Integrand size = 14, antiderivative size = 45 \[ \int \frac {x (c+d x)}{a+b x} \, dx=\frac {(b c-a d) x}{b^2}+\frac {d x^2}{2 b}-\frac {a (b c-a d) \log (a+b x)}{b^3} \]
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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} \[ \int \frac {x (c+d x)}{a+b x} \, dx=-\frac {a (b c-a d) \log (a+b x)}{b^3}+\frac {x (b c-a d)}{b^2}+\frac {d x^2}{2 b} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b c-a d}{b^2}+\frac {d x}{b}+\frac {a (-b c+a d)}{b^2 (a+b x)}\right ) \, dx \\ & = \frac {(b c-a d) x}{b^2}+\frac {d x^2}{2 b}-\frac {a (b c-a d) \log (a+b x)}{b^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {x (c+d x)}{a+b x} \, dx=\frac {b x (2 b c-2 a d+b d x)+2 a (-b c+a d) \log (a+b x)}{2 b^3} \]
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Time = 0.42 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {-\frac {1}{2} b d \,x^{2}+a d x -b c x}{b^{2}}+\frac {a \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{3}}\) | \(43\) |
norman | \(\frac {d \,x^{2}}{2 b}-\frac {\left (a d -b c \right ) x}{b^{2}}+\frac {a \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{3}}\) | \(44\) |
parallelrisch | \(\frac {d \,x^{2} b^{2}+2 \ln \left (b x +a \right ) a^{2} d -2 \ln \left (b x +a \right ) a b c -2 x a b d +2 b^{2} c x}{2 b^{3}}\) | \(51\) |
risch | \(\frac {d \,x^{2}}{2 b}-\frac {a d x}{b^{2}}+\frac {c x}{b}+\frac {a^{2} \ln \left (b x +a \right ) d}{b^{3}}-\frac {a \ln \left (b x +a \right ) c}{b^{2}}\) | \(52\) |
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Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {x (c+d x)}{a+b x} \, dx=\frac {b^{2} d x^{2} + 2 \, {\left (b^{2} c - a b d\right )} x - 2 \, {\left (a b c - a^{2} d\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]
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Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \frac {x (c+d x)}{a+b x} \, dx=\frac {a \left (a d - b c\right ) \log {\left (a + b x \right )}}{b^{3}} + x \left (- \frac {a d}{b^{2}} + \frac {c}{b}\right ) + \frac {d x^{2}}{2 b} \]
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Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int \frac {x (c+d x)}{a+b x} \, dx=\frac {b d x^{2} + 2 \, {\left (b c - a d\right )} x}{2 \, b^{2}} - \frac {{\left (a b c - a^{2} d\right )} \log \left (b x + a\right )}{b^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int \frac {x (c+d x)}{a+b x} \, dx=\frac {b d x^{2} + 2 \, b c x - 2 \, a d x}{2 \, b^{2}} - \frac {{\left (a b c - a^{2} d\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \]
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Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int \frac {x (c+d x)}{a+b x} \, dx=x\,\left (\frac {c}{b}-\frac {a\,d}{b^2}\right )+\frac {d\,x^2}{2\,b}+\frac {\ln \left (a+b\,x\right )\,\left (a^2\,d-a\,b\,c\right )}{b^3} \]
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