Integrand size = 15, antiderivative size = 40 \[ \int \frac {a+b x}{(c+d x)^{5/2}} \, dx=\frac {2 (b c-a d)}{3 d^2 (c+d x)^{3/2}}-\frac {2 b}{d^2 \sqrt {c+d x}} \]
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Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {a+b x}{(c+d x)^{5/2}} \, dx=\frac {2 (b c-a d)}{3 d^2 (c+d x)^{3/2}}-\frac {2 b}{d^2 \sqrt {c+d x}} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-b c+a d}{d (c+d x)^{5/2}}+\frac {b}{d (c+d x)^{3/2}}\right ) \, dx \\ & = \frac {2 (b c-a d)}{3 d^2 (c+d x)^{3/2}}-\frac {2 b}{d^2 \sqrt {c+d x}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72 \[ \int \frac {a+b x}{(c+d x)^{5/2}} \, dx=-\frac {2 (2 b c+a d+3 b d x)}{3 d^2 (c+d x)^{3/2}} \]
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Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.65
method | result | size |
gosper | \(-\frac {2 \left (3 b d x +a d +2 b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}} d^{2}}\) | \(26\) |
trager | \(-\frac {2 \left (3 b d x +a d +2 b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}} d^{2}}\) | \(26\) |
pseudoelliptic | \(-\frac {2 \left (\left (3 b x +a \right ) d +2 b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}} d^{2}}\) | \(26\) |
derivativedivides | \(\frac {-\frac {2 b}{\sqrt {d x +c}}-\frac {2 \left (a d -b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}}}}{d^{2}}\) | \(34\) |
default | \(\frac {-\frac {2 b}{\sqrt {d x +c}}-\frac {2 \left (a d -b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}}}}{d^{2}}\) | \(34\) |
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Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int \frac {a+b x}{(c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (3 \, b d x + 2 \, b c + a d\right )} \sqrt {d x + c}}{3 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (39) = 78\).
Time = 0.42 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.10 \[ \int \frac {a+b x}{(c+d x)^{5/2}} \, dx=\begin {cases} - \frac {2 a d}{3 c d^{2} \sqrt {c + d x} + 3 d^{3} x \sqrt {c + d x}} - \frac {4 b c}{3 c d^{2} \sqrt {c + d x} + 3 d^{3} x \sqrt {c + d x}} - \frac {6 b d x}{3 c d^{2} \sqrt {c + d x} + 3 d^{3} x \sqrt {c + d x}} & \text {for}\: d \neq 0 \\\frac {a x + \frac {b x^{2}}{2}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int \frac {a+b x}{(c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (d x + c\right )} b - b c + a d\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{2}} \]
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Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int \frac {a+b x}{(c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (d x + c\right )} b - b c + a d\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72 \[ \int \frac {a+b x}{(c+d x)^{5/2}} \, dx=-\frac {2\,a\,d-2\,b\,c+6\,b\,\left (c+d\,x\right )}{3\,d^2\,{\left (c+d\,x\right )}^{3/2}} \]
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