Integrand size = 28, antiderivative size = 70 \[ \int \frac {\sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {\sqrt {d+e x}}{b (a+b x)}-\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}} \]
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Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 43, 65, 214} \[ \int \frac {\sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}}-\frac {\sqrt {d+e x}}{b (a+b x)} \]
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Rule 27
Rule 43
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {d+e x}}{(a+b x)^2} \, dx \\ & = -\frac {\sqrt {d+e x}}{b (a+b x)}+\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b} \\ & = -\frac {\sqrt {d+e x}}{b (a+b x)}+\frac {\text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b} \\ & = -\frac {\sqrt {d+e x}}{b (a+b x)}-\frac {e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {\sqrt {d+e x}}{b (a+b x)}+\frac {e \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{3/2} \sqrt {-b d+a e}} \]
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Time = 2.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.83
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {e x +d}}{b x +a}+\frac {e \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}}}{b}\) | \(58\) |
derivativedivides | \(2 e \left (-\frac {\sqrt {e x +d}}{2 b \left (b \left (e x +d \right )+a e -b d \right )}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 b \sqrt {\left (a e -b d \right ) b}}\right )\) | \(73\) |
default | \(2 e \left (-\frac {\sqrt {e x +d}}{2 b \left (b \left (e x +d \right )+a e -b d \right )}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 b \sqrt {\left (a e -b d \right ) b}}\right )\) | \(73\) |
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none
Time = 0.47 (sec) , antiderivative size = 232, normalized size of antiderivative = 3.31 \[ \int \frac {\sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\left [\frac {\sqrt {b^{2} d - a b e} {\left (b e x + a e\right )} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (b^{2} d - a b e\right )} \sqrt {e x + d}}{2 \, {\left (a b^{3} d - a^{2} b^{2} e + {\left (b^{4} d - a b^{3} e\right )} x\right )}}, \frac {\sqrt {-b^{2} d + a b e} {\left (b e x + a e\right )} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (b^{2} d - a b e\right )} \sqrt {e x + d}}{a b^{3} d - a^{2} b^{2} e + {\left (b^{4} d - a b^{3} e\right )} x}\right ] \]
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\[ \int \frac {\sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\int \frac {\sqrt {d + e x}}{\left (a + b x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.32 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {e \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b} - \frac {\sqrt {e x + d} e}{{\left ({\left (e x + d\right )} b - b d + a e\right )} b} \]
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Time = 9.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {e\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{b^{3/2}\,\sqrt {a\,e-b\,d}}-\frac {e\,\sqrt {d+e\,x}}{e\,x\,b^2+a\,e\,b} \]
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