Integrand size = 11, antiderivative size = 31 \[ \int \frac {x^2}{a+b x} \, dx=-\frac {a x}{b^2}+\frac {x^2}{2 b}+\frac {a^2 \log (a+b x)}{b^3} \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^2}{a+b x} \, dx=\frac {a^2 \log (a+b x)}{b^3}-\frac {a x}{b^2}+\frac {x^2}{2 b} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx \\ & = -\frac {a x}{b^2}+\frac {x^2}{2 b}+\frac {a^2 \log (a+b x)}{b^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{a+b x} \, dx=-\frac {a x}{b^2}+\frac {x^2}{2 b}+\frac {a^2 \log (a+b x)}{b^3} \]
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Time = 0.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {-\frac {1}{2} b \,x^{2}+a x}{b^{2}}+\frac {a^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(30\) |
norman | \(-\frac {a x}{b^{2}}+\frac {x^{2}}{2 b}+\frac {a^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(30\) |
risch | \(-\frac {a x}{b^{2}}+\frac {x^{2}}{2 b}+\frac {a^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(30\) |
parallelrisch | \(\frac {b^{2} x^{2}+2 a^{2} \ln \left (b x +a \right )-2 a b x}{2 b^{3}}\) | \(30\) |
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Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{a+b x} \, dx=\frac {b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )}{2 \, b^{3}} \]
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Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{a+b x} \, dx=\frac {a^{2} \log {\left (a + b x \right )}}{b^{3}} - \frac {a x}{b^{2}} + \frac {x^{2}}{2 b} \]
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none
Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{a+b x} \, dx=\frac {a^{2} \log \left (b x + a\right )}{b^{3}} + \frac {b x^{2} - 2 \, a x}{2 \, b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {x^2}{a+b x} \, dx=\frac {a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac {b x^{2} - 2 \, a x}{2 \, b^{2}} \]
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Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{a+b x} \, dx=\frac {2\,a^2\,\ln \left (a+b\,x\right )+b^2\,x^2-2\,a\,b\,x}{2\,b^3} \]
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