Integrand size = 17, antiderivative size = 38 \[ \int \frac {1}{\sqrt {c x^2} (a+b x)} \, dx=\frac {x \log (x)}{a \sqrt {c x^2}}-\frac {x \log (a+b x)}{a \sqrt {c x^2}} \]
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Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {15, 36, 29, 31} \[ \int \frac {1}{\sqrt {c x^2} (a+b x)} \, dx=\frac {x \log (x)}{a \sqrt {c x^2}}-\frac {x \log (a+b x)}{a \sqrt {c x^2}} \]
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Rule 15
Rule 29
Rule 31
Rule 36
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {1}{x (a+b x)} \, dx}{\sqrt {c x^2}} \\ & = \frac {x \int \frac {1}{x} \, dx}{a \sqrt {c x^2}}-\frac {(b x) \int \frac {1}{a+b x} \, dx}{a \sqrt {c x^2}} \\ & = \frac {x \log (x)}{a \sqrt {c x^2}}-\frac {x \log (a+b x)}{a \sqrt {c x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\sqrt {c x^2} (a+b x)} \, dx=\frac {x (\log (x)-\log (a (a+b x)))}{a \sqrt {c x^2}} \]
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Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {x \left (\ln \left (x \right )-\ln \left (b x +a \right )\right )}{\sqrt {c \,x^{2}}\, a}\) | \(24\) |
risch | \(\frac {x \ln \left (-x \right )}{\sqrt {c \,x^{2}}\, a}-\frac {x \ln \left (b x +a \right )}{a \sqrt {c \,x^{2}}}\) | \(37\) |
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none
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.84 \[ \int \frac {1}{\sqrt {c x^2} (a+b x)} \, dx=\left [\frac {\sqrt {c x^{2}} \log \left (\frac {x}{b x + a}\right )}{a c x}, \frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )} \sqrt {-c}}{a c x}\right )}{a c}\right ] \]
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\[ \int \frac {1}{\sqrt {c x^2} (a+b x)} \, dx=\int \frac {1}{\sqrt {c x^{2}} \left (a + b x\right )}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {c x^2} (a+b x)} \, dx=-\frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a \sqrt {c}} \]
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Exception generated. \[ \int \frac {1}{\sqrt {c x^2} (a+b x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\sqrt {c x^2} (a+b x)} \, dx=\int \frac {1}{\sqrt {c\,x^2}\,\left (a+b\,x\right )} \,d x \]
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