a^2+b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{4 a^{7/4}}-\frac {\text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a^3 \log (x)-a^2 b \log (x)+a^3 \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )+a^2 b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4-a b \log (x) \text {$\#$1}^4-a^2 \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+a b \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{4 a} \]
[Out]
Time = 0.98 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.99, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {2081, 6857, 285, 335, 338, 304, 209, 212, 477, 525, 524} \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^4} \, dx=-\frac {b \sqrt [4]{a x^4+b x^2} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 a^{7/4} \sqrt {x} \sqrt [4]{a x^2+b}}+\frac {b \sqrt [4]{a x^4+b x^2} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 a^{7/4} \sqrt {x} \sqrt [4]{a x^2+b}}-\frac {x \left (\frac {\sqrt {-a} a}{\sqrt {b}}+a+1\right ) \sqrt [4]{a x^4+b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a \sqrt [4]{\frac {a x^2}{b}+1}}-\frac {x \left (\frac {(-a)^{3/2}}{\sqrt {b}}+a+1\right ) \sqrt [4]{a x^4+b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a \sqrt [4]{\frac {a x^2}{b}+1}}+\frac {x \sqrt [4]{a x^4+b x^2}}{2 a} \]
[In]
[Out]
Rule 209
Rule 212
Rule 285
Rule 304
Rule 335
Rule 338
Rule 477
Rule 524
Rule 525
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{b x^2+a x^4} \int \frac {\sqrt {x} \sqrt [4]{b+a x^2} \left (-b-a x^2+x^4\right )}{b+a x^4} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {\sqrt [4]{b x^2+a x^4} \int \left (\frac {\sqrt {x} \sqrt [4]{b+a x^2}}{a}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \left ((1+a) b+a^2 x^2\right )}{a \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 \sqrt {x} \sqrt [4]{b+a x^2} \, dx}{a \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\sqrt [4]{b x^2+a x^4} \int \frac {\sqrt {x} \sqrt [4]{b+a x^2} \left ((1+a) b+a^2 x^2\right )}{b+a x^4} \, dx}{a \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {x \sqrt [4]{b x^2+a x^4}}{2 a}-\frac {\sqrt [4]{b x^2+a x^4} \int \left (-\frac {\sqrt {-a} \left (a^2 \sqrt {b}+\sqrt {-a} (1+a) b\right ) \sqrt {x} \sqrt [4]{b+a x^2}}{2 a \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^2\right )}+\frac {\sqrt {-a} \left (a^2 \sqrt {b}-\sqrt {-a} (1+a) b\right ) \sqrt {x} \sqrt [4]{b+a x^2}}{2 a \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^2\right )}\right ) \, dx}{a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x}}{\left (b+a x^2\right )^{3/4}} \, dx}{4 a \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {x \sqrt [4]{b x^2+a x^4}}{2 a}-\frac {\left (\left (a \left (\sqrt {-a}+\sqrt {b}\right )+\sqrt {b}\right ) \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{b+a x^2}}{\sqrt {b}+\sqrt {-a} x^2} \, dx}{2 a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (\left (\sqrt {-a} a-(1+a) \sqrt {b}\right ) \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{b+a x^2}}{\sqrt {b}-\sqrt {-a} x^2} \, dx}{2 a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 a \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {x \sqrt [4]{b x^2+a x^4}}{2 a}-\frac {\left (\left (a \left (\sqrt {-a}+\sqrt {b}\right )+\sqrt {b}\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {b}+\sqrt {-a} x^4} \, dx,x,\sqrt {x}\right )}{a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (\left (\sqrt {-a} a-(1+a) \sqrt {b}\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {b}-\sqrt {-a} x^4} \, dx,x,\sqrt {x}\right )}{a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {x \sqrt [4]{b x^2+a x^4}}{2 a}+\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/2} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/2} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (\left (a \left (\sqrt {-a}+\sqrt {b}\right )+\sqrt {b}\right ) \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 {b}+\sqrt {-a} x^4} \, dx,x,\sqrt {x}\right )}{a \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}}+\frac {\left (\left (\sqrt {-a} a-(1+a) \sqrt {b}\right ) \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 {b}-\sqrt {-a} x^4} \, dx,x,\sqrt {x}\right )}{a \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}} \\ & = \frac {x \sqrt [4]{b x^2+a x^4}}{2 a}-\frac {\left (1+\frac {1}{a}+\frac {\sqrt {-a}}{\sqrt {b}}\right ) 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 \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {\left (1+\frac {1}{a}-\frac {\sqrt {-a}}{\sqrt {b}}\right ) 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 \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {b \sqrt [4]{b x^2+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{7/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {b \sqrt [4]{b x^2+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{7/4} \sqrt {x} \sqrt [4]{b+a x^2}} \\ \end{align*}
Time = 0.99 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.15 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^4} \, dx=\frac {x^{3/2} \left (b+a x^2\right )^{3/4} \left (4 a^{3/4} x^{3/2} \sqrt [4]{b+a x^2}+8 a^2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )-2 b \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+\left (-8 a^2+2 b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+a^{3/4} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a^3 \log (x)-a^2 b \log (x)+2 a^3 \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )+2 a^2 b \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4-a b \log (x) \text {$\#$1}^4-2 a^2 \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+2 b \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+2 a b \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{8 a^{7/4} \left (x^2 \left (b+a x^2\right )\right )^{3/4}} \]
[In]
[Out]
Time = 0.71 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.75
method | result | size |
pseudoelliptic | \(\frac {4 x \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}} a^{\frac {3}{4}}-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}+a b \right )}{\sum }\frac {\left (\left (-a^{2}+a b +b \right ) \textit {\_R}^{4}+a^{2} \left (a +b \right )\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (-\textit {\_R}^{4}+a \right )}\right ) a^{\frac {3}{4}}-4 \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right ) a^{2}-8 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) a^{2}+\ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right ) b +2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b}{8 a^{\frac {7}{4}}}\) | \(252\) |
[In]
[Out]
Timed out. \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^4} \, dx=\text {Timed out} \]
[In]
[Out]
Not integrable
Time = 15.34 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.09 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^4} \, dx=\int \frac {\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (- a x^{2} - b + x^{4}\right )}{a x^{4} + b}\, dx \]
[In]
[Out]
Not integrable
Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^4} \, dx=\int { \frac {{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} - b\right )}}{a x^{4} + b} \,d x } \]
[In]
[Out]
Not integrable
Time = 1.42 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^4} \, dx=\int { \frac {{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} - b\right )}}{a x^{4} + b} \,d x } \]
[In]
[Out]
Not integrable
Time = 7.54 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (-b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^4} \, dx=\int -\frac {{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (-x^4+a\,x^2+b\right )}{a\,x^4+b} \,d x \]
[In]
[Out]