Integrand size = 20, antiderivative size = 31 \[ \int \frac {x^3 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\frac {x (a+b x)^{1+n}}{b c (1+n) \sqrt {c x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32} \[ \int \frac {x^3 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\frac {x (a+b x)^{n+1}}{b c (n+1) \sqrt {c x^2}} \]
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Rule 15
Rule 32
Rubi steps \begin{align*} \text {integral}& = \frac {x \int (a+b x)^n \, dx}{c \sqrt {c x^2}} \\ & = \frac {x (a+b x)^{1+n}}{b c (1+n) \sqrt {c x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\frac {x^3 (a+b x)^{1+n}}{b (1+n) \left (c x^2\right )^{3/2}} \]
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(\frac {x^{3} \left (b x +a \right )^{1+n}}{b \left (1+n \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(29\) |
risch | \(\frac {x \left (b x +a \right ) \left (b x +a \right )^{n}}{c \sqrt {c \,x^{2}}\, b \left (1+n \right )}\) | \(33\) |
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Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {x^3 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c x^{2}} {\left (b x + a\right )} {\left (b x + a\right )}^{n}}{{\left (b c^{2} n + b c^{2}\right )} x} \]
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\[ \int \frac {x^3 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\begin {cases} \frac {x^{4}}{a \left (c x^{2}\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \wedge n = -1 \\\frac {a^{n} x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \\\int \frac {x^{3}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\\frac {a x^{3} \left (a + b x\right )^{n}}{b n \left (c x^{2}\right )^{\frac {3}{2}} + b \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b x^{4} \left (a + b x\right )^{n}}{b n \left (c x^{2}\right )^{\frac {3}{2}} + b \left (c x^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\frac {{\left (b \sqrt {c} x + a \sqrt {c}\right )} {\left (b x + a\right )}^{n}}{b c^{2} {\left (n + 1\right )}} \]
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Exception generated. \[ \int \frac {x^3 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {x^3 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\frac {\left (\frac {x^2}{c\,\left (n+1\right )}+\frac {a\,x}{b\,c\,\left (n+1\right )}\right )\,{\left (a+b\,x\right )}^n}{\sqrt {c\,x^2}} \]
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