Integrand size = 16, antiderivative size = 44 \[ \int \frac {(a+b x)^2 (A+B x)}{x^2} \, dx=-\frac {a^2 A}{x}+b (A b+2 a B) x+\frac {1}{2} b^2 B x^2+a (2 A b+a B) \log (x) \]
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Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int \frac {(a+b x)^2 (A+B x)}{x^2} \, dx=-\frac {a^2 A}{x}+b x (2 a B+A b)+a \log (x) (a B+2 A b)+\frac {1}{2} b^2 B x^2 \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (b (A b+2 a B)+\frac {a^2 A}{x^2}+\frac {a (2 A b+a B)}{x}+b^2 B x\right ) \, dx \\ & = -\frac {a^2 A}{x}+b (A b+2 a B) x+\frac {1}{2} b^2 B x^2+a (2 A b+a B) \log (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^2 (A+B x)}{x^2} \, dx=-\frac {a^2 A}{x}+2 a b B x+\frac {1}{2} b^2 x (2 A+B x)+a (2 A b+a B) \log (x) \]
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Time = 0.39 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {b^{2} B \,x^{2}}{2}+A \,b^{2} x +2 B a b x +a \left (2 A b +B a \right ) \ln \left (x \right )-\frac {a^{2} A}{x}\) | \(44\) |
risch | \(\frac {b^{2} B \,x^{2}}{2}+A \,b^{2} x +2 B a b x -\frac {a^{2} A}{x}+2 A \ln \left (x \right ) a b +B \ln \left (x \right ) a^{2}\) | \(46\) |
norman | \(\frac {\left (b^{2} A +2 a b B \right ) x^{2}-a^{2} A +\frac {b^{2} B \,x^{3}}{2}}{x}+\left (2 a b A +a^{2} B \right ) \ln \left (x \right )\) | \(51\) |
parallelrisch | \(\frac {b^{2} B \,x^{3}+4 A \ln \left (x \right ) x a b +2 A \,b^{2} x^{2}+2 B \ln \left (x \right ) x \,a^{2}+4 B a b \,x^{2}-2 a^{2} A}{2 x}\) | \(55\) |
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none
Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b x)^2 (A+B x)}{x^2} \, dx=\frac {B b^{2} x^{3} - 2 \, A a^{2} + 2 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} x \log \left (x\right )}{2 \, x} \]
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Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^2 (A+B x)}{x^2} \, dx=- \frac {A a^{2}}{x} + \frac {B b^{2} x^{2}}{2} + a \left (2 A b + B a\right ) \log {\left (x \right )} + x \left (A b^{2} + 2 B a b\right ) \]
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none
Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.05 \[ \int \frac {(a+b x)^2 (A+B x)}{x^2} \, dx=\frac {1}{2} \, B b^{2} x^{2} - \frac {A a^{2}}{x} + {\left (2 \, B a b + A b^{2}\right )} x + {\left (B a^{2} + 2 \, A a b\right )} \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.05 \[ \int \frac {(a+b x)^2 (A+B x)}{x^2} \, dx=\frac {1}{2} \, B b^{2} x^{2} + 2 \, B a b x + A b^{2} x - \frac {A a^{2}}{x} + {\left (B a^{2} + 2 \, A a b\right )} \log \left ({\left | x \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.05 \[ \int \frac {(a+b x)^2 (A+B x)}{x^2} \, dx=\ln \left (x\right )\,\left (B\,a^2+2\,A\,b\,a\right )+x\,\left (A\,b^2+2\,B\,a\,b\right )-\frac {A\,a^2}{x}+\frac {B\,b^2\,x^2}{2} \]
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