Integrand size = 20, antiderivative size = 98 \[ \int \frac {x^6 \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {a^2 (A b-a B) x}{b^4}-\frac {a (A b-a B) x^3}{3 b^3}+\frac {(A b-a B) x^5}{5 b^2}+\frac {B x^7}{7 b}-\frac {a^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {470, 308, 211} \[ \int \frac {x^6 \left (A+B x^2\right )}{a+b x^2} \, dx=-\frac {a^{5/2} (A b-a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}}+\frac {a^2 x (A b-a B)}{b^4}-\frac {a x^3 (A b-a B)}{3 b^3}+\frac {x^5 (A b-a B)}{5 b^2}+\frac {B x^7}{7 b} \]
[In]
[Out]
Rule 211
Rule 308
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {B x^7}{7 b}-\frac {(-7 A b+7 a B) \int \frac {x^6}{a+b x^2} \, dx}{7 b} \\ & = \frac {B x^7}{7 b}-\frac {(-7 A b+7 a B) \int \left (\frac {a^2}{b^3}-\frac {a x^2}{b^2}+\frac {x^4}{b}-\frac {a^3}{b^3 \left (a+b x^2\right )}\right ) \, dx}{7 b} \\ & = \frac {a^2 (A b-a B) x}{b^4}-\frac {a (A b-a B) x^3}{3 b^3}+\frac {(A b-a B) x^5}{5 b^2}+\frac {B x^7}{7 b}-\frac {\left (a^3 (A b-a B)\right ) \int \frac {1}{a+b x^2} \, dx}{b^4} \\ & = \frac {a^2 (A b-a B) x}{b^4}-\frac {a (A b-a B) x^3}{3 b^3}+\frac {(A b-a B) x^5}{5 b^2}+\frac {B x^7}{7 b}-\frac {a^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {x^6 \left (A+B x^2\right )}{a+b x^2} \, dx=-\frac {a^2 (-A b+a B) x}{b^4}+\frac {a (-A b+a B) x^3}{3 b^3}+\frac {(A b-a B) x^5}{5 b^2}+\frac {B x^7}{7 b}+\frac {a^{5/2} (-A b+a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}} \]
[In]
[Out]
Time = 2.61 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {\frac {1}{7} b^{3} B \,x^{7}+\frac {1}{5} A \,b^{3} x^{5}-\frac {1}{5} B a \,b^{2} x^{5}-\frac {1}{3} a A \,b^{2} x^{3}+\frac {1}{3} B \,a^{2} b \,x^{3}+a^{2} A b x -a^{3} B x}{b^{4}}-\frac {a^{3} \left (A b -B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{4} \sqrt {a b}}\) | \(99\) |
risch | \(\frac {B \,x^{7}}{7 b}+\frac {A \,x^{5}}{5 b}-\frac {B a \,x^{5}}{5 b^{2}}-\frac {a A \,x^{3}}{3 b^{2}}+\frac {B \,a^{2} x^{3}}{3 b^{3}}+\frac {a^{2} A x}{b^{3}}-\frac {a^{3} B x}{b^{4}}+\frac {\sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x -a \right ) A}{2 b^{4}}-\frac {\sqrt {-a b}\, a^{3} \ln \left (-\sqrt {-a b}\, x -a \right ) B}{2 b^{5}}-\frac {\sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x -a \right ) A}{2 b^{4}}+\frac {\sqrt {-a b}\, a^{3} \ln \left (\sqrt {-a b}\, x -a \right ) B}{2 b^{5}}\) | \(185\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.33 \[ \int \frac {x^6 \left (A+B x^2\right )}{a+b x^2} \, dx=\left [\frac {30 \, B b^{3} x^{7} - 42 \, {\left (B a b^{2} - A b^{3}\right )} x^{5} + 70 \, {\left (B a^{2} b - A a b^{2}\right )} x^{3} - 105 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 210 \, {\left (B a^{3} - A a^{2} b\right )} x}{210 \, b^{4}}, \frac {15 \, B b^{3} x^{7} - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{5} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{3} + 105 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 105 \, {\left (B a^{3} - A a^{2} b\right )} x}{105 \, b^{4}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (87) = 174\).
Time = 0.23 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.84 \[ \int \frac {x^6 \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {B x^{7}}{7 b} + x^{5} \left (\frac {A}{5 b} - \frac {B a}{5 b^{2}}\right ) + x^{3} \left (- \frac {A a}{3 b^{2}} + \frac {B a^{2}}{3 b^{3}}\right ) + x \left (\frac {A a^{2}}{b^{3}} - \frac {B a^{3}}{b^{4}}\right ) - \frac {\sqrt {- \frac {a^{5}}{b^{9}}} \left (- A b + B a\right ) \log {\left (- \frac {b^{4} \sqrt {- \frac {a^{5}}{b^{9}}} \left (- A b + B a\right )}{- A a^{2} b + B a^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {a^{5}}{b^{9}}} \left (- A b + B a\right ) \log {\left (\frac {b^{4} \sqrt {- \frac {a^{5}}{b^{9}}} \left (- A b + B a\right )}{- A a^{2} b + B a^{3}} + x \right )}}{2} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02 \[ \int \frac {x^6 \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {{\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, B b^{3} x^{7} - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{5} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{3} - 105 \, {\left (B a^{3} - A a^{2} b\right )} x}{105 \, b^{4}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.10 \[ \int \frac {x^6 \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {{\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, B b^{6} x^{7} - 21 \, B a b^{5} x^{5} + 21 \, A b^{6} x^{5} + 35 \, B a^{2} b^{4} x^{3} - 35 \, A a b^{5} x^{3} - 105 \, B a^{3} b^{3} x + 105 \, A a^{2} b^{4} x}{105 \, b^{7}} \]
[In]
[Out]
Time = 5.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.20 \[ \int \frac {x^6 \left (A+B x^2\right )}{a+b x^2} \, dx=x^5\,\left (\frac {A}{5\,b}-\frac {B\,a}{5\,b^2}\right )+\frac {B\,x^7}{7\,b}+\frac {a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,x\,\left (A\,b-B\,a\right )}{B\,a^4-A\,a^3\,b}\right )\,\left (A\,b-B\,a\right )}{b^{9/2}}-\frac {a\,x^3\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{3\,b}+\frac {a^2\,x\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b^2} \]
[In]
[Out]