Integrand size = 29, antiderivative size = 43 \[ \int \frac {1}{\sqrt {\frac {-b+b c}{d}+b x} \sqrt {c+d x}} \, dx=\frac {2 \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {-\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]
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Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {65, 221} \[ \int \frac {1}{\sqrt {\frac {-b+b c}{d}+b x} \sqrt {c+d x}} \, dx=\frac {2 \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {b x-\frac {b (1-c)}{d}}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rule 65
Rule 221
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {-b+b c}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {\frac {-b+b c}{d}+b x}\right )}{b} \\ & = \frac {2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {-\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\sqrt {\frac {-b+b c}{d}+b x} \sqrt {c+d x}} \, dx=\frac {2 \sqrt {-1+c+d x} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {-1+c+d x}}\right )}{d \sqrt {\frac {b (-1+c+d x)}{d}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs. \(2(33)=66\).
Time = 0.60 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.33
method | result | size |
default | \(\frac {\sqrt {\left (b x +\frac {b \left (c -1\right )}{d}\right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {b \left (c -1\right )}{2}+\frac {b c}{2}+b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (b \left (c -1\right )+b c \right ) x +\frac {b \left (c -1\right ) c}{d}}\right )}{\sqrt {b x +\frac {b \left (c -1\right )}{d}}\, \sqrt {d x +c}\, \sqrt {b d}}\) | \(100\) |
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (30) = 60\).
Time = 0.24 (sec) , antiderivative size = 175, normalized size of antiderivative = 4.07 \[ \int \frac {1}{\sqrt {\frac {-b+b c}{d}+b x} \sqrt {c+d x}} \, dx=\left [\frac {\sqrt {b d} \log \left (8 \, b d^{2} x^{2} + 8 \, b c^{2} + 8 \, {\left (2 \, b c - b\right )} d x + 4 \, \sqrt {b d} {\left (2 \, d x + 2 \, c - 1\right )} \sqrt {d x + c} \sqrt {\frac {b d x + b c - b}{d}} - 8 \, b c + b\right )}{2 \, b d}, -\frac {\sqrt {-b d} \arctan \left (\frac {\sqrt {-b d} {\left (2 \, d x + 2 \, c - 1\right )} \sqrt {d x + c} \sqrt {\frac {b d x + b c - b}{d}}}{2 \, {\left (b d^{2} x^{2} + b c^{2} + {\left (2 \, b c - b\right )} d x - b c\right )}}\right )}{b d}\right ] \]
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\[ \int \frac {1}{\sqrt {\frac {-b+b c}{d}+b x} \sqrt {c+d x}} \, dx=\int \frac {1}{\sqrt {b \left (\frac {c}{d} + x - \frac {1}{d}\right )} \sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {1}{\sqrt {\frac {-b+b c}{d}+b x} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (30) = 60\).
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\sqrt {\frac {-b+b c}{d}+b x} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left | b \right |} \log \left (-\sqrt {b d^{2} x + b c d - b d} \sqrt {b d} + \sqrt {b^{2} d^{2} + {\left (b d^{2} x + b c d - b d\right )} b d}\right )}{\sqrt {b d} b} \]
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Time = 0.48 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\sqrt {\frac {-b+b c}{d}+b x} \sqrt {c+d x}} \, dx=\frac {4\,\mathrm {atan}\left (-\frac {d\,\left (\sqrt {b\,x-\frac {b-b\,c}{d}}-\sqrt {-\frac {b-b\,c}{d}}\right )}{\sqrt {-b\,d}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )}{\sqrt {-b\,d}} \]
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