Integrand size = 13, antiderivative size = 30 \[ \int \frac {1}{\left (c (a+b x)^2\right )^{3/2}} \, dx=-\frac {1}{2 b c (a+b x) \sqrt {c (a+b x)^2}} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 15, 30} \[ \int \frac {1}{\left (c (a+b x)^2\right )^{3/2}} \, dx=-\frac {1}{2 b c (a+b x) \sqrt {c (a+b x)^2}} \]
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Rule 15
Rule 30
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (c x^2\right )^{3/2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,a+b x\right )}{b c \sqrt {c (a+b x)^2}} \\ & = -\frac {1}{2 b c (a+b x) \sqrt {c (a+b x)^2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (c (a+b x)^2\right )^{3/2}} \, dx=-\frac {a+b x}{2 b \left (c (a+b x)^2\right )^{3/2}} \]
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Time = 3.99 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73
method | result | size |
gosper | \(-\frac {b x +a}{2 b \left (c \left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(22\) |
default | \(-\frac {b x +a}{2 b \left (c \left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(22\) |
risch | \(-\frac {1}{2 b c \left (b x +a \right ) \sqrt {c \left (b x +a \right )^{2}}}\) | \(27\) |
trager | \(\frac {\left (b x +2 a \right ) x \sqrt {b^{2} c \,x^{2}+2 a b c x +a^{2} c}}{2 a^{2} c^{2} \left (b x +a \right )^{3}}\) | \(46\) |
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (26) = 52\).
Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.30 \[ \int \frac {1}{\left (c (a+b x)^2\right )^{3/2}} \, dx=-\frac {\sqrt {b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{2 \, {\left (b^{4} c^{2} x^{3} + 3 \, a b^{3} c^{2} x^{2} + 3 \, a^{2} b^{2} c^{2} x + a^{3} b c^{2}\right )}} \]
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Time = 1.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (c (a+b x)^2\right )^{3/2}} \, dx=\begin {cases} - \frac {\frac {a}{b} + x}{2 \left (c \left (a + b x\right )^{2}\right )^{\frac {3}{2}}} & \text {for}\: b \neq 0 \\\frac {x}{\left (a^{2} c\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (c (a+b x)^2\right )^{3/2}} \, dx=-\frac {1}{2 \, b^{3} c^{\frac {3}{2}} {\left (x + \frac {a}{b}\right )}^{2}} \]
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none
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (c (a+b x)^2\right )^{3/2}} \, dx=-\frac {1}{2 \, {\left (b x + a\right )}^{2} b c^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right )} \]
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Time = 6.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (c (a+b x)^2\right )^{3/2}} \, dx=-\frac {\sqrt {c\,{\left (a+b\,x\right )}^2}}{2\,b\,c^2\,{\left (a+b\,x\right )}^3} \]
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