Integrand size = 28, antiderivative size = 76 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {\sqrt {d+e x}}{(b d-a e) (a+b x)}+\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 76, 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, 44, 65, 214} \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)} \]
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Rule 27
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx \\ & = -\frac {\sqrt {d+e x}}{(b d-a e) (a+b x)}-\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 (b d-a e)} \\ & = -\frac {\sqrt {d+e x}}{(b d-a e) (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 d-a e} \\ & = -\frac {\sqrt {d+e x}}{(b d-a e) (a+b x)}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {\sqrt {d+e x}}{(-b d+a e) (a+b x)}+\frac {e \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{\sqrt {b} (-b d+a e)^{3/2}} \]
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Time = 2.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {\frac {\sqrt {e x +d}}{b x +a}+\frac {e \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}}}{a e -b d}\) | \(64\) |
derivativedivides | \(2 e \left (\frac {\sqrt {e x +d}}{2 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}\right )\) | \(87\) |
default | \(2 e \left (\frac {\sqrt {e x +d}}{2 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}\right )\) | \(87\) |
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Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (64) = 128\).
Time = 0.38 (sec) , antiderivative size = 280, normalized size of antiderivative = 3.68 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\left [-\frac {\sqrt {b^{2} d - a b e} {\left (b e x + a e\right )} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) + 2 \, {\left (b^{2} d - a b e\right )} \sqrt {e x + d}}{2 \, {\left (a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2} + {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x\right )}}, -\frac {\sqrt {-b^{2} d + a b e} {\left (b e x + a e\right )} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left (b^{2} d - a b e\right )} \sqrt {e x + d}}{a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2} + {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x}\right ] \]
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\[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\int \frac {1}{\left (a + b x\right )^{2} \sqrt {d + e x}}\, dx \]
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Exception generated. \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {e \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} {\left (b d - a e\right )}} - \frac {\sqrt {e x + d} e}{{\left ({\left (e x + d\right )} b - b d + a e\right )} {\left (b d - a e\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {e\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{\sqrt {b}\,{\left (a\,e-b\,d\right )}^{3/2}}+\frac {e\,\sqrt {d+e\,x}}{\left (a\,e-b\,d\right )\,\left (a\,e-b\,d+b\,\left (d+e\,x\right )\right )} \]
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