Integrand size = 20, antiderivative size = 69 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\frac {x^2}{2 a}+\frac {(a-b) \arctan \left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}-\frac {\log \left (a+b+2 a x^2+a x^4\right )}{2 a} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1128, 717, 648, 632, 210, 642} \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\frac {(a-b) \arctan \left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}-\frac {\log \left (a x^4+2 a x^2+a+b\right )}{2 a}+\frac {x^2}{2 a} \]
[In]
[Out]
Rule 210
Rule 632
Rule 642
Rule 648
Rule 717
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{a+b+2 a x+a x^2} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2 a}+\frac {\text {Subst}\left (\int \frac {-a-b-2 a x}{a+b+2 a x+a x^2} \, dx,x,x^2\right )}{2 a} \\ & = \frac {x^2}{2 a}-\frac {\text {Subst}\left (\int \frac {2 a+2 a x}{a+b+2 a x+a x^2} \, dx,x,x^2\right )}{2 a}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{a+b+2 a x+a x^2} \, dx,x,x^2\right )}{2 a} \\ & = \frac {x^2}{2 a}-\frac {\log \left (a+b+2 a x^2+a x^4\right )}{2 a}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{-4 a b-x^2} \, dx,x,2 a \left (1+x^2\right )\right )}{a} \\ & = \frac {x^2}{2 a}+\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}-\frac {\log \left (a+b+2 a x^2+a x^4\right )}{2 a} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\frac {\frac {(a-b) \arctan \left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{\sqrt {b}}+\sqrt {a} \left (x^2-\log \left (b+a \left (1+x^2\right )^2\right )\right )}{2 a^{3/2}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {x^{2}}{2 a}+\frac {-\ln \left (a \,x^{4}+2 a \,x^{2}+a +b \right )+\frac {\left (a -b \right ) \arctan \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{\sqrt {a b}}}{2 a}\) | \(62\) |
risch | \(\frac {x^{2}}{2 a}-\frac {\ln \left (\left (a^{2} b -b^{2} a +\sqrt {-a b \left (a -b \right )^{2}}\, a \right ) x^{2}+\sqrt {-a b \left (a -b \right )^{2}}\, a +\sqrt {-a b \left (a -b \right )^{2}}\, b \right )}{2 a}+\frac {\ln \left (\left (a^{2} b -b^{2} a +\sqrt {-a b \left (a -b \right )^{2}}\, a \right ) x^{2}+\sqrt {-a b \left (a -b \right )^{2}}\, a +\sqrt {-a b \left (a -b \right )^{2}}\, b \right ) \sqrt {-a b \left (a -b \right )^{2}}}{4 a^{2} b}-\frac {\ln \left (\left (a^{2} b -b^{2} a -\sqrt {-a b \left (a -b \right )^{2}}\, a \right ) x^{2}-\sqrt {-a b \left (a -b \right )^{2}}\, a -\sqrt {-a b \left (a -b \right )^{2}}\, b \right )}{2 a}-\frac {\ln \left (\left (a^{2} b -b^{2} a -\sqrt {-a b \left (a -b \right )^{2}}\, a \right ) x^{2}-\sqrt {-a b \left (a -b \right )^{2}}\, a -\sqrt {-a b \left (a -b \right )^{2}}\, b \right ) \sqrt {-a b \left (a -b \right )^{2}}}{4 a^{2} b}\) | \(320\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.28 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\left [\frac {2 \, a b x^{2} - 2 \, a b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + \sqrt {-a b} {\left (a - b\right )} \log \left (\frac {a x^{4} + 2 \, a x^{2} + 2 \, \sqrt {-a b} {\left (x^{2} + 1\right )} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right )}{4 \, a^{2} b}, \frac {a b x^{2} - a b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) - \sqrt {a b} {\left (a - b\right )} \arctan \left (\frac {\sqrt {a b}}{a x^{2} + a}\right )}{2 \, a^{2} b}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (60) = 120\).
Time = 0.63 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.09 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\left (- \frac {1}{2 a} - \frac {\sqrt {- a^{3} b} \left (a - b\right )}{4 a^{3} b}\right ) \log {\left (x^{2} + \frac {4 a b \left (- \frac {1}{2 a} - \frac {\sqrt {- a^{3} b} \left (a - b\right )}{4 a^{3} b}\right ) + a + b}{a - b} \right )} + \left (- \frac {1}{2 a} + \frac {\sqrt {- a^{3} b} \left (a - b\right )}{4 a^{3} b}\right ) \log {\left (x^{2} + \frac {4 a b \left (- \frac {1}{2 a} + \frac {\sqrt {- a^{3} b} \left (a - b\right )}{4 a^{3} b}\right ) + a + b}{a - b} \right )} + \frac {x^{2}}{2 a} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\frac {x^{2}}{2 \, a} + \frac {{\left (a - b\right )} \arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} - \frac {\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{2 \, a} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\frac {x^{2}}{2 \, a} + \frac {{\left (a - b\right )} \arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} - \frac {\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{2 \, a} \]
[In]
[Out]
Time = 13.21 (sec) , antiderivative size = 302, normalized size of antiderivative = 4.38 \[ \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx=\frac {x^2}{2\,a}-\frac {\ln \left (a\,x^4+2\,a\,x^2+a+b\right )}{2\,a}-\frac {\mathrm {atan}\left (\frac {a\,b\,\left (x^2\,\left (\frac {\frac {\sqrt {a}\,\left (2\,a-2\,b\right )}{\sqrt {b}}+\frac {\left (a-b\right )\,\left (4\,a\,b-12\,a^2\right )}{4\,a^{3/2}\,\sqrt {b}}}{a+b}+\frac {\sqrt {a}\,\left (6\,a-2\,b-\frac {{\left (a-b\right )}^2}{b}+\frac {2\,a\,b-6\,a^2}{a}\right )}{\sqrt {b}\,\left (a+b\right )}\right )-\frac {\frac {\left (a-b\right )\,\left (16\,a\,b-\frac {8\,a^3+8\,b\,a^2}{a}+16\,a^2\right )}{4\,a^{3/2}\,\sqrt {b}}-\frac {\left (16\,a^3+16\,b\,a^2\right )\,\left (a-b\right )}{8\,a^{5/2}\,\sqrt {b}}}{a+b}+\frac {\sqrt {a}\,\left (4\,a+4\,b-\frac {8\,a\,b-\frac {8\,a^3+8\,b\,a^2}{2\,a}+8\,a^2}{a}-\frac {{\left (a-b\right )}^2\,\left (a^3+b\,a^2\right )}{a^3\,b}\right )}{\sqrt {b}\,\left (a+b\right )}\right )}{a^2-2\,a\,b+b^2}\right )\,\left (a-b\right )}{2\,a^{3/2}\,\sqrt {b}} \]
[In]
[Out]