Integrand size = 18, antiderivative size = 38 \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {a x^2}{c \sqrt {c x^2}}+\frac {b x^3}{2 c \sqrt {c x^2}} \]
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Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {15} \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {a x^2}{c \sqrt {c x^2}}+\frac {b x^3}{2 c \sqrt {c x^2}} \]
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Rule 15
Rubi steps \begin{align*} \text {integral}& = \frac {x \int (a+b x) \, dx}{c \sqrt {c x^2}} \\ & = \frac {a x^2}{c \sqrt {c x^2}}+\frac {b x^3}{2 c \sqrt {c x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61 \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {x^4 (2 a+b x)}{2 \left (c x^2\right )^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(\frac {x^{4} \left (b x +2 a \right )}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(20\) |
default | \(\frac {x^{4} \left (b x +2 a \right )}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(20\) |
trager | \(\frac {\left (b x +2 a +b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{2 c^{2} x}\) | \(27\) |
risch | \(\frac {a \,x^{2}}{c \sqrt {c \,x^{2}}}+\frac {b \,x^{3}}{2 c \sqrt {c \,x^{2}}}\) | \(33\) |
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none
Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.50 \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c x^{2}} {\left (b x + 2 \, a\right )}}{2 \, c^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {a x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b x^{5}}{2 \left (c x^{2}\right )^{\frac {3}{2}}} \]
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none
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {b x^{3}}{2 \, \sqrt {c x^{2}} c} + \frac {a x^{2}}{\sqrt {c x^{2}} c} \]
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none
Time = 0.36 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.50 \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {b x^{2} + 2 \, a x}{2 \, c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\int \frac {x^3\,\left (a+b\,x\right )}{{\left (c\,x^2\right )}^{3/2}} \,d x \]
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