Integrand size = 17, antiderivative size = 62 \[ \int \frac {\sqrt {c+d x}}{a+b x} \, dx=\frac {2 \sqrt {c+d x}}{b}-\frac {2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {52, 65, 214} \[ \int \frac {\sqrt {c+d x}}{a+b x} \, dx=\frac {2 \sqrt {c+d x}}{b}-\frac {2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2}} \]
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Rule 52
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {c+d x}}{b}+\frac {(b c-a d) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{b} \\ & = \frac {2 \sqrt {c+d x}}{b}+\frac {(2 (b c-a d)) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b d} \\ & = \frac {2 \sqrt {c+d x}}{b}-\frac {2 \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+d x}}{a+b x} \, dx=\frac {2 \sqrt {c+d x}}{b}-\frac {2 \sqrt {-b c+a d} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{3/2}} \]
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Time = 0.72 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {d x +c}-\frac {2 \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{b}\) | \(58\) |
derivativedivides | \(\frac {2 \sqrt {d x +c}}{b}+\frac {2 \left (-a d +b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b \sqrt {\left (a d -b c \right ) b}}\) | \(61\) |
default | \(\frac {2 \sqrt {d x +c}}{b}+\frac {2 \left (-a d +b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b \sqrt {\left (a d -b c \right ) b}}\) | \(61\) |
risch | \(\frac {2 \sqrt {d x +c}}{b}-\frac {2 \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b \sqrt {\left (a d -b c \right ) b}}\) | \(61\) |
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none
Time = 0.24 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.31 \[ \int \frac {\sqrt {c+d x}}{a+b x} \, dx=\left [\frac {\sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, \sqrt {d x + c}}{b}, -\frac {2 \, {\left (\sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - \sqrt {d x + c}\right )}}{b}\right ] \]
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Time = 1.79 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {c+d x}}{a+b x} \, dx=\begin {cases} \frac {2 \left (\frac {d \sqrt {c + d x}}{b} - \frac {d \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{2} \sqrt {\frac {a d - b c}{b}}}\right )}{d} & \text {for}\: d \neq 0 \\\sqrt {c} \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {\sqrt {c+d x}}{a+b x} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+d x}}{a+b x} \, dx=\frac {2 \, {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b} + \frac {2 \, \sqrt {d x + c}}{b} \]
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Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {c+d x}}{a+b x} \, dx=\frac {2\,\sqrt {c+d\,x}}{b}-\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )\,\sqrt {a\,d-b\,c}}{b^{3/2}} \]
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