Integrand size = 20, antiderivative size = 101 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=-\frac {A}{2 a^3 x^2}-\frac {A b-a B}{4 a^2 \left (a+b x^2\right )^2}-\frac {2 A b-a B}{2 a^3 \left (a+b x^2\right )}-\frac {(3 A b-a B) \log (x)}{a^4}+\frac {(3 A b-a B) \log \left (a+b x^2\right )}{2 a^4} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 101, 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} \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {(3 A b-a B) \log \left (a+b x^2\right )}{2 a^4}-\frac {\log (x) (3 A b-a B)}{a^4}-\frac {2 A b-a B}{2 a^3 \left (a+b x^2\right )}-\frac {A}{2 a^3 x^2}-\frac {A b-a B}{4 a^2 \left (a+b x^2\right )^2} \]
[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)^3} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{a^3 x^2}+\frac {-3 A b+a B}{a^4 x}-\frac {b (-A b+a B)}{a^2 (a+b x)^3}-\frac {b (-2 A b+a B)}{a^3 (a+b x)^2}-\frac {b (-3 A b+a B)}{a^4 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {A}{2 a^3 x^2}-\frac {A b-a B}{4 a^2 \left (a+b x^2\right )^2}-\frac {2 A b-a B}{2 a^3 \left (a+b x^2\right )}-\frac {(3 A b-a B) \log (x)}{a^4}+\frac {(3 A b-a B) \log \left (a+b x^2\right )}{2 a^4} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {-\frac {2 a A}{x^2}+\frac {a^2 (-A b+a B)}{\left (a+b x^2\right )^2}+\frac {2 a (-2 A b+a B)}{a+b x^2}+4 (-3 A b+a B) \log (x)+2 (3 A b-a B) \log \left (a+b x^2\right )}{4 a^4} \]
[In]
[Out]
Time = 2.53 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {\frac {b \left (3 A b -B a \right ) x^{4}}{a^{3}}-\frac {A}{2 a}+\frac {b^{2} \left (9 A b -3 B a \right ) x^{6}}{4 a^{4}}}{x^{2} \left (b \,x^{2}+a \right )^{2}}-\frac {\left (3 A b -B a \right ) \ln \left (x \right )}{a^{4}}+\frac {\left (3 A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{4}}\) | \(97\) |
default | \(-\frac {A}{2 a^{3} x^{2}}+\frac {\left (-3 A b +B a \right ) \ln \left (x \right )}{a^{4}}+\frac {b \left (\frac {\left (3 A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {a^{2} \left (A b -B a \right )}{2 b \left (b \,x^{2}+a \right )^{2}}-\frac {a \left (2 A b -B a \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{4}}\) | \(102\) |
risch | \(\frac {-\frac {b \left (3 A b -B a \right ) x^{4}}{2 a^{3}}-\frac {3 \left (3 A b -B a \right ) x^{2}}{4 a^{2}}-\frac {A}{2 a}}{x^{2} \left (b \,x^{2}+a \right )^{2}}-\frac {3 \ln \left (x \right ) A b}{a^{4}}+\frac {\ln \left (x \right ) B}{a^{3}}+\frac {3 \ln \left (-b \,x^{2}-a \right ) A b}{2 a^{4}}-\frac {\ln \left (-b \,x^{2}-a \right ) B}{2 a^{3}}\) | \(108\) |
parallelrisch | \(-\frac {12 A \ln \left (x \right ) x^{6} b^{3}-6 A \ln \left (b \,x^{2}+a \right ) x^{6} b^{3}-4 B \ln \left (x \right ) x^{6} a \,b^{2}+2 B \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{2}-9 A \,x^{6} b^{3}+3 B \,x^{6} a \,b^{2}+24 A \ln \left (x \right ) x^{4} a \,b^{2}-12 A \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{2}-8 B \ln \left (x \right ) x^{4} a^{2} b +4 B \ln \left (b \,x^{2}+a \right ) x^{4} a^{2} b -12 A a \,b^{2} x^{4}+4 B \,a^{2} b \,x^{4}+12 A \ln \left (x \right ) x^{2} a^{2} b -6 A \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b -4 B \ln \left (x \right ) x^{2} a^{3}+2 B \ln \left (b \,x^{2}+a \right ) x^{2} a^{3}+2 a^{3} A}{4 a^{4} x^{2} \left (b \,x^{2}+a \right )^{2}}\) | \(240\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (89) = 178\).
Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.95 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} - 2 \, A a^{3} + 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2} - 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{6} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{6} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \]
[In]
[Out]
Time = 0.65 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {- 2 A a^{2} + x^{4} \left (- 6 A b^{2} + 2 B a b\right ) + x^{2} \left (- 9 A a b + 3 B a^{2}\right )}{4 a^{5} x^{2} + 8 a^{4} b x^{4} + 4 a^{3} b^{2} x^{6}} + \frac {\left (- 3 A b + B a\right ) \log {\left (x \right )}}{a^{4}} - \frac {\left (- 3 A b + B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{4}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {2 \, {\left (B a b - 3 \, A b^{2}\right )} x^{4} - 2 \, A a^{2} + 3 \, {\left (B a^{2} - 3 \, A a b\right )} x^{2}}{4 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} - \frac {{\left (B a - 3 \, A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac {{\left (B a - 3 \, A b\right )} \log \left (x^{2}\right )}{2 \, a^{4}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.37 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {{\left (B a - 3 \, A b\right )} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {{\left (B a b - 3 \, A b^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4} b} + \frac {3 \, B a b^{2} x^{4} - 9 \, A b^{3} x^{4} + 8 \, B a^{2} b x^{2} - 22 \, A a b^{2} x^{2} + 6 \, B a^{3} - 14 \, A a^{2} b}{4 \, {\left (b x^{2} + a\right )}^{2} a^{4}} - \frac {B a x^{2} - 3 \, A b x^{2} + A a}{2 \, a^{4} x^{2}} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (3\,A\,b-B\,a\right )}{2\,a^4}-\frac {\frac {A}{2\,a}+\frac {3\,x^2\,\left (3\,A\,b-B\,a\right )}{4\,a^2}+\frac {b\,x^4\,\left (3\,A\,b-B\,a\right )}{2\,a^3}}{a^2\,x^2+2\,a\,b\,x^4+b^2\,x^6}-\frac {\ln \left (x\right )\,\left (3\,A\,b-B\,a\right )}{a^4} \]
[In]
[Out]