Integrand size = 18, antiderivative size = 41 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^3} \, dx=\frac {1}{3} a c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b c^2 x^3 \sqrt {c x^2} \]
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Time = 0.01 (sec) , antiderivative size = 41, 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} \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^3} \, dx=\frac {1}{3} a c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b c^2 x^3 \sqrt {c x^2} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^2 \sqrt {c x^2}\right ) \int x^2 (a+b x) \, dx}{x} \\ & = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a x^2+b x^3\right ) \, dx}{x} \\ & = \frac {1}{3} a c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b c^2 x^3 \sqrt {c x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^3} \, dx=\frac {1}{12} c^2 x^2 \sqrt {c x^2} (4 a+3 b x) \]
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Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(\frac {\left (3 b x +4 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{12 x^{2}}\) | \(21\) |
default | \(\frac {\left (3 b x +4 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{12 x^{2}}\) | \(21\) |
risch | \(\frac {a \,c^{2} x^{2} \sqrt {c \,x^{2}}}{3}+\frac {b \,c^{2} x^{3} \sqrt {c \,x^{2}}}{4}\) | \(34\) |
trager | \(\frac {c^{2} \left (3 b \,x^{3}+4 a \,x^{2}+3 b \,x^{2}+4 a x +3 b x +4 a +3 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{12 x}\) | \(52\) |
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Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.68 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^3} \, dx=\frac {1}{12} \, {\left (3 \, b c^{2} x^{3} + 4 \, a c^{2} x^{2}\right )} \sqrt {c x^{2}} \]
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Time = 0.46 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^3} \, dx=\frac {a \left (c x^{2}\right )^{\frac {5}{2}}}{3 x^{2}} + \frac {b \left (c x^{2}\right )^{\frac {5}{2}}}{4 x} \]
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Exception generated. \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^3} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.68 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^3} \, dx=\frac {1}{12} \, {\left (3 \, b c^{2} x^{4} \mathrm {sgn}\left (x\right ) + 4 \, a c^{2} x^{3} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \]
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Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^3} \, dx=\frac {c^{5/2}\,\left (4\,a\,\sqrt {x^6}+3\,b\,x^3\,\sqrt {x^2}\right )}{12} \]
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