Integrand size = 30, antiderivative size = 112 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 \sqrt {b d-a e} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {660, 52, 65, 214} \[ \int \frac {\sqrt {d+e x}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (a+b x) \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 52
Rule 65
Rule 214
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right ) \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 )}{b^2 e \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 \sqrt {b d-a e} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) \left (\sqrt {b} \sqrt {d+e x}-\sqrt {-b d+a e} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{b^{3/2} \sqrt {(a+b x)^2}} \]
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Time = 2.16 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {2 \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{b \left (b x +a \right )}-\frac {2 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )}\) | \(93\) |
default | \(\frac {2 \left (b x +a \right ) \left (-\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a e +\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b d +\sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\right )}{\sqrt {\left (b x +a \right )^{2}}\, b \sqrt {\left (a e -b d \right ) b}}\) | \(104\) |
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Time = 0.64 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\left [\frac {\sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, \sqrt {e x + d}}{b}, -\frac {2 \, {\left (\sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - \sqrt {e x + d}\right )}}{b}\right ] \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {\left (a + b x\right )^{2}}}\, dx \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {{\left (b x + a\right )}^{2}}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 \, {\left (b d \mathrm {sgn}\left (b x + a\right ) - a e \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b} + \frac {2 \, \sqrt {e x + d} \mathrm {sgn}\left (b x + a\right )}{b} \]
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Timed out. \[ \int \frac {\sqrt {d+e x}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {\sqrt {d+e\,x}}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
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