Integrand size = 29, antiderivative size = 42 \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=\frac {2 \arcsin \left (\frac {\sqrt {d} \sqrt {\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 42, 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, 222} \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=\frac {2 \arcsin \left (\frac {\sqrt {d} \sqrt {\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]
[In]
[Out]
Rule 65
Rule 222
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 \sin ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=\frac {2 \sqrt {1-c+d x} \arctan \left (\frac {\sqrt {1-c+d x}}{\sqrt {c-d x}}\right )}{d \sqrt {\frac {b (1-c+d x)}{d}}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(32)=64\).
Time = 0.60 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.60
method | result | size |
default | \(\frac {\sqrt {\left (b x -\frac {b \left (c -1\right )}{d}\right ) \left (-d x +c \right )}\, \arctan \left (\frac {\sqrt {b d}\, \left (x -\frac {b \left (c -1\right )+b c}{2 b d}\right )}{\sqrt {-b d \,x^{2}+\left (b \left (c -1\right )+b c \right ) x -\frac {b \left (c -1\right ) c}{d}}}\right )}{\sqrt {b x -\frac {b \left (c -1\right )}{d}}\, \sqrt {-d x +c}\, \sqrt {b d}}\) | \(109\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (31) = 62\).
Time = 0.24 (sec) , antiderivative size = 176, normalized size of antiderivative = 4.19 \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=\left [-\frac {\sqrt {-b d} \log \left (8 \, b d^{2} x^{2} + 8 \, b c^{2} - 8 \, {\left (2 \, b c - b\right )} d x - 4 \, \sqrt {-b d} {\left (2 \, d x - 2 \, c + 1\right )} \sqrt {-d x + c} \sqrt {\frac {b d x - b c + b}{d}} - 8 \, b c + b\right )}{2 \, b d}, -\frac {\sqrt {b d} \arctan \left (\frac {\sqrt {b d} {\left (2 \, d x - 2 \, c + 1\right )} \sqrt {-d x + c} \sqrt {\frac {b d x - b c + b}{d}}}{2 \, {\left (b d^{2} x^{2} + b c^{2} - {\left (2 \, b c - b\right )} d x - b c\right )}}\right )}{b d}\right ] \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=\int \frac {1}{\sqrt {b \left (- \frac {c}{d} + x + \frac {1}{d}\right )} \sqrt {c - d x}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (31) = 62\).
Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.64 \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=-\frac {2 \, {\left | b \right |} \log \left (-\sqrt {b d^{2} x - b c d + b d} \sqrt {-b d} + \sqrt {b^{2} d^{2} - {\left (b d^{2} x - b c d + b d\right )} b d}\right )}{\sqrt {-b d} b} \]
[In]
[Out]
Time = 0.54 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=-\frac {4\,\mathrm {atan}\left (-\frac {d\,\left (\sqrt {\frac {b-b\,c}{d}+b\,x}-\sqrt {\frac {b-b\,c}{d}}\right )}{\sqrt {b\,d}\,\left (\sqrt {c-d\,x}-\sqrt {c}\right )}\right )}{\sqrt {b\,d}} \]
[In]
[Out]