Integrand size = 15, antiderivative size = 44 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2 A \sqrt {a+b x}}{b}+\frac {2 B \sqrt {a+b x} \left (-a+\frac {1}{3} (a+b x)\right )}{b^2} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} (A b-a B)}{b^2}+\frac {2 B (a+b x)^{3/2}}{3 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A b-a B}{b \sqrt {a+b x}}+\frac {B \sqrt {a+b x}}{b}\right ) \, dx \\ & = \frac {2 (A b-a B) \sqrt {a+b x}}{b^2}+\frac {2 B (a+b x)^{3/2}}{3 b^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.66 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} (3 A b-2 a B+b B x)}{3 b^2} \]
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Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-b \beta x +2 a \beta -3 \alpha b \right )}{3 b^{2}}\) | \(27\) |
trager | \(-\frac {2 \sqrt {b x +a}\, \left (-b \beta x +2 a \beta -3 \alpha b \right )}{3 b^{2}}\) | \(27\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-b \beta x +2 a \beta -3 \alpha b \right )}{3 b^{2}}\) | \(27\) |
pseudoelliptic | \(-\frac {2 \sqrt {b x +a}\, \left (-b \beta x +2 a \beta -3 \alpha b \right )}{3 b^{2}}\) | \(27\) |
derivativedivides | \(\frac {\frac {2 \beta \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \beta \sqrt {b x +a}+2 \alpha b \sqrt {b x +a}}{b^{2}}\) | \(38\) |
default | \(\frac {\frac {2 \beta \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \beta \sqrt {b x +a}+2 \alpha b \sqrt {b x +a}}{b^{2}}\) | \(38\) |
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none
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.57 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (b \beta x + 3 \, \alpha b - 2 \, a \beta \right )} \sqrt {b x + a}}{3 \, b^{2}} \]
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Time = 0.41 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.20 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\begin {cases} \frac {2 A \sqrt {a + b x} + \frac {2 B \left (- a \sqrt {a + b x} + \frac {\left (a + b x\right )^{\frac {3}{2}}}{3}\right )}{b}}{b} & \text {for}\: b \neq 0 \\\frac {A x + \frac {B x^{2}}{2}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {b x + a} \alpha + \frac {{\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} \beta }{b}\right )}}{3 \, b} \]
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Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {b x + a} \alpha + \frac {{\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} \beta }{b}\right )}}{3 \, b} \]
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Time = 16.96 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.64 \[ \int \frac {A+B x}{\sqrt {a+b x}} \, dx=\frac {2\,\sqrt {a+b\,x}\,\left (3\,\alpha \,b+\left (a+b\,x\right )\,\beta -3\,a\,\beta \right )}{3\,b^2} \]
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