Integrand size = 17, antiderivative size = 18 \[ \int \frac {x^3}{a x^2+b x^3} \, dx=\frac {x}{b}-\frac {a \log (a+b x)}{b^2} \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 45} \[ \int \frac {x^3}{a x^2+b x^3} \, dx=\frac {x}{b}-\frac {a \log (a+b x)}{b^2} \]
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Rule 45
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x}{a+b x} \, dx \\ & = \int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx \\ & = \frac {x}{b}-\frac {a \log (a+b x)}{b^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{a x^2+b x^3} \, dx=\frac {x}{b}-\frac {a \log (a+b x)}{b^2} \]
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Time = 1.82 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {x}{b}-\frac {a \ln \left (b x +a \right )}{b^{2}}\) | \(19\) |
norman | \(\frac {x}{b}-\frac {a \ln \left (b x +a \right )}{b^{2}}\) | \(19\) |
risch | \(\frac {x}{b}-\frac {a \ln \left (b x +a \right )}{b^{2}}\) | \(19\) |
parallelrisch | \(-\frac {a \ln \left (b x +a \right )-b x}{b^{2}}\) | \(19\) |
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none
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {x^3}{a x^2+b x^3} \, dx=\frac {b x - a \log \left (b x + a\right )}{b^{2}} \]
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Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{a x^2+b x^3} \, dx=- \frac {a \log {\left (a + b x \right )}}{b^{2}} + \frac {x}{b} \]
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none
Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{a x^2+b x^3} \, dx=\frac {x}{b} - \frac {a \log \left (b x + a\right )}{b^{2}} \]
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none
Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {x^3}{a x^2+b x^3} \, dx=\frac {x}{b} - \frac {a \log \left ({\left | b x + a \right |}\right )}{b^{2}} \]
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Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{a x^2+b x^3} \, dx=-\frac {a\,\ln \left (a+b\,x\right )-b\,x}{b^2} \]
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