Integrand size = 30, antiderivative size = 72 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} \sqrt {b d-a e} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {660, 65, 214} \[ \int \frac {1}{\sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {b d-a e}} \]
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Rule 65
Rule 214
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (2 \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} \sqrt {b d-a e} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{\sqrt {b} \sqrt {-b d+a e} \sqrt {(a+b x)^2}} \]
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Time = 2.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {2 \left (b x +a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (b x +a \right )^{2}}\, \sqrt {\left (a e -b d \right ) b}}\) | \(51\) |
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Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.65 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\left [\frac {\log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right )}{\sqrt {b^{2} d - a b e}}, \frac {2 \, \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right )}{b^{2} d - a b e}\right ] \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {1}{\sqrt {d + e x} \sqrt {\left (a + b x\right )^{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {1}{\sqrt {{\left (b x + a\right )}^{2}} \sqrt {e x + d}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right ) \mathrm {sgn}\left (b x + a\right )}{\sqrt {-b^{2} d + a b e}} \]
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Timed out. \[ \int \frac {1}{\sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {1}{\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d+e\,x}} \,d x \]
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