Integrand size = 16, antiderivative size = 40 \[ \int \frac {(a+b x)^2 (A+B x)}{x} \, dx=2 a A b x+\frac {1}{2} A b^2 x^2+\frac {B (a+b x)^3}{3 b}+a^2 A \log (x) \]
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Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {81, 45} \[ \int \frac {(a+b x)^2 (A+B x)}{x} \, dx=a^2 A \log (x)+2 a A b x+\frac {B (a+b x)^3}{3 b}+\frac {1}{2} A b^2 x^2 \]
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Rule 45
Rule 81
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+b x)^3}{3 b}+A \int \frac {(a+b x)^2}{x} \, dx \\ & = \frac {B (a+b x)^3}{3 b}+A \int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx \\ & = 2 a A b x+\frac {1}{2} A b^2 x^2+\frac {B (a+b x)^3}{3 b}+a^2 A \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^2 (A+B x)}{x} \, dx=a^2 B x+a b x (2 A+B x)+\frac {1}{6} b^2 x^2 (3 A+2 B x)+a^2 A \log (x) \]
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Time = 0.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {b^{2} B \,x^{3}}{3}+\frac {A \,b^{2} x^{2}}{2}+B a b \,x^{2}+2 a A b x +a^{2} B x +a^{2} A \ln \left (x \right )\) | \(46\) |
norman | \(\left (\frac {1}{2} b^{2} A +a b B \right ) x^{2}+\left (2 a b A +a^{2} B \right ) x +\frac {b^{2} B \,x^{3}}{3}+a^{2} A \ln \left (x \right )\) | \(46\) |
risch | \(\frac {b^{2} B \,x^{3}}{3}+\frac {A \,b^{2} x^{2}}{2}+B a b \,x^{2}+2 a A b x +a^{2} B x +a^{2} A \ln \left (x \right )\) | \(46\) |
parallelrisch | \(\frac {b^{2} B \,x^{3}}{3}+\frac {A \,b^{2} x^{2}}{2}+B a b \,x^{2}+2 a A b x +a^{2} B x +a^{2} A \ln \left (x \right )\) | \(46\) |
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Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^2 (A+B x)}{x} \, dx=\frac {1}{3} \, B b^{2} x^{3} + A a^{2} \log \left (x\right ) + \frac {1}{2} \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + {\left (B a^{2} + 2 \, A a b\right )} x \]
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Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^2 (A+B x)}{x} \, dx=A a^{2} \log {\left (x \right )} + \frac {B b^{2} x^{3}}{3} + x^{2} \left (\frac {A b^{2}}{2} + B a b\right ) + x \left (2 A a b + B a^{2}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^2 (A+B x)}{x} \, dx=\frac {1}{3} \, B b^{2} x^{3} + A a^{2} \log \left (x\right ) + \frac {1}{2} \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + {\left (B a^{2} + 2 \, A a b\right )} x \]
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Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^2 (A+B x)}{x} \, dx=\frac {1}{3} \, B b^{2} x^{3} + B a b x^{2} + \frac {1}{2} \, A b^{2} x^{2} + B a^{2} x + 2 \, A a b x + A a^{2} \log \left ({\left | x \right |}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^2 (A+B x)}{x} \, dx=x^2\,\left (\frac {A\,b^2}{2}+B\,a\,b\right )+x\,\left (B\,a^2+2\,A\,b\,a\right )+\frac {B\,b^2\,x^3}{3}+A\,a^2\,\ln \left (x\right ) \]
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