Integrand size = 11, antiderivative size = 61 \[ \int x^m (a+b x)^3 \, dx=\frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{2+m}}{2+m}+\frac {3 a b^2 x^{3+m}}{3+m}+\frac {b^3 x^{4+m}}{4+m} \]
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Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int x^m (a+b x)^3 \, dx=\frac {a^3 x^{m+1}}{m+1}+\frac {3 a^2 b x^{m+2}}{m+2}+\frac {3 a b^2 x^{m+3}}{m+3}+\frac {b^3 x^{m+4}}{m+4} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 x^m+3 a^2 b x^{1+m}+3 a b^2 x^{2+m}+b^3 x^{3+m}\right ) \, dx \\ & = \frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{2+m}}{2+m}+\frac {3 a b^2 x^{3+m}}{3+m}+\frac {b^3 x^{4+m}}{4+m} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int x^m (a+b x)^3 \, dx=x^{1+m} \left (\frac {a^3}{1+m}+\frac {3 a^2 b x}{2+m}+\frac {3 a b^2 x^2}{3+m}+\frac {b^3 x^3}{4+m}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.18
method | result | size |
norman | \(\frac {a^{3} x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b^{3} x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}+\frac {3 a \,b^{2} x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {3 a^{2} b \,x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}\) | \(72\) |
risch | \(\frac {x \left (b^{3} m^{3} x^{3}+3 a \,b^{2} m^{3} x^{2}+6 b^{3} m^{2} x^{3}+3 a^{2} b \,m^{3} x +21 a \,b^{2} m^{2} x^{2}+11 m \,x^{3} b^{3}+a^{3} m^{3}+24 a^{2} b \,m^{2} x +42 m \,x^{2} a \,b^{2}+6 b^{3} x^{3}+9 a^{3} m^{2}+57 m x \,a^{2} b +24 a \,b^{2} x^{2}+26 m \,a^{3}+36 a^{2} b x +24 a^{3}\right ) x^{m}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(169\) |
gosper | \(\frac {x^{1+m} \left (b^{3} m^{3} x^{3}+3 a \,b^{2} m^{3} x^{2}+6 b^{3} m^{2} x^{3}+3 a^{2} b \,m^{3} x +21 a \,b^{2} m^{2} x^{2}+11 m \,x^{3} b^{3}+a^{3} m^{3}+24 a^{2} b \,m^{2} x +42 m \,x^{2} a \,b^{2}+6 b^{3} x^{3}+9 a^{3} m^{2}+57 m x \,a^{2} b +24 a \,b^{2} x^{2}+26 m \,a^{3}+36 a^{2} b x +24 a^{3}\right )}{\left (1+m \right ) \left (2+m \right ) \left (3+m \right ) \left (4+m \right )}\) | \(170\) |
parallelrisch | \(\frac {x^{4} x^{m} b^{3} m^{3}+6 x^{4} x^{m} b^{3} m^{2}+3 x^{3} x^{m} a \,b^{2} m^{3}+11 x^{4} x^{m} b^{3} m +21 x^{3} x^{m} a \,b^{2} m^{2}+3 x^{2} x^{m} a^{2} b \,m^{3}+6 x^{4} x^{m} b^{3}+42 x^{3} x^{m} a \,b^{2} m +24 x^{2} x^{m} a^{2} b \,m^{2}+x \,x^{m} a^{3} m^{3}+24 x^{3} x^{m} a \,b^{2}+57 x^{2} x^{m} a^{2} b m +9 x \,x^{m} a^{3} m^{2}+36 x^{2} x^{m} a^{2} b +26 x \,x^{m} a^{3} m +24 x \,x^{m} a^{3}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(225\) |
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Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (61) = 122\).
Time = 0.23 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.57 \[ \int x^m (a+b x)^3 \, dx=\frac {{\left ({\left (b^{3} m^{3} + 6 \, b^{3} m^{2} + 11 \, b^{3} m + 6 \, b^{3}\right )} x^{4} + 3 \, {\left (a b^{2} m^{3} + 7 \, a b^{2} m^{2} + 14 \, a b^{2} m + 8 \, a b^{2}\right )} x^{3} + 3 \, {\left (a^{2} b m^{3} + 8 \, a^{2} b m^{2} + 19 \, a^{2} b m + 12 \, a^{2} b\right )} x^{2} + {\left (a^{3} m^{3} + 9 \, a^{3} m^{2} + 26 \, a^{3} m + 24 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]
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Leaf count of result is larger than twice the leaf count of optimal. 663 vs. \(2 (53) = 106\).
Time = 0.39 (sec) , antiderivative size = 663, normalized size of antiderivative = 10.87 \[ \int x^m (a+b x)^3 \, dx=\begin {cases} - \frac {a^{3}}{3 x^{3}} - \frac {3 a^{2} b}{2 x^{2}} - \frac {3 a b^{2}}{x} + b^{3} \log {\left (x \right )} & \text {for}\: m = -4 \\- \frac {a^{3}}{2 x^{2}} - \frac {3 a^{2} b}{x} + 3 a b^{2} \log {\left (x \right )} + b^{3} x & \text {for}\: m = -3 \\- \frac {a^{3}}{x} + 3 a^{2} b \log {\left (x \right )} + 3 a b^{2} x + \frac {b^{3} x^{2}}{2} & \text {for}\: m = -2 \\a^{3} \log {\left (x \right )} + 3 a^{2} b x + \frac {3 a b^{2} x^{2}}{2} + \frac {b^{3} x^{3}}{3} & \text {for}\: m = -1 \\\frac {a^{3} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {9 a^{3} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {26 a^{3} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 a^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {3 a^{2} b m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 a^{2} b m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {57 a^{2} b m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {36 a^{2} b x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {3 a b^{2} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {21 a b^{2} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {42 a b^{2} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 a b^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {b^{3} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 b^{3} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {11 b^{3} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 b^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int x^m (a+b x)^3 \, dx=\frac {b^{3} x^{m + 4}}{m + 4} + \frac {3 \, a b^{2} x^{m + 3}}{m + 3} + \frac {3 \, a^{2} b x^{m + 2}}{m + 2} + \frac {a^{3} x^{m + 1}}{m + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (61) = 122\).
Time = 0.29 (sec) , antiderivative size = 224, normalized size of antiderivative = 3.67 \[ \int x^m (a+b x)^3 \, dx=\frac {b^{3} m^{3} x^{4} x^{m} + 3 \, a b^{2} m^{3} x^{3} x^{m} + 6 \, b^{3} m^{2} x^{4} x^{m} + 3 \, a^{2} b m^{3} x^{2} x^{m} + 21 \, a b^{2} m^{2} x^{3} x^{m} + 11 \, b^{3} m x^{4} x^{m} + a^{3} m^{3} x x^{m} + 24 \, a^{2} b m^{2} x^{2} x^{m} + 42 \, a b^{2} m x^{3} x^{m} + 6 \, b^{3} x^{4} x^{m} + 9 \, a^{3} m^{2} x x^{m} + 57 \, a^{2} b m x^{2} x^{m} + 24 \, a b^{2} x^{3} x^{m} + 26 \, a^{3} m x x^{m} + 36 \, a^{2} b x^{2} x^{m} + 24 \, a^{3} x x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]
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Time = 0.44 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.74 \[ \int x^m (a+b x)^3 \, dx=x^m\,\left (\frac {a^3\,x\,\left (m^3+9\,m^2+26\,m+24\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {b^3\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {3\,a\,b^2\,x^3\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {3\,a^2\,b\,x^2\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}\right ) \]
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