Integrand size = 24, antiderivative size = 83 \[ \int \frac {x^6}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {x^5}{6 b \left (a+b x^2\right )^3}-\frac {5 x^3}{24 b^2 \left (a+b x^2\right )^2}-\frac {5 x}{16 b^3 \left (a+b x^2\right )}+\frac {5 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 \sqrt {a} b^{7/2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 294, 211} \[ \int \frac {x^6}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {5 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 \sqrt {a} b^{7/2}}-\frac {5 x}{16 b^3 \left (a+b x^2\right )}-\frac {5 x^3}{24 b^2 \left (a+b x^2\right )^2}-\frac {x^5}{6 b \left (a+b x^2\right )^3} \]
[In]
[Out]
Rule 28
Rule 211
Rule 294
Rubi steps \begin{align*} \text {integral}& = b^4 \int \frac {x^6}{\left (a b+b^2 x^2\right )^4} \, dx \\ & = -\frac {x^5}{6 b \left (a+b x^2\right )^3}+\frac {1}{6} \left (5 b^2\right ) \int \frac {x^4}{\left (a b+b^2 x^2\right )^3} \, dx \\ & = -\frac {x^5}{6 b \left (a+b x^2\right )^3}-\frac {5 x^3}{24 b^2 \left (a+b x^2\right )^2}+\frac {5}{8} \int \frac {x^2}{\left (a b+b^2 x^2\right )^2} \, dx \\ & = -\frac {x^5}{6 b \left (a+b x^2\right )^3}-\frac {5 x^3}{24 b^2 \left (a+b x^2\right )^2}-\frac {5 x}{16 b^3 \left (a+b x^2\right )}+\frac {5 \int \frac {1}{a b+b^2 x^2} \, dx}{16 b^2} \\ & = -\frac {x^5}{6 b \left (a+b x^2\right )^3}-\frac {5 x^3}{24 b^2 \left (a+b x^2\right )^2}-\frac {5 x}{16 b^3 \left (a+b x^2\right )}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 \sqrt {a} b^{7/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80 \[ \int \frac {x^6}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {x \left (15 a^2+40 a b x^2+33 b^2 x^4\right )}{48 b^3 \left (a+b x^2\right )^3}+\frac {5 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{16 \sqrt {a} b^{7/2}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {-\frac {11 x^{5}}{16 b}-\frac {5 a \,x^{3}}{6 b^{2}}-\frac {5 a^{2} x}{16 b^{3}}}{\left (b \,x^{2}+a \right )^{3}}+\frac {5 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 b^{3} \sqrt {a b}}\) | \(58\) |
risch | \(\frac {-\frac {11 x^{5}}{16 b}-\frac {5 a \,x^{3}}{6 b^{2}}-\frac {5 a^{2} x}{16 b^{3}}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}-\frac {5 \ln \left (b x +\sqrt {-a b}\right )}{32 \sqrt {-a b}\, b^{3}}+\frac {5 \ln \left (-b x +\sqrt {-a b}\right )}{32 \sqrt {-a b}\, b^{3}}\) | \(104\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 254, normalized size of antiderivative = 3.06 \[ \int \frac {x^6}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\left [-\frac {66 \, a b^{3} x^{5} + 80 \, a^{2} b^{2} x^{3} + 30 \, a^{3} b x + 15 \, {\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{96 \, {\left (a b^{7} x^{6} + 3 \, a^{2} b^{6} x^{4} + 3 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}}, -\frac {33 \, a b^{3} x^{5} + 40 \, a^{2} b^{2} x^{3} + 15 \, a^{3} b x - 15 \, {\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{48 \, {\left (a b^{7} x^{6} + 3 \, a^{2} b^{6} x^{4} + 3 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}}\right ] \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.61 \[ \int \frac {x^6}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=- \frac {5 \sqrt {- \frac {1}{a b^{7}}} \log {\left (- a b^{3} \sqrt {- \frac {1}{a b^{7}}} + x \right )}}{32} + \frac {5 \sqrt {- \frac {1}{a b^{7}}} \log {\left (a b^{3} \sqrt {- \frac {1}{a b^{7}}} + x \right )}}{32} + \frac {- 15 a^{2} x - 40 a b x^{3} - 33 b^{2} x^{5}}{48 a^{3} b^{3} + 144 a^{2} b^{4} x^{2} + 144 a b^{5} x^{4} + 48 b^{6} x^{6}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.98 \[ \int \frac {x^6}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {33 \, b^{2} x^{5} + 40 \, a b x^{3} + 15 \, a^{2} x}{48 \, {\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} + \frac {5 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} b^{3}} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.67 \[ \int \frac {x^6}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {5 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} b^{3}} - \frac {33 \, b^{2} x^{5} + 40 \, a b x^{3} + 15 \, a^{2} x}{48 \, {\left (b x^{2} + a\right )}^{3} b^{3}} \]
[In]
[Out]
Time = 13.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94 \[ \int \frac {x^6}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {5\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{16\,\sqrt {a}\,b^{7/2}}-\frac {\frac {11\,x^5}{16\,b}+\frac {5\,a\,x^3}{6\,b^2}+\frac {5\,a^2\,x}{16\,b^3}}{a^3+3\,a^2\,b\,x^2+3\,a\,b^2\,x^4+b^3\,x^6} \]
[In]
[Out]