Integrand size = 20, antiderivative size = 97 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^2} \, dx=-\frac {A}{4 a^2 x^4}+\frac {2 A b-a B}{2 a^3 x^2}+\frac {b (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac {b (3 A b-2 a B) \log (x)}{a^4}-\frac {b (3 A b-2 a B) \log \left (a+b x^2\right )}{2 a^4} \]
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Time = 0.07 (sec) , antiderivative size = 97, 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} \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^2} \, dx=-\frac {b (3 A b-2 a B) \log \left (a+b x^2\right )}{2 a^4}+\frac {b \log (x) (3 A b-2 a B)}{a^4}+\frac {b (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac {2 A b-a B}{2 a^3 x^2}-\frac {A}{4 a^2 x^4} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^3 (a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{a^2 x^3}+\frac {-2 A b+a B}{a^3 x^2}-\frac {b (-3 A b+2 a B)}{a^4 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)^2}+\frac {b^2 (-3 A b+2 a B)}{a^4 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {A}{4 a^2 x^4}+\frac {2 A b-a B}{2 a^3 x^2}+\frac {b (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac {b (3 A b-2 a B) \log (x)}{a^4}-\frac {b (3 A b-2 a B) \log \left (a+b x^2\right )}{2 a^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^2} \, dx=-\frac {\frac {a^2 A}{x^4}+\frac {2 a (-2 A b+a B)}{x^2}+\frac {2 a b (-A b+a B)}{a+b x^2}-4 b (3 A b-2 a B) \log (x)+2 b (3 A b-2 a B) \log \left (a+b x^2\right )}{4 a^4} \]
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Time = 2.51 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {A}{4 a^{2} x^{4}}-\frac {-2 A b +B a}{2 x^{2} a^{3}}+\frac {b \left (3 A b -2 B a \right ) \ln \left (x \right )}{a^{4}}-\frac {b^{2} \left (\frac {\left (3 A b -2 B a \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {a \left (A b -B a \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{4}}\) | \(96\) |
norman | \(\frac {-\frac {A}{4 a}+\frac {\left (3 A b -2 B a \right ) x^{2}}{4 a^{2}}-\frac {b \left (3 b^{2} A -2 a b B \right ) x^{6}}{2 a^{4}}}{x^{4} \left (b \,x^{2}+a \right )}+\frac {b \left (3 A b -2 B a \right ) \ln \left (x \right )}{a^{4}}-\frac {b \left (3 A b -2 B a \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{4}}\) | \(99\) |
risch | \(\frac {\frac {b \left (3 A b -2 B a \right ) x^{4}}{2 a^{3}}+\frac {\left (3 A b -2 B a \right ) x^{2}}{4 a^{2}}-\frac {A}{4 a}}{x^{4} \left (b \,x^{2}+a \right )}+\frac {3 b^{2} \ln \left (x \right ) A}{a^{4}}-\frac {2 b \ln \left (x \right ) B}{a^{3}}-\frac {3 b^{2} \ln \left (b \,x^{2}+a \right ) A}{2 a^{4}}+\frac {b \ln \left (b \,x^{2}+a \right ) B}{a^{3}}\) | \(108\) |
parallelrisch | \(\frac {12 A \ln \left (x \right ) x^{6} b^{3}-6 A \ln \left (b \,x^{2}+a \right ) x^{6} b^{3}-8 B \ln \left (x \right ) x^{6} a \,b^{2}+4 B \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{2}-6 A \,x^{6} b^{3}+4 B \,x^{6} a \,b^{2}+12 A \ln \left (x \right ) x^{4} a \,b^{2}-6 A \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{2}-8 B \ln \left (x \right ) x^{4} a^{2} b +4 B \ln \left (b \,x^{2}+a \right ) x^{4} a^{2} b +3 A \,a^{2} b \,x^{2}-2 B \,a^{3} x^{2}-a^{3} A}{4 a^{4} x^{4} \left (b \,x^{2}+a \right )}\) | \(181\) |
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Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.59 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^2} \, dx=-\frac {2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{4} + A a^{3} + {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x^{2} - 2 \, {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{6} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{4}\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{6} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{4}\right )} \log \left (x\right )}{4 \, {\left (a^{4} b x^{6} + a^{5} x^{4}\right )}} \]
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Time = 0.58 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^2} \, dx=\frac {- A a^{2} + x^{4} \cdot \left (6 A b^{2} - 4 B a b\right ) + x^{2} \cdot \left (3 A a b - 2 B a^{2}\right )}{4 a^{4} x^{4} + 4 a^{3} b x^{6}} - \frac {b \left (- 3 A b + 2 B a\right ) \log {\left (x \right )}}{a^{4}} + \frac {b \left (- 3 A b + 2 B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{4}} \]
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Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^2} \, dx=-\frac {2 \, {\left (2 \, B a b - 3 \, A b^{2}\right )} x^{4} + A a^{2} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}}{4 \, {\left (a^{3} b x^{6} + a^{4} x^{4}\right )}} + \frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{4}} - \frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.55 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{4}} + \frac {{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4} b} - \frac {2 \, B a b^{2} x^{2} - 3 \, A b^{3} x^{2} + 3 \, B a^{2} b - 4 \, A a b^{2}}{2 \, {\left (b x^{2} + a\right )} a^{4}} + \frac {6 \, B a b x^{4} - 9 \, A b^{2} x^{4} - 2 \, B a^{2} x^{2} + 4 \, A a b x^{2} - A a^{2}}{4 \, a^{4} x^{4}} \]
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Time = 4.84 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^2} \, dx=\frac {\frac {x^2\,\left (3\,A\,b-2\,B\,a\right )}{4\,a^2}-\frac {A}{4\,a}+\frac {b\,x^4\,\left (3\,A\,b-2\,B\,a\right )}{2\,a^3}}{b\,x^6+a\,x^4}-\frac {\ln \left (b\,x^2+a\right )\,\left (3\,A\,b^2-2\,B\,a\,b\right )}{2\,a^4}+\frac {\ln \left (x\right )\,\left (3\,A\,b^2-2\,B\,a\,b\right )}{a^4} \]
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