Integrand size = 18, antiderivative size = 39 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^4} \, dx=\frac {1}{2} a c^2 x \sqrt {c x^2}+\frac {1}{3} b c^2 x^2 \sqrt {c x^2} \]
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Time = 0.01 (sec) , antiderivative size = 39, 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} \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^4} \, dx=\frac {1}{2} a c^2 x \sqrt {c x^2}+\frac {1}{3} b c^2 x^2 \sqrt {c x^2} \]
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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 (a+b x) \, dx}{x} \\ & = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a x+b x^2\right ) \, dx}{x} \\ & = \frac {1}{2} a c^2 x \sqrt {c x^2}+\frac {1}{3} b c^2 x^2 \sqrt {c x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.64 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^4} \, dx=\frac {1}{6} c^2 x \sqrt {c x^2} (3 a+2 b x) \]
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Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(\frac {\left (2 b x +3 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{6 x^{3}}\) | \(21\) |
default | \(\frac {\left (2 b x +3 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{6 x^{3}}\) | \(21\) |
risch | \(\frac {a \,c^{2} x \sqrt {c \,x^{2}}}{2}+\frac {b \,c^{2} x^{2} \sqrt {c \,x^{2}}}{3}\) | \(32\) |
trager | \(\frac {c^{2} \left (2 b \,x^{2}+3 a x +2 b x +3 a +2 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{6 x}\) | \(40\) |
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none
Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.67 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^4} \, dx=\frac {1}{6} \, {\left (2 \, b c^{2} x^{2} + 3 \, a c^{2} x\right )} \sqrt {c x^{2}} \]
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Time = 0.84 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^4} \, dx=\frac {a \left (c x^{2}\right )^{\frac {5}{2}}}{2 x^{3}} + \frac {b \left (c x^{2}\right )^{\frac {5}{2}}}{3 x^{2}} \]
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Exception generated. \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^4} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^4} \, dx=\frac {1}{6} \, {\left (2 \, b c^{2} x^{3} \mathrm {sgn}\left (x\right ) + 3 \, a c^{2} x^{2} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \]
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Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.51 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^4} \, dx=\frac {c^{5/2}\,\left (2\,b\,\sqrt {x^6}+3\,a\,x\,\left |x\right |\right )}{6} \]
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