Integrand size = 20, antiderivative size = 48 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^2} \, dx=-\frac {a^2 A}{x}+a (2 A b+a B) x+\frac {1}{3} b (A b+2 a B) x^3+\frac {1}{5} b^2 B x^5 \]
[Out]
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^2} \, dx=-\frac {a^2 A}{x}+\frac {1}{3} b x^3 (2 a B+A b)+a x (a B+2 A b)+\frac {1}{5} b^2 B x^5 \]
[In]
[Out]
Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a (2 A b+a B)+\frac {a^2 A}{x^2}+b (A b+2 a B) x^2+b^2 B x^4\right ) \, dx \\ & = -\frac {a^2 A}{x}+a (2 A b+a B) x+\frac {1}{3} b (A b+2 a B) x^3+\frac {1}{5} b^2 B x^5 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^2} \, dx=-\frac {a^2 A}{x}+a (2 A b+a B) x+\frac {1}{3} b (A b+2 a B) x^3+\frac {1}{5} b^2 B x^5 \]
[In]
[Out]
Time = 2.47 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {b^{2} B \,x^{5}}{5}+\frac {A \,b^{2} x^{3}}{3}+\frac {2 B a b \,x^{3}}{3}+2 a A b x +a^{2} B x -\frac {a^{2} A}{x}\) | \(49\) |
risch | \(\frac {b^{2} B \,x^{5}}{5}+\frac {A \,b^{2} x^{3}}{3}+\frac {2 B a b \,x^{3}}{3}+2 a A b x +a^{2} B x -\frac {a^{2} A}{x}\) | \(49\) |
norman | \(\frac {\frac {b^{2} B \,x^{6}}{5}+\left (\frac {1}{3} b^{2} A +\frac {2}{3} a b B \right ) x^{4}+\left (2 a b A +a^{2} B \right ) x^{2}-a^{2} A}{x}\) | \(52\) |
gosper | \(-\frac {-3 b^{2} B \,x^{6}-5 A \,b^{2} x^{4}-10 B a b \,x^{4}-30 a A b \,x^{2}-15 a^{2} B \,x^{2}+15 a^{2} A}{15 x}\) | \(56\) |
parallelrisch | \(\frac {3 b^{2} B \,x^{6}+5 A \,b^{2} x^{4}+10 B a b \,x^{4}+30 a A b \,x^{2}+15 a^{2} B \,x^{2}-15 a^{2} A}{15 x}\) | \(56\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^2} \, dx=\frac {3 \, B b^{2} x^{6} + 5 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} - 15 \, A a^{2} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{15 \, x} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^2} \, dx=- \frac {A a^{2}}{x} + \frac {B b^{2} x^{5}}{5} + x^{3} \left (\frac {A b^{2}}{3} + \frac {2 B a b}{3}\right ) + x \left (2 A a b + B a^{2}\right ) \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^2} \, dx=\frac {1}{5} \, B b^{2} x^{5} + \frac {1}{3} \, {\left (2 \, B a b + A b^{2}\right )} x^{3} - \frac {A a^{2}}{x} + {\left (B a^{2} + 2 \, A a b\right )} x \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^2} \, dx=\frac {1}{5} \, B b^{2} x^{5} + \frac {2}{3} \, B a b x^{3} + \frac {1}{3} \, A b^{2} x^{3} + B a^{2} x + 2 \, A a b x - \frac {A a^{2}}{x} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^2} \, dx=x^3\,\left (\frac {A\,b^2}{3}+\frac {2\,B\,a\,b}{3}\right )+x\,\left (B\,a^2+2\,A\,b\,a\right )-\frac {A\,a^2}{x}+\frac {B\,b^2\,x^5}{5} \]
[In]
[Out]