Integrand size = 14, antiderivative size = 45 \[ \int \frac {x (A+B x)}{a+b x} \, dx=\frac {(A b-a B) x}{b^2}+\frac {B x^2}{2 b}-\frac {a (A b-a B) \log (a+b x)}{b^3} \]
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Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {78} \[ \int \frac {x (A+B x)}{a+b x} \, dx=-\frac {a (A b-a B) \log (a+b x)}{b^3}+\frac {x (A b-a B)}{b^2}+\frac {B x^2}{2 b} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A b-a B}{b^2}+\frac {B x}{b}+\frac {a (-A b+a B)}{b^2 (a+b x)}\right ) \, dx \\ & = \frac {(A b-a B) x}{b^2}+\frac {B x^2}{2 b}-\frac {a (A b-a B) \log (a+b x)}{b^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {x (A+B x)}{a+b x} \, dx=\frac {b x (2 A b-2 a B+b B x)+2 a (-A b+a B) \log (a+b x)}{2 b^3} \]
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Time = 0.96 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {\frac {1}{2} b B \,x^{2}+A b x -B a x}{b^{2}}-\frac {a \left (A b -B a \right ) \ln \left (b x +a \right )}{b^{3}}\) | \(43\) |
norman | \(\frac {\left (A b -B a \right ) x}{b^{2}}+\frac {B \,x^{2}}{2 b}-\frac {a \left (A b -B a \right ) \ln \left (b x +a \right )}{b^{3}}\) | \(44\) |
risch | \(\frac {B \,x^{2}}{2 b}+\frac {A x}{b}-\frac {B a x}{b^{2}}-\frac {a \ln \left (b x +a \right ) A}{b^{2}}+\frac {a^{2} \ln \left (b x +a \right ) B}{b^{3}}\) | \(52\) |
parallelrisch | \(-\frac {-b^{2} B \,x^{2}+2 A \ln \left (b x +a \right ) a b -2 A \,b^{2} x -2 B \ln \left (b x +a \right ) a^{2}+2 B a b x}{2 b^{3}}\) | \(52\) |
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Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {x (A+B x)}{a+b x} \, dx=\frac {B b^{2} x^{2} - 2 \, {\left (B a b - A b^{2}\right )} x + 2 \, {\left (B a^{2} - A a b\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]
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Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \frac {x (A+B x)}{a+b x} \, dx=\frac {B x^{2}}{2 b} + \frac {a \left (- A b + B a\right ) \log {\left (a + b x \right )}}{b^{3}} + x \left (\frac {A}{b} - \frac {B a}{b^{2}}\right ) \]
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none
Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {x (A+B x)}{a+b x} \, dx=\frac {B b x^{2} - 2 \, {\left (B a - A b\right )} x}{2 \, b^{2}} + \frac {{\left (B a^{2} - A a b\right )} \log \left (b x + a\right )}{b^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {x (A+B x)}{a+b x} \, dx=\frac {B b x^{2} - 2 \, B a x + 2 \, A b x}{2 \, b^{2}} + \frac {{\left (B a^{2} - A a b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \]
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Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int \frac {x (A+B x)}{a+b x} \, dx=x\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )+\frac {B\,x^2}{2\,b}+\frac {\ln \left (a+b\,x\right )\,\left (B\,a^2-A\,a\,b\right )}{b^3} \]
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