Integrand size = 13, antiderivative size = 28 \[ \int \frac {1}{\sqrt {c (a+b x)^2}} \, dx=\frac {(a+b x) \log (a+b x)}{b \sqrt {c (a+b x)^2}} \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 15, 29} \[ \int \frac {1}{\sqrt {c (a+b x)^2}} \, dx=\frac {(a+b x) \log (a+b x)}{b \sqrt {c (a+b x)^2}} \]
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Rule 15
Rule 29
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {c x^2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \text {Subst}\left (\int \frac {1}{x} \, dx,x,a+b x\right )}{b \sqrt {c (a+b x)^2}} \\ & = \frac {(a+b x) \log (a+b x)}{b \sqrt {c (a+b x)^2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {c (a+b x)^2}} \, dx=\frac {(a+b x) \log (a+b x)}{b \sqrt {c (a+b x)^2}} \]
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Time = 3.77 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {\left (b x +a \right ) \ln \left (b x +a \right )}{b \sqrt {c \left (b x +a \right )^{2}}}\) | \(27\) |
risch | \(\frac {\left (b x +a \right ) \ln \left (b x +a \right )}{b \sqrt {c \left (b x +a \right )^{2}}}\) | \(27\) |
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none
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\sqrt {c (a+b x)^2}} \, dx=\frac {\sqrt {b^{2} c x^{2} + 2 \, a b c x + a^{2} c} \log \left (b x + a\right )}{b^{2} c x + a b c} \]
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Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {c (a+b x)^2}} \, dx=\frac {\left (\frac {a}{b} + x\right ) \log {\left (\frac {a}{b} + x \right )}}{\sqrt {b^{2} c \left (\frac {a}{b} + x\right )^{2}}} \]
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none
Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.54 \[ \int \frac {1}{\sqrt {c (a+b x)^2}} \, dx=\frac {\log \left (x + \frac {a}{b}\right )}{b \sqrt {c}} \]
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none
Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\sqrt {c (a+b x)^2}} \, dx=\frac {\log \left ({\left | b x + a \right |} \sqrt {{\left | c \right |}} {\left | \mathrm {sgn}\left (b x + a\right ) \right |}\right )}{b \sqrt {c} \mathrm {sgn}\left (b x + a\right )} \]
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Time = 6.76 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\sqrt {c (a+b x)^2}} \, dx=\frac {\ln \left (\sqrt {c}\,\left (a+b\,x\right )+\sqrt {c\,{\left (a+b\,x\right )}^2}\right )}{b\,\sqrt {c}} \]
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