Integrand size = 18, antiderivative size = 35 \[ \int x^2 \sqrt {c x^2} (a+b x) \, dx=\frac {1}{4} a x^3 \sqrt {c x^2}+\frac {1}{5} b x^4 \sqrt {c x^2} \]
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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} \[ \int x^2 \sqrt {c x^2} (a+b x) \, dx=\frac {1}{4} a x^3 \sqrt {c x^2}+\frac {1}{5} b x^4 \sqrt {c x^2} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c x^2} \int x^3 (a+b x) \, dx}{x} \\ & = \frac {\sqrt {c x^2} \int \left (a x^3+b x^4\right ) \, dx}{x} \\ & = \frac {1}{4} a x^3 \sqrt {c x^2}+\frac {1}{5} b x^4 \sqrt {c x^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.69 \[ \int x^2 \sqrt {c x^2} (a+b x) \, dx=\frac {1}{20} x^3 \sqrt {c x^2} (5 a+4 b x) \]
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Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(\frac {x^{3} \left (4 b x +5 a \right ) \sqrt {c \,x^{2}}}{20}\) | \(21\) |
default | \(\frac {x^{3} \left (4 b x +5 a \right ) \sqrt {c \,x^{2}}}{20}\) | \(21\) |
risch | \(\frac {a \,x^{3} \sqrt {c \,x^{2}}}{4}+\frac {b \,x^{4} \sqrt {c \,x^{2}}}{5}\) | \(28\) |
trager | \(\frac {\left (4 b \,x^{4}+5 a \,x^{3}+4 b \,x^{3}+5 a \,x^{2}+4 b \,x^{2}+5 a x +4 b x +5 a +4 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{20 x}\) | \(61\) |
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Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int x^2 \sqrt {c x^2} (a+b x) \, dx=\frac {1}{20} \, {\left (4 \, b x^{4} + 5 \, a x^{3}\right )} \sqrt {c x^{2}} \]
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Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int x^2 \sqrt {c x^2} (a+b x) \, dx=\frac {a x^{3} \sqrt {c x^{2}}}{4} + \frac {b x^{4} \sqrt {c x^{2}}}{5} \]
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Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int x^2 \sqrt {c x^2} (a+b x) \, dx=\frac {\left (c x^{2}\right )^{\frac {3}{2}} b x^{2}}{5 \, c} + \frac {\left (c x^{2}\right )^{\frac {3}{2}} a x}{4 \, c} \]
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Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int x^2 \sqrt {c x^2} (a+b x) \, dx=\frac {1}{20} \, {\left (4 \, b x^{5} \mathrm {sgn}\left (x\right ) + 5 \, a x^{4} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \]
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Timed out. \[ \int x^2 \sqrt {c x^2} (a+b x) \, dx=\int x^2\,\sqrt {c\,x^2}\,\left (a+b\,x\right ) \,d x \]
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