Integrand size = 17, antiderivative size = 81 \[ \int (d x)^m \left (b x+c x^2\right )^3 \, dx=\frac {b^3 (d x)^{4+m}}{d^4 (4+m)}+\frac {3 b^2 c (d x)^{5+m}}{d^5 (5+m)}+\frac {3 b c^2 (d x)^{6+m}}{d^6 (6+m)}+\frac {c^3 (d x)^{7+m}}{d^7 (7+m)} \]
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Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {661, 45} \[ \int (d x)^m \left (b x+c x^2\right )^3 \, dx=\frac {b^3 (d x)^{m+4}}{d^4 (m+4)}+\frac {3 b^2 c (d x)^{m+5}}{d^5 (m+5)}+\frac {3 b c^2 (d x)^{m+6}}{d^6 (m+6)}+\frac {c^3 (d x)^{m+7}}{d^7 (m+7)} \]
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Rule 45
Rule 661
Rubi steps \begin{align*} \text {integral}& = \frac {\int (d x)^{3+m} (b+c x)^3 \, dx}{d^3} \\ & = \frac {\int \left (b^3 (d x)^{3+m}+\frac {3 b^2 c (d x)^{4+m}}{d}+\frac {3 b c^2 (d x)^{5+m}}{d^2}+\frac {c^3 (d x)^{6+m}}{d^3}\right ) \, dx}{d^3} \\ & = \frac {b^3 (d x)^{4+m}}{d^4 (4+m)}+\frac {3 b^2 c (d x)^{5+m}}{d^5 (5+m)}+\frac {3 b c^2 (d x)^{6+m}}{d^6 (6+m)}+\frac {c^3 (d x)^{7+m}}{d^7 (7+m)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.70 \[ \int (d x)^m \left (b x+c x^2\right )^3 \, dx=x^4 (d x)^m \left (\frac {b^3}{4+m}+\frac {3 b^2 c x}{5+m}+\frac {3 b c^2 x^2}{6+m}+\frac {c^3 x^3}{7+m}\right ) \]
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Time = 2.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.01
method | result | size |
norman | \(\frac {b^{3} x^{4} {\mathrm e}^{m \ln \left (d x \right )}}{4+m}+\frac {c^{3} x^{7} {\mathrm e}^{m \ln \left (d x \right )}}{7+m}+\frac {3 b \,c^{2} x^{6} {\mathrm e}^{m \ln \left (d x \right )}}{6+m}+\frac {3 b^{2} c \,x^{5} {\mathrm e}^{m \ln \left (d x \right )}}{5+m}\) | \(82\) |
gosper | \(\frac {\left (d x \right )^{m} \left (c^{3} m^{3} x^{3}+3 b \,c^{2} m^{3} x^{2}+15 c^{3} m^{2} x^{3}+3 b^{2} c \,m^{3} x +48 b \,c^{2} m^{2} x^{2}+74 m \,x^{3} c^{3}+b^{3} m^{3}+51 b^{2} c \,m^{2} x +249 b \,c^{2} m \,x^{2}+120 c^{3} x^{3}+18 b^{3} m^{2}+282 b^{2} c m x +420 b \,c^{2} x^{2}+107 b^{3} m +504 b^{2} c x +210 b^{3}\right ) x^{4}}{\left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right )}\) | \(173\) |
risch | \(\frac {\left (d x \right )^{m} \left (c^{3} m^{3} x^{3}+3 b \,c^{2} m^{3} x^{2}+15 c^{3} m^{2} x^{3}+3 b^{2} c \,m^{3} x +48 b \,c^{2} m^{2} x^{2}+74 m \,x^{3} c^{3}+b^{3} m^{3}+51 b^{2} c \,m^{2} x +249 b \,c^{2} m \,x^{2}+120 c^{3} x^{3}+18 b^{3} m^{2}+282 b^{2} c m x +420 b \,c^{2} x^{2}+107 b^{3} m +504 b^{2} c x +210 b^{3}\right ) x^{4}}{\left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right )}\) | \(173\) |
parallelrisch | \(\frac {x^{7} \left (d x \right )^{m} c^{3} m^{3}+15 x^{7} \left (d x \right )^{m} c^{3} m^{2}+3 x^{6} \left (d x \right )^{m} b \,c^{2} m^{3}+74 x^{7} \left (d x \right )^{m} c^{3} m +48 x^{6} \left (d x \right )^{m} b \,c^{2} m^{2}+3 x^{5} \left (d x \right )^{m} b^{2} c \,m^{3}+120 x^{7} \left (d x \right )^{m} c^{3}+249 x^{6} \left (d x \right )^{m} b \,c^{2} m +51 x^{5} \left (d x \right )^{m} b^{2} c \,m^{2}+x^{4} \left (d x \right )^{m} b^{3} m^{3}+420 x^{6} \left (d x \right )^{m} b \,c^{2}+282 x^{5} \left (d x \right )^{m} b^{2} c m +18 x^{4} \left (d x \right )^{m} b^{3} m^{2}+504 x^{5} \left (d x \right )^{m} b^{2} c +107 x^{4} \left (d x \right )^{m} b^{3} m +210 x^{4} \left (d x \right )^{m} b^{3}}{\left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right )}\) | \(265\) |
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Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.99 \[ \int (d x)^m \left (b x+c x^2\right )^3 \, dx=\frac {{\left ({\left (c^{3} m^{3} + 15 \, c^{3} m^{2} + 74 \, c^{3} m + 120 \, c^{3}\right )} x^{7} + 3 \, {\left (b c^{2} m^{3} + 16 \, b c^{2} m^{2} + 83 \, b c^{2} m + 140 \, b c^{2}\right )} x^{6} + 3 \, {\left (b^{2} c m^{3} + 17 \, b^{2} c m^{2} + 94 \, b^{2} c m + 168 \, b^{2} c\right )} x^{5} + {\left (b^{3} m^{3} + 18 \, b^{3} m^{2} + 107 \, b^{3} m + 210 \, b^{3}\right )} x^{4}\right )} \left (d x\right )^{m}}{m^{4} + 22 \, m^{3} + 179 \, m^{2} + 638 \, m + 840} \]
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Leaf count of result is larger than twice the leaf count of optimal. 711 vs. \(2 (73) = 146\).
Time = 0.43 (sec) , antiderivative size = 711, normalized size of antiderivative = 8.78 \[ \int (d x)^m \left (b x+c x^2\right )^3 \, dx=\begin {cases} \frac {- \frac {b^{3}}{3 x^{3}} - \frac {3 b^{2} c}{2 x^{2}} - \frac {3 b c^{2}}{x} + c^{3} \log {\left (x \right )}}{d^{7}} & \text {for}\: m = -7 \\\frac {- \frac {b^{3}}{2 x^{2}} - \frac {3 b^{2} c}{x} + 3 b c^{2} \log {\left (x \right )} + c^{3} x}{d^{6}} & \text {for}\: m = -6 \\\frac {- \frac {b^{3}}{x} + 3 b^{2} c \log {\left (x \right )} + 3 b c^{2} x + \frac {c^{3} x^{2}}{2}}{d^{5}} & \text {for}\: m = -5 \\\frac {b^{3} \log {\left (x \right )} + 3 b^{2} c x + \frac {3 b c^{2} x^{2}}{2} + \frac {c^{3} x^{3}}{3}}{d^{4}} & \text {for}\: m = -4 \\\frac {b^{3} m^{3} x^{4} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {18 b^{3} m^{2} x^{4} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {107 b^{3} m x^{4} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {210 b^{3} x^{4} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {3 b^{2} c m^{3} x^{5} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {51 b^{2} c m^{2} x^{5} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {282 b^{2} c m x^{5} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {504 b^{2} c x^{5} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {3 b c^{2} m^{3} x^{6} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {48 b c^{2} m^{2} x^{6} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {249 b c^{2} m x^{6} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {420 b c^{2} x^{6} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {c^{3} m^{3} x^{7} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {15 c^{3} m^{2} x^{7} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {74 c^{3} m x^{7} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {120 c^{3} x^{7} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int (d x)^m \left (b x+c x^2\right )^3 \, dx=\frac {c^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, b c^{2} d^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, b^{2} c d^{m} x^{5} x^{m}}{m + 5} + \frac {b^{3} d^{m} x^{4} x^{m}}{m + 4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (81) = 162\).
Time = 0.27 (sec) , antiderivative size = 264, normalized size of antiderivative = 3.26 \[ \int (d x)^m \left (b x+c x^2\right )^3 \, dx=\frac {\left (d x\right )^{m} c^{3} m^{3} x^{7} + 3 \, \left (d x\right )^{m} b c^{2} m^{3} x^{6} + 15 \, \left (d x\right )^{m} c^{3} m^{2} x^{7} + 3 \, \left (d x\right )^{m} b^{2} c m^{3} x^{5} + 48 \, \left (d x\right )^{m} b c^{2} m^{2} x^{6} + 74 \, \left (d x\right )^{m} c^{3} m x^{7} + \left (d x\right )^{m} b^{3} m^{3} x^{4} + 51 \, \left (d x\right )^{m} b^{2} c m^{2} x^{5} + 249 \, \left (d x\right )^{m} b c^{2} m x^{6} + 120 \, \left (d x\right )^{m} c^{3} x^{7} + 18 \, \left (d x\right )^{m} b^{3} m^{2} x^{4} + 282 \, \left (d x\right )^{m} b^{2} c m x^{5} + 420 \, \left (d x\right )^{m} b c^{2} x^{6} + 107 \, \left (d x\right )^{m} b^{3} m x^{4} + 504 \, \left (d x\right )^{m} b^{2} c x^{5} + 210 \, \left (d x\right )^{m} b^{3} x^{4}}{m^{4} + 22 \, m^{3} + 179 \, m^{2} + 638 \, m + 840} \]
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Time = 9.39 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.11 \[ \int (d x)^m \left (b x+c x^2\right )^3 \, dx={\left (d\,x\right )}^m\,\left (\frac {b^3\,x^4\,\left (m^3+18\,m^2+107\,m+210\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {c^3\,x^7\,\left (m^3+15\,m^2+74\,m+120\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {3\,b\,c^2\,x^6\,\left (m^3+16\,m^2+83\,m+140\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {3\,b^2\,c\,x^5\,\left (m^3+17\,m^2+94\,m+168\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}\right ) \]
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