Integrand size = 19, antiderivative size = 30 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {a+b x}}{(b c-a d) \sqrt {c+d x}} \]
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Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {a+b x}}{\sqrt {c+d x} (b c-a d)} \]
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Rule 37
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {a+b x}}{(b c-a d) \sqrt {c+d x}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {a+b x}}{(b c-a d) \sqrt {c+d x}} \]
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}}{\sqrt {d x +c}\, \left (a d -b c \right )}\) | \(27\) |
default | \(-\frac {2 \sqrt {b x +a}}{\sqrt {d x +c}\, \left (a d -b c \right )}\) | \(27\) |
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none
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {2 \, \sqrt {b x + a} \sqrt {d x + c}}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x} \]
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\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\int \frac {1}{\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.34 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {2 \, \sqrt {b x + a} b^{2}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (b c {\left | b \right |} - a d {\left | b \right |}\right )}} \]
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Time = 0.74 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=-\frac {2\,\sqrt {a+b\,x}}{\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}} \]
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