Integrand size = 13, antiderivative size = 23 \[ \int \frac {1}{(c+d (a+b x))^{5/2}} \, dx=-\frac {2}{3 b d (c+d (a+b x))^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32} \[ \int \frac {1}{(c+d (a+b x))^{5/2}} \, dx=-\frac {2}{3 b d (d (a+b x)+c)^{3/2}} \]
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Rule 32
Rule 33
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(c+d x)^{5/2}} \, dx,x,a+b x\right )}{b} \\ & = -\frac {2}{3 b d (c+d (a+b x))^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d (a+b x))^{5/2}} \, dx=-\frac {2}{3 b d (c+a d+b d x)^{3/2}} \]
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Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
gosper | \(-\frac {2}{3 \left (b d x +a d +c \right )^{\frac {3}{2}} d b}\) | \(20\) |
derivativedivides | \(-\frac {2}{3 \left (b d x +a d +c \right )^{\frac {3}{2}} d b}\) | \(20\) |
default | \(-\frac {2}{3 \left (b d x +a d +c \right )^{\frac {3}{2}} d b}\) | \(20\) |
trager | \(-\frac {2}{3 \left (b d x +a d +c \right )^{\frac {3}{2}} d b}\) | \(20\) |
pseudoelliptic | \(-\frac {2}{3 b d \left (c +d \left (b x +a \right )\right )^{\frac {3}{2}}}\) | \(20\) |
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (19) = 38\).
Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.96 \[ \int \frac {1}{(c+d (a+b x))^{5/2}} \, dx=-\frac {2 \, \sqrt {b d x + a d + c}}{3 \, {\left (b^{3} d^{3} x^{2} + a^{2} b d^{3} + 2 \, a b c d^{2} + b c^{2} d + 2 \, {\left (a b^{2} d^{3} + b^{2} c d^{2}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (19) = 38\).
Time = 0.70 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.43 \[ \int \frac {1}{(c+d (a+b x))^{5/2}} \, dx=\begin {cases} \frac {x}{c^{\frac {5}{2}}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x}{\left (a d + c\right )^{\frac {5}{2}}} & \text {for}\: b = 0 \\\frac {x}{c^{\frac {5}{2}}} & \text {for}\: d = 0 \\- \frac {2 \sqrt {a d + b d x + c}}{3 a^{2} b d^{3} + 6 a b^{2} d^{3} x + 6 a b c d^{2} + 3 b^{3} d^{3} x^{2} + 6 b^{2} c d^{2} x + 3 b c^{2} d} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(c+d (a+b x))^{5/2}} \, dx=-\frac {2}{3 \, {\left ({\left (b x + a\right )} d + c\right )}^{\frac {3}{2}} b d} \]
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Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(c+d (a+b x))^{5/2}} \, dx=-\frac {2}{3 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}} b d} \]
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Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(c+d (a+b x))^{5/2}} \, dx=-\frac {2}{3\,b\,d\,{\left (c+d\,\left (a+b\,x\right )\right )}^{3/2}} \]
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