Integrand size = 20, antiderivative size = 54 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}}+\frac {\log \left (a+b+2 a x^2+a x^4\right )}{4 a} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1128, 648, 632, 210, 642} \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=\frac {\log \left (a x^4+2 a x^2+a+b\right )}{4 a}-\frac {\arctan \left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}} \]
[In]
[Out]
Rule 210
Rule 632
Rule 642
Rule 648
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{a+b+2 a x+a x^2} \, dx,x,x^2\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{a+b+2 a x+a x^2} \, dx,x,x^2\right )\right )+\frac {\text {Subst}\left (\int \frac {2 a+2 a x}{a+b+2 a x+a x^2} \, dx,x,x^2\right )}{4 a} \\ & = \frac {\log \left (a+b+2 a x^2+a x^4\right )}{4 a}+\text {Subst}\left (\int \frac {1}{-4 a b-x^2} \, dx,x,2 a \left (1+x^2\right )\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}}+\frac {\log \left (a+b+2 a x^2+a x^4\right )}{4 a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=\frac {-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{\sqrt {b}}+\log \left (b+a \left (1+x^2\right )^2\right )}{4 a} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {\ln \left (a \,x^{4}+2 a \,x^{2}+a +b \right )}{4 a}-\frac {\arctan \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{2 \sqrt {a b}}\) | \(47\) |
risch | \(\frac {\ln \left (\left (a \sqrt {-a b}-a b \right ) x^{2}+a \sqrt {-a b}+\sqrt {-a b}\, b \right )}{4 a}+\frac {\ln \left (\left (a \sqrt {-a b}-a b \right ) x^{2}+a \sqrt {-a b}+\sqrt {-a b}\, b \right ) \sqrt {-a b}}{4 b a}+\frac {\ln \left (\left (-a \sqrt {-a b}-a b \right ) x^{2}-a \sqrt {-a b}-\sqrt {-a b}\, b \right )}{4 a}-\frac {\ln \left (\left (-a \sqrt {-a b}-a b \right ) x^{2}-a \sqrt {-a b}-\sqrt {-a b}\, b \right ) \sqrt {-a b}}{4 b a}\) | \(186\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.43 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=\left [\frac {b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) - \sqrt {-a b} \log \left (\frac {a x^{4} + 2 \, a x^{2} + 2 \, \sqrt {-a b} {\left (x^{2} + 1\right )} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right )}{4 \, a b}, \frac {b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 2 \, \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{a x^{2} + a}\right )}{4 \, a b}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (48) = 96\).
Time = 0.35 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.17 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=\left (\frac {1}{4 a} - \frac {\sqrt {- a^{3} b}}{4 a^{2} b}\right ) \log {\left (x^{2} + \frac {- 4 a b \left (\frac {1}{4 a} - \frac {\sqrt {- a^{3} b}}{4 a^{2} b}\right ) + a + b}{a} \right )} + \left (\frac {1}{4 a} + \frac {\sqrt {- a^{3} b}}{4 a^{2} b}\right ) \log {\left (x^{2} + \frac {- 4 a b \left (\frac {1}{4 a} + \frac {\sqrt {- a^{3} b}}{4 a^{2} b}\right ) + a + b}{a} \right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=-\frac {\arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b}} + \frac {\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{4 \, a} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=-\frac {\arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b}} + \frac {\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{4 \, a} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.57 \[ \int \frac {x^3}{a+b+2 a x^2+a x^4} \, dx=\frac {\ln \left (a\,x^4+2\,a\,x^2+a+b\right )}{4\,a}-\frac {\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}}{a+b}+\frac {a^{3/2}}{\sqrt {b}\,\left (a+b\right )}+\frac {\sqrt {a}\,\sqrt {b}\,x^2}{a+b}+\frac {a^{3/2}\,x^2}{\sqrt {b}\,\left (a+b\right )}\right )}{2\,\sqrt {a}\,\sqrt {b}} \]
[In]
[Out]