Integrand size = 20, antiderivative size = 43 \[ \int \frac {1}{\sqrt {a-b x} \sqrt {c+d x}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {d} \sqrt {a-b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}} \]
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Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {65, 223, 209} \[ \int \frac {1}{\sqrt {a-b x} \sqrt {c+d x}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {d} \sqrt {a-b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {a d}{b}-\frac {d x^2}{b}}} \, dx,x,\sqrt {a-b x}\right )}{b} \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{1+\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a-b x}}{\sqrt {c+d x}}\right )}{b} \\ & = -\frac {2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a-b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a-b x} \sqrt {c+d x}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a-b x}}\right )}{\sqrt {b} \sqrt {d}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(31)=62\).
Time = 0.56 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.95
method | result | size |
default | \(\frac {\sqrt {\left (-b x +a \right ) \left (d x +c \right )}\, \arctan \left (\frac {\sqrt {b d}\, \left (x -\frac {a d -b c}{2 b d}\right )}{\sqrt {-b d \,x^{2}+\left (a d -b c \right ) x +a c}}\right )}{\sqrt {-b x +a}\, \sqrt {d x +c}\, \sqrt {b d}}\) | \(84\) |
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (31) = 62\).
Time = 0.24 (sec) , antiderivative size = 185, normalized size of antiderivative = 4.30 \[ \int \frac {1}{\sqrt {a-b x} \sqrt {c+d x}} \, dx=\left [-\frac {\sqrt {-b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c - a d\right )} \sqrt {-b d} \sqrt {-b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d - a b d^{2}\right )} x\right )}{2 \, b d}, -\frac {\sqrt {b d} \arctan \left (\frac {{\left (2 \, b d x + b c - a d\right )} \sqrt {b d} \sqrt {-b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x\right )}}\right )}{b d}\right ] \]
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\[ \int \frac {1}{\sqrt {a-b x} \sqrt {c+d x}} \, dx=\int \frac {1}{\sqrt {a - b x} \sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {1}{\sqrt {a-b x} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {a-b x} \sqrt {c+d x}} \, dx=\frac {2 \, b \log \left ({\left | -\sqrt {-b d} \sqrt {-b x + a} + \sqrt {b^{2} c + {\left (b x - a\right )} b d + a b d} \right |}\right )}{\sqrt {-b d} {\left | b \right |}} \]
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Time = 0.34 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\sqrt {a-b x} \sqrt {c+d x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {d\,\left (\sqrt {a-b\,x}-\sqrt {a}\right )}{\sqrt {b\,d}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )}{\sqrt {b\,d}} \]
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