Integrand size = 20, antiderivative size = 50 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {A}{2 a x^2}-\frac {(A b-a B) \log (x)}{a^2}+\frac {(A b-a B) \log \left (a+b x^2\right )}{2 a^2} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 78} \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )} \, dx=\frac {(A b-a B) \log \left (a+b x^2\right )}{2 a^2}-\frac {\log (x) (A b-a B)}{a^2}-\frac {A}{2 a x^2} \]
[In]
[Out]
Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^2 (a+b x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{a x^2}+\frac {-A b+a B}{a^2 x}-\frac {b (-A b+a B)}{a^2 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {A}{2 a x^2}-\frac {(A b-a B) \log (x)}{a^2}+\frac {(A b-a B) \log \left (a+b x^2\right )}{2 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {A}{2 a x^2}+\frac {(-A b+a B) \log (x)}{a^2}+\frac {(A b-a B) \log \left (a+b x^2\right )}{2 a^2} \]
[In]
[Out]
Time = 2.50 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {A}{2 a \,x^{2}}+\frac {\left (-A b +B a \right ) \ln \left (x \right )}{a^{2}}+\frac {\left (A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\) | \(46\) |
norman | \(-\frac {A}{2 a \,x^{2}}-\frac {\left (A b -B a \right ) \ln \left (x \right )}{a^{2}}+\frac {\left (A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\) | \(47\) |
parallelrisch | \(-\frac {2 A \ln \left (x \right ) x^{2} b -A \ln \left (b \,x^{2}+a \right ) x^{2} b -2 B \ln \left (x \right ) x^{2} a +B \ln \left (b \,x^{2}+a \right ) x^{2} a +A a}{2 a^{2} x^{2}}\) | \(60\) |
risch | \(-\frac {A}{2 a \,x^{2}}-\frac {\ln \left (x \right ) A b}{a^{2}}+\frac {\ln \left (x \right ) B}{a}+\frac {\ln \left (-b \,x^{2}-a \right ) A b}{2 a^{2}}-\frac {\ln \left (-b \,x^{2}-a \right ) B}{2 a}\) | \(62\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {{\left (B a - A b\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \, {\left (B a - A b\right )} x^{2} \log \left (x\right ) + A a}{2 \, a^{2} x^{2}} \]
[In]
[Out]
Time = 0.43 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )} \, dx=- \frac {A}{2 a x^{2}} + \frac {\left (- A b + B a\right ) \log {\left (x \right )}}{a^{2}} - \frac {\left (- A b + B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {{\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac {{\left (B a - A b\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {A}{2 \, a x^{2}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.42 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )} \, dx=\frac {{\left (B a - A b\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {{\left (B a b - A b^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} - \frac {B a x^{2} - A b x^{2} + A a}{2 \, a^{2} x^{2}} \]
[In]
[Out]
Time = 5.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (A\,b-B\,a\right )}{2\,a^2}-\frac {A}{2\,a\,x^2}-\frac {\ln \left (x\right )\,\left (A\,b-B\,a\right )}{a^2} \]
[In]
[Out]