Integrand size = 18, antiderivative size = 35 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^2} \, dx=\frac {1}{2} a c x \sqrt {c x^2}+\frac {1}{3} b c x^2 \sqrt {c x^2} \]
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Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^2} \, dx=\frac {1}{2} a c x \sqrt {c x^2}+\frac {1}{3} b c x^2 \sqrt {c x^2} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int x (a+b x) \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \left (a x+b x^2\right ) \, dx}{x} \\ & = \frac {1}{2} a c x \sqrt {c x^2}+\frac {1}{3} b c x^2 \sqrt {c x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.66 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^2} \, dx=\frac {1}{6} c x \sqrt {c x^2} (3 a+2 b x) \]
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Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(\frac {\left (2 b x +3 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{6 x}\) | \(21\) |
default | \(\frac {\left (2 b x +3 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{6 x}\) | \(21\) |
risch | \(\frac {a c x \sqrt {c \,x^{2}}}{2}+\frac {b c \,x^{2} \sqrt {c \,x^{2}}}{3}\) | \(28\) |
trager | \(\frac {c \left (2 b \,x^{2}+3 a x +2 b x +3 a +2 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{6 x}\) | \(38\) |
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none
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^2} \, dx=\frac {1}{6} \, {\left (2 \, b c x^{2} + 3 \, a c x\right )} \sqrt {c x^{2}} \]
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.69 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^2} \, dx=\frac {a \left (c x^{2}\right )^{\frac {3}{2}}}{2 x} + \frac {b \left (c x^{2}\right )^{\frac {3}{2}}}{3} \]
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Exception generated. \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^2} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^2} \, dx=\frac {1}{6} \, {\left (2 \, b x^{3} \mathrm {sgn}\left (x\right ) + 3 \, a x^{2} \mathrm {sgn}\left (x\right )\right )} c^{\frac {3}{2}} \]
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Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^2} \, dx=\frac {c^{3/2}\,\left (2\,b\,\sqrt {x^6}+3\,a\,x\,\left |x\right |\right )}{6} \]
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