Integrand size = 13, antiderivative size = 25 \[ \int \sqrt {c (a+b x)^2} \, dx=\frac {(a+b x) \sqrt {c (a+b x)^2}}{2 b} \]
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Time = 0.01 (sec) , antiderivative size = 25, 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} \[ \int \sqrt {c (a+b x)^2} \, dx=\frac {(a+b x) \sqrt {c (a+b x)^2}}{2 b} \]
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Rule 15
Rule 30
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {c x^2} \, dx,x,a+b x\right )}{b} \\ & = \frac {\sqrt {c (a+b x)^2} \text {Subst}(\int x \, dx,x,a+b x)}{b (a+b x)} \\ & = \frac {(a+b x) \sqrt {c (a+b x)^2}}{2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \sqrt {c (a+b x)^2} \, dx=\frac {c x (a+b x) (2 a+b x)}{2 \sqrt {c (a+b x)^2}} \]
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Time = 3.86 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16
method | result | size |
gosper | \(\frac {x \left (b x +2 a \right ) \sqrt {c \left (b x +a \right )^{2}}}{2 b x +2 a}\) | \(29\) |
default | \(\frac {x \left (b x +2 a \right ) \sqrt {c \left (b x +a \right )^{2}}}{2 b x +2 a}\) | \(29\) |
trager | \(\frac {x \left (b x +2 a \right ) \sqrt {b^{2} c \,x^{2}+2 a b c x +a^{2} c}}{2 b x +2 a}\) | \(40\) |
risch | \(\frac {\sqrt {c \left (b x +a \right )^{2}}\, a x}{b x +a}+\frac {\sqrt {c \left (b x +a \right )^{2}}\, b \,x^{2}}{2 b x +2 a}\) | \(47\) |
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none
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \sqrt {c (a+b x)^2} \, dx=\frac {\sqrt {b^{2} c x^{2} + 2 \, a b c x + a^{2} c} {\left (b x^{2} + 2 \, a x\right )}}{2 \, {\left (b x + a\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (20) = 40\).
Time = 0.45 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.84 \[ \int \sqrt {c (a+b x)^2} \, dx=\begin {cases} \left (\frac {a}{2 b} + \frac {x}{2}\right ) \sqrt {a^{2} c + 2 a b c x + b^{2} c x^{2}} & \text {for}\: b^{2} c \neq 0 \\\frac {\left (a^{2} c + 2 a b c x\right )^{\frac {3}{2}}}{3 a b c} & \text {for}\: a b c \neq 0 \\x \sqrt {a^{2} c} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (21) = 42\).
Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \sqrt {c (a+b x)^2} \, dx=\frac {1}{2} \, \sqrt {b^{2} c x^{2} + 2 \, a b c x + a^{2} c} x + \frac {\sqrt {b^{2} c x^{2} + 2 \, a b c x + a^{2} c} a}{2 \, b} \]
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \sqrt {c (a+b x)^2} \, dx=\frac {1}{2} \, {\left ({\left (b x^{2} + 2 \, a x\right )} \mathrm {sgn}\left (b x + a\right ) + \frac {a^{2} \mathrm {sgn}\left (b x + a\right )}{b}\right )} \sqrt {c} \]
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Time = 6.74 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \sqrt {c (a+b x)^2} \, dx=\frac {\sqrt {c\,{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{2\,b} \]
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