Integrand size = 21, antiderivative size = 47 \[ \int \frac {1}{\sqrt {a+b x}+\sqrt {c+b x}} \, dx=\frac {2 (a+b x)^{3/2}}{3 b (a-c)}-\frac {2 (c+b x)^{3/2}}{3 b (a-c)} \]
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Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6821} \[ \int \frac {1}{\sqrt {a+b x}+\sqrt {c+b x}} \, dx=\frac {2 (a+b x)^{3/2}}{3 b (a-c)}-\frac {2 (b x+c)^{3/2}}{3 b (a-c)} \]
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Rule 6821
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (\sqrt {a+b x}-\sqrt {c+b x}\right ) \, dx}{a-c} \\ & = \frac {2 (a+b x)^{3/2}}{3 b (a-c)}-\frac {2 (c+b x)^{3/2}}{3 b (a-c)} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {a+b x}+\sqrt {c+b x}} \, dx=\frac {2 \left ((a+b x)^{3/2}-(c+b x)^{3/2}\right )}{3 b (a-c)} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3 b \left (a -c \right )}-\frac {2 \left (b x +c \right )^{\frac {3}{2}}}{3 b \left (a -c \right )}\) | \(40\) |
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none
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\sqrt {a+b x}+\sqrt {c+b x}} \, dx=\frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - {\left (b x + c\right )}^{\frac {3}{2}}\right )}}{3 \, {\left (a b - b c\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (32) = 64\).
Time = 0.36 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.89 \[ \int \frac {1}{\sqrt {a+b x}+\sqrt {c+b x}} \, dx=\begin {cases} \frac {2 a}{3 b \sqrt {a + b x} + 3 b \sqrt {b x + c}} + \frac {4 b x}{3 b \sqrt {a + b x} + 3 b \sqrt {b x + c}} + \frac {2 c}{3 b \sqrt {a + b x} + 3 b \sqrt {b x + c}} + \frac {2 \sqrt {a + b x} \sqrt {b x + c}}{3 b \sqrt {a + b x} + 3 b \sqrt {b x + c}} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {a} + \sqrt {c}} & \text {otherwise} \end {cases} \]
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\[ \int \frac {1}{\sqrt {a+b x}+\sqrt {c+b x}} \, dx=\int { \frac {1}{\sqrt {b x + a} + \sqrt {b x + c}} \,d x } \]
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none
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\sqrt {a+b x}+\sqrt {c+b x}} \, dx=-\frac {2}{3} \, \sqrt {b x + c} {\left (\frac {{\left (b x + a\right )} b}{a b^{2} - b^{2} c} - \frac {a b - b c}{a b^{2} - b^{2} c}\right )} + \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, {\left (a b - b c\right )}} \]
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Time = 16.85 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.68 \[ \int \frac {1}{\sqrt {a+b x}+\sqrt {c+b x}} \, dx=\frac {2\,x\,\sqrt {a+b\,x}}{3\,\left (a-c\right )}-\frac {2\,x\,\sqrt {c+b\,x}}{3\,\left (a-c\right )}+\frac {2\,a\,\sqrt {a+b\,x}}{3\,b\,\left (a-c\right )}-\frac {2\,c\,\sqrt {c+b\,x}}{3\,b\,\left (a-c\right )} \]
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