Integrand size = 20, antiderivative size = 26 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x} \, dx=\frac {\sqrt {c x^2} (a+b x)^3}{3 b x} \]
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Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32} \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x} \, dx=\frac {\sqrt {c x^2} (a+b x)^3}{3 b x} \]
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Rule 15
Rule 32
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c x^2} \int (a+b x)^2 \, dx}{x} \\ & = \frac {\sqrt {c x^2} (a+b x)^3}{3 b x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x} \, dx=\frac {c x (a+b x)^3}{3 b \sqrt {c x^2}} \]
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {\left (b x +a \right )^{3} \sqrt {c \,x^{2}}}{3 b x}\) | \(23\) |
risch | \(\frac {\left (b x +a \right )^{3} \sqrt {c \,x^{2}}}{3 b x}\) | \(23\) |
gosper | \(\frac {\left (b^{2} x^{2}+3 a b x +3 a^{2}\right ) \sqrt {c \,x^{2}}}{3}\) | \(28\) |
trager | \(\frac {\left (b^{2} x^{2}+3 a b x +b^{2} x +3 a^{2}+3 a b +b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{3 x}\) | \(46\) |
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none
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x} \, dx=\frac {1}{3} \, {\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt {c x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x} \, dx=a^{2} \sqrt {c x^{2}} + a b x \sqrt {c x^{2}} + \frac {b^{2} x^{2} \sqrt {c x^{2}}}{3} \]
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Exception generated. \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x} \, dx=\frac {1}{3} \, {\left (\frac {{\left (b x + a\right )}^{3} \mathrm {sgn}\left (x\right )}{b} - \frac {a^{3} \mathrm {sgn}\left (x\right )}{b}\right )} \sqrt {c} \]
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Timed out. \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x} \, dx=\int \frac {\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2}{x} \,d x \]
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