Integrand size = 20, antiderivative size = 42 \[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)} \, dx=\frac {\sqrt {c x^2} \log (x)}{a x}-\frac {\sqrt {c x^2} \log (a+b x)}{a x} \]
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Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 36, 29, 31} \[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)} \, dx=\frac {\sqrt {c x^2} \log (x)}{a x}-\frac {\sqrt {c x^2} \log (a+b x)}{a x} \]
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Rule 15
Rule 29
Rule 31
Rule 36
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c x^2} \int \frac {1}{x (a+b x)} \, dx}{x} \\ & = \frac {\sqrt {c x^2} \int \frac {1}{x} \, dx}{a x}-\frac {\left (b \sqrt {c x^2}\right ) \int \frac {1}{a+b x} \, dx}{a x} \\ & = \frac {\sqrt {c x^2} \log (x)}{a x}-\frac {\sqrt {c x^2} \log (a+b x)}{a x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)} \, dx=\frac {c x (\log (x)-\log (a (a+b x)))}{a \sqrt {c x^2}} \]
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Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.62
method | result | size |
default | \(\frac {\sqrt {c \,x^{2}}\, \left (\ln \left (x \right )-\ln \left (b x +a \right )\right )}{a x}\) | \(26\) |
risch | \(\frac {\sqrt {c \,x^{2}}\, \ln \left (-x \right )}{x a}-\frac {\ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{a x}\) | \(41\) |
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Time = 0.23 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)} \, dx=\left [\frac {\sqrt {c x^{2}} \log \left (\frac {x}{b x + a}\right )}{a x}, \frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )} \sqrt {-c}}{a c x}\right )}{a}\right ] \]
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\[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)} \, dx=\int \frac {\sqrt {c x^{2}}}{x^{2} \left (a + b x\right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)} \, dx=-\frac {\sqrt {c} \log \left (b x + a\right )}{a} + \frac {\sqrt {c} \log \left (x\right )}{a} \]
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Exception generated. \[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {c x^2}}{x^2 (a+b x)} \, dx=\int \frac {\sqrt {c\,x^2}}{x^2\,\left (a+b\,x\right )} \,d x \]
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