Integrand size = 19, antiderivative size = 42 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}} \]
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Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {65, 223, 212} \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rule 65
Rule 212
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 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(30)=60\).
Time = 0.68 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.81
method | result | size |
default | \(\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {b d}}\) | \(76\) |
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (30) = 60\).
Time = 0.24 (sec) , antiderivative size = 178, normalized size of antiderivative = 4.24 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\left [\frac {\sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )}{2 \, b d}, -\frac {\sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right )}{b d}\right ] \]
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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 = 0.34 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \, b \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} {\left | b \right |}} \]
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Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}{\sqrt {-b\,d}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}\right )}{\sqrt {-b\,d}} \]
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