Integrand size = 22, antiderivative size = 66 \[ \int \frac {a+b x^2+c x^4}{d+e x^2} \, dx=-\frac {(c d-b e) x}{e^2}+\frac {c x^3}{3 e}+\frac {\left (c d^2-b d e+a e^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{5/2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1167, 211} \[ \int \frac {a+b x^2+c x^4}{d+e x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt {d} e^{5/2}}-\frac {x (c d-b e)}{e^2}+\frac {c x^3}{3 e} \]
[In]
[Out]
Rule 211
Rule 1167
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {c d-b e}{e^2}+\frac {c x^2}{e}+\frac {c d^2-b d e+a e^2}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = -\frac {(c d-b e) x}{e^2}+\frac {c x^3}{3 e}+\frac {\left (c d^2-b d e+a e^2\right ) \int \frac {1}{d+e x^2} \, dx}{e^2} \\ & = -\frac {(c d-b e) x}{e^2}+\frac {c x^3}{3 e}+\frac {\left (c d^2-b d e+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{5/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int \frac {a+b x^2+c x^4}{d+e x^2} \, dx=\frac {(-c d+b e) x}{e^2}+\frac {c x^3}{3 e}+\frac {\left (c d^2-b d e+a e^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{5/2}} \]
[In]
[Out]
Time = 0.51 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {\frac {1}{3} c \,x^{3} e +b e x -c d x}{e^{2}}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{e^{2} \sqrt {e d}}\) | \(57\) |
risch | \(\frac {c \,x^{3}}{3 e}+\frac {b x}{e}-\frac {c d x}{e^{2}}-\frac {\ln \left (e x +\sqrt {-e d}\right ) a}{2 \sqrt {-e d}}+\frac {\ln \left (e x +\sqrt {-e d}\right ) b d}{2 e \sqrt {-e d}}-\frac {\ln \left (e x +\sqrt {-e d}\right ) c \,d^{2}}{2 e^{2} \sqrt {-e d}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) a}{2 \sqrt {-e d}}-\frac {\ln \left (-e x +\sqrt {-e d}\right ) b d}{2 e \sqrt {-e d}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) c \,d^{2}}{2 e^{2} \sqrt {-e d}}\) | \(168\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.41 \[ \int \frac {a+b x^2+c x^4}{d+e x^2} \, dx=\left [\frac {2 \, c d e^{2} x^{3} - 3 \, {\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 6 \, {\left (c d^{2} e - b d e^{2}\right )} x}{6 \, d e^{3}}, \frac {c d e^{2} x^{3} + 3 \, {\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - 3 \, {\left (c d^{2} e - b d e^{2}\right )} x}{3 \, d e^{3}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (58) = 116\).
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.77 \[ \int \frac {a+b x^2+c x^4}{d+e x^2} \, dx=\frac {c x^{3}}{3 e} + x \left (\frac {b}{e} - \frac {c d}{e^{2}}\right ) - \frac {\sqrt {- \frac {1}{d e^{5}}} \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (- d e^{2} \sqrt {- \frac {1}{d e^{5}}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{d e^{5}}} \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (d e^{2} \sqrt {- \frac {1}{d e^{5}}} + x \right )}}{2} \]
[In]
[Out]
Exception generated. \[ \int \frac {a+b x^2+c x^4}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x^2+c x^4}{d+e x^2} \, dx=\frac {{\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e^{2}} + \frac {c e^{2} x^{3} - 3 \, c d e x + 3 \, b e^{2} x}{3 \, e^{3}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86 \[ \int \frac {a+b x^2+c x^4}{d+e x^2} \, dx=x\,\left (\frac {b}{e}-\frac {c\,d}{e^2}\right )+\frac {c\,x^3}{3\,e}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{\sqrt {d}\,e^{5/2}} \]
[In]
[Out]