Integrand size = 18, antiderivative size = 52 \[ \int (d x)^m \left (a+b x^3+c x^6\right ) \, dx=\frac {a (d x)^{1+m}}{d (1+m)}+\frac {b (d x)^{4+m}}{d^4 (4+m)}+\frac {c (d x)^{7+m}}{d^7 (7+m)} \]
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Time = 0.01 (sec) , antiderivative size = 52, 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 = {14} \[ \int (d x)^m \left (a+b x^3+c x^6\right ) \, dx=\frac {a (d x)^{m+1}}{d (m+1)}+\frac {b (d x)^{m+4}}{d^4 (m+4)}+\frac {c (d x)^{m+7}}{d^7 (m+7)} \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \left (a (d x)^m+\frac {b (d x)^{3+m}}{d^3}+\frac {c (d x)^{6+m}}{d^6}\right ) \, dx \\ & = \frac {a (d x)^{1+m}}{d (1+m)}+\frac {b (d x)^{4+m}}{d^4 (4+m)}+\frac {c (d x)^{7+m}}{d^7 (7+m)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.67 \[ \int (d x)^m \left (a+b x^3+c x^6\right ) \, dx=x (d x)^m \left (\frac {a}{1+m}+\frac {b x^3}{4+m}+\frac {c x^6}{7+m}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.98
method | result | size |
norman | \(\frac {a x \,{\mathrm e}^{m \ln \left (d x \right )}}{1+m}+\frac {b \,x^{4} {\mathrm e}^{m \ln \left (d x \right )}}{4+m}+\frac {c \,x^{7} {\mathrm e}^{m \ln \left (d x \right )}}{7+m}\) | \(51\) |
gosper | \(\frac {x \left (c \,m^{2} x^{6}+5 c m \,x^{6}+4 c \,x^{6}+b \,m^{2} x^{3}+8 b m \,x^{3}+7 b \,x^{3}+a \,m^{2}+11 a m +28 a \right ) \left (d x \right )^{m}}{\left (7+m \right ) \left (4+m \right ) \left (1+m \right )}\) | \(78\) |
risch | \(\frac {x \left (c \,m^{2} x^{6}+5 c m \,x^{6}+4 c \,x^{6}+b \,m^{2} x^{3}+8 b m \,x^{3}+7 b \,x^{3}+a \,m^{2}+11 a m +28 a \right ) \left (d x \right )^{m}}{\left (7+m \right ) \left (4+m \right ) \left (1+m \right )}\) | \(78\) |
parallelrisch | \(\frac {x^{7} \left (d x \right )^{m} c \,m^{2}+5 x^{7} \left (d x \right )^{m} c m +4 x^{7} \left (d x \right )^{m} c +x^{4} \left (d x \right )^{m} b \,m^{2}+8 x^{4} \left (d x \right )^{m} b m +7 x^{4} \left (d x \right )^{m} b +x \left (d x \right )^{m} a \,m^{2}+11 x \left (d x \right )^{m} a m +28 x \left (d x \right )^{m} a}{\left (7+m \right ) \left (4+m \right ) \left (1+m \right )}\) | \(120\) |
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none
Time = 0.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.37 \[ \int (d x)^m \left (a+b x^3+c x^6\right ) \, dx=\frac {{\left ({\left (c m^{2} + 5 \, c m + 4 \, c\right )} x^{7} + {\left (b m^{2} + 8 \, b m + 7 \, b\right )} x^{4} + {\left (a m^{2} + 11 \, a m + 28 \, a\right )} x\right )} \left (d x\right )^{m}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \]
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Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (42) = 84\).
Time = 0.37 (sec) , antiderivative size = 299, normalized size of antiderivative = 5.75 \[ \int (d x)^m \left (a+b x^3+c x^6\right ) \, dx=\begin {cases} \frac {- \frac {a}{6 x^{6}} - \frac {b}{3 x^{3}} + c \log {\left (x \right )}}{d^{7}} & \text {for}\: m = -7 \\\frac {- \frac {a}{3 x^{3}} + b \log {\left (x \right )} + \frac {c x^{3}}{3}}{d^{4}} & \text {for}\: m = -4 \\\frac {a \log {\left (x \right )} + \frac {b x^{3}}{3} + \frac {c x^{6}}{6}}{d} & \text {for}\: m = -1 \\\frac {a m^{2} x \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {11 a m x \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {28 a x \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {b m^{2} x^{4} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {8 b m x^{4} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {7 b x^{4} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {c m^{2} x^{7} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {5 c m x^{7} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {4 c x^{7} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96 \[ \int (d x)^m \left (a+b x^3+c x^6\right ) \, dx=\frac {c d^{m} x^{7} x^{m}}{m + 7} + \frac {b d^{m} x^{4} x^{m}}{m + 4} + \frac {\left (d x\right )^{m + 1} a}{d {\left (m + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (52) = 104\).
Time = 0.34 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.29 \[ \int (d x)^m \left (a+b x^3+c x^6\right ) \, dx=\frac {\left (d x\right )^{m} c m^{2} x^{7} + 5 \, \left (d x\right )^{m} c m x^{7} + 4 \, \left (d x\right )^{m} c x^{7} + \left (d x\right )^{m} b m^{2} x^{4} + 8 \, \left (d x\right )^{m} b m x^{4} + 7 \, \left (d x\right )^{m} b x^{4} + \left (d x\right )^{m} a m^{2} x + 11 \, \left (d x\right )^{m} a m x + 28 \, \left (d x\right )^{m} a x}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \]
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Time = 8.35 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.71 \[ \int (d x)^m \left (a+b x^3+c x^6\right ) \, dx={\left (d\,x\right )}^m\,\left (\frac {b\,x^4\,\left (m^2+8\,m+7\right )}{m^3+12\,m^2+39\,m+28}+\frac {c\,x^7\,\left (m^2+5\,m+4\right )}{m^3+12\,m^2+39\,m+28}+\frac {a\,x\,\left (m^2+11\,m+28\right )}{m^3+12\,m^2+39\,m+28}\right ) \]
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