Integrand size = 18, antiderivative size = 96 \[ \int x \sqrt {c x^2} (a+b x)^n \, dx=\frac {a^2 \sqrt {c x^2} (a+b x)^{1+n}}{b^3 (1+n) x}-\frac {2 a \sqrt {c x^2} (a+b x)^{2+n}}{b^3 (2+n) x}+\frac {\sqrt {c x^2} (a+b x)^{3+n}}{b^3 (3+n) x} \]
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Time = 0.02 (sec) , antiderivative size = 96, 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} \[ \int x \sqrt {c x^2} (a+b x)^n \, dx=\frac {a^2 \sqrt {c x^2} (a+b x)^{n+1}}{b^3 (n+1) x}-\frac {2 a \sqrt {c x^2} (a+b x)^{n+2}}{b^3 (n+2) x}+\frac {\sqrt {c x^2} (a+b x)^{n+3}}{b^3 (n+3) x} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c x^2} \int x^2 (a+b x)^n \, dx}{x} \\ & = \frac {\sqrt {c x^2} \int \left (\frac {a^2 (a+b x)^n}{b^2}-\frac {2 a (a+b x)^{1+n}}{b^2}+\frac {(a+b x)^{2+n}}{b^2}\right ) \, dx}{x} \\ & = \frac {a^2 \sqrt {c x^2} (a+b x)^{1+n}}{b^3 (1+n) x}-\frac {2 a \sqrt {c x^2} (a+b x)^{2+n}}{b^3 (2+n) x}+\frac {\sqrt {c x^2} (a+b x)^{3+n}}{b^3 (3+n) x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.71 \[ \int x \sqrt {c x^2} (a+b x)^n \, dx=\frac {c x (a+b x)^{1+n} \left (2 a^2-2 a b (1+n) x+b^2 \left (2+3 n+n^2\right ) x^2\right )}{b^3 (1+n) (2+n) (3+n) \sqrt {c x^2}} \]
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Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86
method | result | size |
gosper | \(\frac {\sqrt {c \,x^{2}}\, \left (b x +a \right )^{1+n} \left (b^{2} n^{2} x^{2}+3 b^{2} n \,x^{2}-2 a b n x +2 b^{2} x^{2}-2 a b x +2 a^{2}\right )}{b^{3} x \left (n^{3}+6 n^{2}+11 n +6\right )}\) | \(83\) |
risch | \(\frac {\sqrt {c \,x^{2}}\, \left (b^{3} n^{2} x^{3}+a \,b^{2} n^{2} x^{2}+3 b^{3} n \,x^{3}+a \,b^{2} n \,x^{2}+2 b^{3} x^{3}-2 a^{2} b n x +2 a^{3}\right ) \left (b x +a \right )^{n}}{x \left (2+n \right ) \left (3+n \right ) \left (1+n \right ) b^{3}}\) | \(98\) |
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Time = 0.23 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.10 \[ \int x \sqrt {c x^2} (a+b x)^n \, dx=-\frac {{\left (2 \, a^{2} b n x - {\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{3} - 2 \, a^{3} - {\left (a b^{2} n^{2} + a b^{2} n\right )} x^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}\right )} x} \]
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\[ \int x \sqrt {c x^2} (a+b x)^n \, dx=\begin {cases} \frac {a^{n} x^{2} \sqrt {c x^{2}}}{3} & \text {for}\: b = 0 \\\int \frac {x \sqrt {c x^{2}}}{\left (a + b x\right )^{3}}\, dx & \text {for}\: n = -3 \\\int \frac {x \sqrt {c x^{2}}}{\left (a + b x\right )^{2}}\, dx & \text {for}\: n = -2 \\\int \frac {x \sqrt {c x^{2}}}{a + b x}\, dx & \text {for}\: n = -1 \\\frac {2 a^{3} \sqrt {c x^{2}} \left (a + b x\right )^{n}}{b^{3} n^{3} x + 6 b^{3} n^{2} x + 11 b^{3} n x + 6 b^{3} x} - \frac {2 a^{2} b n x \sqrt {c x^{2}} \left (a + b x\right )^{n}}{b^{3} n^{3} x + 6 b^{3} n^{2} x + 11 b^{3} n x + 6 b^{3} x} + \frac {a b^{2} n^{2} x^{2} \sqrt {c x^{2}} \left (a + b x\right )^{n}}{b^{3} n^{3} x + 6 b^{3} n^{2} x + 11 b^{3} n x + 6 b^{3} x} + \frac {a b^{2} n x^{2} \sqrt {c x^{2}} \left (a + b x\right )^{n}}{b^{3} n^{3} x + 6 b^{3} n^{2} x + 11 b^{3} n x + 6 b^{3} x} + \frac {b^{3} n^{2} x^{3} \sqrt {c x^{2}} \left (a + b x\right )^{n}}{b^{3} n^{3} x + 6 b^{3} n^{2} x + 11 b^{3} n x + 6 b^{3} x} + \frac {3 b^{3} n x^{3} \sqrt {c x^{2}} \left (a + b x\right )^{n}}{b^{3} n^{3} x + 6 b^{3} n^{2} x + 11 b^{3} n x + 6 b^{3} x} + \frac {2 b^{3} x^{3} \sqrt {c x^{2}} \left (a + b x\right )^{n}}{b^{3} n^{3} x + 6 b^{3} n^{2} x + 11 b^{3} n x + 6 b^{3} x} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.83 \[ \int x \sqrt {c x^2} (a+b x)^n \, dx=\frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} \sqrt {c} x^{3} + {\left (n^{2} + n\right )} a b^{2} \sqrt {c} x^{2} - 2 \, a^{2} b \sqrt {c} n x + 2 \, a^{3} \sqrt {c}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (90) = 180\).
Time = 0.30 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.08 \[ \int x \sqrt {c x^2} (a+b x)^n \, dx=-{\left (\frac {2 \, a^{3} a^{n} \mathrm {sgn}\left (x\right )}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} - \frac {{\left (b x + a\right )}^{n} b^{3} n^{2} x^{3} \mathrm {sgn}\left (x\right ) + {\left (b x + a\right )}^{n} a b^{2} n^{2} x^{2} \mathrm {sgn}\left (x\right ) + 3 \, {\left (b x + a\right )}^{n} b^{3} n x^{3} \mathrm {sgn}\left (x\right ) + {\left (b x + a\right )}^{n} a b^{2} n x^{2} \mathrm {sgn}\left (x\right ) + 2 \, {\left (b x + a\right )}^{n} b^{3} x^{3} \mathrm {sgn}\left (x\right ) - 2 \, {\left (b x + a\right )}^{n} a^{2} b n x \mathrm {sgn}\left (x\right ) + 2 \, {\left (b x + a\right )}^{n} a^{3} \mathrm {sgn}\left (x\right )}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}}\right )} \sqrt {c} \]
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Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.48 \[ \int x \sqrt {c x^2} (a+b x)^n \, dx=\frac {{\left (a+b\,x\right )}^n\,\left (\frac {2\,a^3\,\sqrt {c\,x^2}}{b^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {x^3\,\sqrt {c\,x^2}\,\left (n^2+3\,n+2\right )}{n^3+6\,n^2+11\,n+6}-\frac {2\,a^2\,n\,x\,\sqrt {c\,x^2}}{b^2\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {a\,n\,x^2\,\sqrt {c\,x^2}\,\left (n+1\right )}{b\,\left (n^3+6\,n^2+11\,n+6\right )}\right )}{x} \]
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