Integrand size = 19, antiderivative size = 73 \[ \int (c x)^m \left (b x^2+c x^4\right )^3 \, dx=\frac {b^3 x^7 (c x)^m}{7+m}+\frac {3 b^2 c x^9 (c x)^m}{9+m}+\frac {3 b c^2 x^{11} (c x)^m}{11+m}+\frac {c^3 x^{13} (c x)^m}{13+m} \]
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Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1156, 1598, 276} \[ \int (c x)^m \left (b x^2+c x^4\right )^3 \, dx=\frac {b^3 x^7 (c x)^m}{m+7}+\frac {3 b^2 c x^9 (c x)^m}{m+9}+\frac {3 b c^2 x^{11} (c x)^m}{m+11}+\frac {c^3 x^{13} (c x)^m}{m+13} \]
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Rule 276
Rule 1156
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \left (x^{-m} (c x)^m\right ) \text {Subst}\left (\int x^m \left (b x^2+c x^4\right )^3 \, dx,x,x\right ) \\ & = \left (x^{-m} (c x)^m\right ) \text {Subst}\left (\int x^{6+m} \left (b+c x^2\right )^3 \, dx,x,x\right ) \\ & = \left (x^{-m} (c x)^m\right ) \text {Subst}\left (\int \left (b^3 x^{6+m}+3 b^2 c x^{8+m}+3 b c^2 x^{10+m}+c^3 x^{12+m}\right ) \, dx,x,x\right ) \\ & = \frac {b^3 x^7 (c x)^m}{7+m}+\frac {3 b^2 c x^9 (c x)^m}{9+m}+\frac {3 b c^2 x^{11} (c x)^m}{11+m}+\frac {c^3 x^{13} (c x)^m}{13+m} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int (c x)^m \left (b x^2+c x^4\right )^3 \, dx=x^7 (c x)^m \left (\frac {b^3}{7+m}+\frac {3 b^2 c x^2}{9+m}+\frac {3 b c^2 x^4}{11+m}+\frac {c^3 x^6}{13+m}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(180\) vs. \(2(73)=146\).
Time = 0.24 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.48
method | result | size |
gosper | \(\frac {\left (c x \right )^{m} \left (c^{3} m^{3} x^{6}+27 c^{3} m^{2} x^{6}+3 b \,c^{2} m^{3} x^{4}+239 m \,x^{6} c^{3}+87 b \,c^{2} m^{2} x^{4}+693 c^{3} x^{6}+3 b^{2} c \,m^{3} x^{2}+813 b \,c^{2} m \,x^{4}+93 b^{2} c \,m^{2} x^{2}+2457 b \,c^{2} x^{4}+b^{3} m^{3}+933 b^{2} c m \,x^{2}+33 b^{3} m^{2}+3003 b^{2} c \,x^{2}+359 b^{3} m +1287 b^{3}\right ) x^{7}}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right )}\) | \(181\) |
risch | \(\frac {\left (c x \right )^{m} \left (c^{3} m^{3} x^{6}+27 c^{3} m^{2} x^{6}+3 b \,c^{2} m^{3} x^{4}+239 m \,x^{6} c^{3}+87 b \,c^{2} m^{2} x^{4}+693 c^{3} x^{6}+3 b^{2} c \,m^{3} x^{2}+813 b \,c^{2} m \,x^{4}+93 b^{2} c \,m^{2} x^{2}+2457 b \,c^{2} x^{4}+b^{3} m^{3}+933 b^{2} c m \,x^{2}+33 b^{3} m^{2}+3003 b^{2} c \,x^{2}+359 b^{3} m +1287 b^{3}\right ) x^{7}}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right )}\) | \(181\) |
parallelrisch | \(\frac {x^{13} \left (c x \right )^{m} c^{3} m^{3}+27 x^{13} \left (c x \right )^{m} c^{3} m^{2}+239 x^{13} \left (c x \right )^{m} c^{3} m +3 x^{11} \left (c x \right )^{m} b \,c^{2} m^{3}+693 x^{13} \left (c x \right )^{m} c^{3}+87 x^{11} \left (c x \right )^{m} b \,c^{2} m^{2}+813 x^{11} \left (c x \right )^{m} b \,c^{2} m +3 x^{9} \left (c x \right )^{m} b^{2} c \,m^{3}+2457 x^{11} \left (c x \right )^{m} b \,c^{2}+93 x^{9} \left (c x \right )^{m} b^{2} c \,m^{2}+933 x^{9} \left (c x \right )^{m} b^{2} c m +x^{7} \left (c x \right )^{m} b^{3} m^{3}+3003 x^{9} \left (c x \right )^{m} b^{2} c +33 x^{7} \left (c x \right )^{m} b^{3} m^{2}+359 x^{7} \left (c x \right )^{m} b^{3} m +1287 x^{7} \left (c x \right )^{m} b^{3}}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right )}\) | \(265\) |
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Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (73) = 146\).
Time = 0.25 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.21 \[ \int (c x)^m \left (b x^2+c x^4\right )^3 \, dx=\frac {{\left ({\left (c^{3} m^{3} + 27 \, c^{3} m^{2} + 239 \, c^{3} m + 693 \, c^{3}\right )} x^{13} + 3 \, {\left (b c^{2} m^{3} + 29 \, b c^{2} m^{2} + 271 \, b c^{2} m + 819 \, b c^{2}\right )} x^{11} + 3 \, {\left (b^{2} c m^{3} + 31 \, b^{2} c m^{2} + 311 \, b^{2} c m + 1001 \, b^{2} c\right )} x^{9} + {\left (b^{3} m^{3} + 33 \, b^{3} m^{2} + 359 \, b^{3} m + 1287 \, b^{3}\right )} x^{7}\right )} \left (c x\right )^{m}}{m^{4} + 40 \, m^{3} + 590 \, m^{2} + 3800 \, m + 9009} \]
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Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (66) = 132\).
Time = 0.68 (sec) , antiderivative size = 731, normalized size of antiderivative = 10.01 \[ \int (c x)^m \left (b x^2+c x^4\right )^3 \, dx=\begin {cases} \frac {- \frac {b^{3}}{6 x^{6}} - \frac {3 b^{2} c}{4 x^{4}} - \frac {3 b c^{2}}{2 x^{2}} + c^{3} \log {\left (x \right )}}{c^{13}} & \text {for}\: m = -13 \\\frac {- \frac {b^{3}}{4 x^{4}} - \frac {3 b^{2} c}{2 x^{2}} + 3 b c^{2} \log {\left (x \right )} + \frac {c^{3} x^{2}}{2}}{c^{11}} & \text {for}\: m = -11 \\\frac {- \frac {b^{3}}{2 x^{2}} + 3 b^{2} c \log {\left (x \right )} + \frac {3 b c^{2} x^{2}}{2} + \frac {c^{3} x^{4}}{4}}{c^{9}} & \text {for}\: m = -9 \\\frac {b^{3} \log {\left (x \right )} + \frac {3 b^{2} c x^{2}}{2} + \frac {3 b c^{2} x^{4}}{4} + \frac {c^{3} x^{6}}{6}}{c^{7}} & \text {for}\: m = -7 \\\frac {b^{3} m^{3} x^{7} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {33 b^{3} m^{2} x^{7} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {359 b^{3} m x^{7} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {1287 b^{3} x^{7} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {3 b^{2} c m^{3} x^{9} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {93 b^{2} c m^{2} x^{9} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {933 b^{2} c m x^{9} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {3003 b^{2} c x^{9} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {3 b c^{2} m^{3} x^{11} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {87 b c^{2} m^{2} x^{11} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {813 b c^{2} m x^{11} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {2457 b c^{2} x^{11} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {c^{3} m^{3} x^{13} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {27 c^{3} m^{2} x^{13} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {239 c^{3} m x^{13} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {693 c^{3} x^{13} \left (c x\right )^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04 \[ \int (c x)^m \left (b x^2+c x^4\right )^3 \, dx=\frac {c^{m + 3} x^{13} x^{m}}{m + 13} + \frac {3 \, b c^{m + 2} x^{11} x^{m}}{m + 11} + \frac {3 \, b^{2} c^{m + 1} x^{9} x^{m}}{m + 9} + \frac {b^{3} c^{m} x^{7} x^{m}}{m + 7} \]
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Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (73) = 146\).
Time = 0.28 (sec) , antiderivative size = 264, normalized size of antiderivative = 3.62 \[ \int (c x)^m \left (b x^2+c x^4\right )^3 \, dx=\frac {\left (c x\right )^{m} c^{3} m^{3} x^{13} + 27 \, \left (c x\right )^{m} c^{3} m^{2} x^{13} + 3 \, \left (c x\right )^{m} b c^{2} m^{3} x^{11} + 239 \, \left (c x\right )^{m} c^{3} m x^{13} + 87 \, \left (c x\right )^{m} b c^{2} m^{2} x^{11} + 693 \, \left (c x\right )^{m} c^{3} x^{13} + 3 \, \left (c x\right )^{m} b^{2} c m^{3} x^{9} + 813 \, \left (c x\right )^{m} b c^{2} m x^{11} + 93 \, \left (c x\right )^{m} b^{2} c m^{2} x^{9} + 2457 \, \left (c x\right )^{m} b c^{2} x^{11} + \left (c x\right )^{m} b^{3} m^{3} x^{7} + 933 \, \left (c x\right )^{m} b^{2} c m x^{9} + 33 \, \left (c x\right )^{m} b^{3} m^{2} x^{7} + 3003 \, \left (c x\right )^{m} b^{2} c x^{9} + 359 \, \left (c x\right )^{m} b^{3} m x^{7} + 1287 \, \left (c x\right )^{m} b^{3} x^{7}}{m^{4} + 40 \, m^{3} + 590 \, m^{2} + 3800 \, m + 9009} \]
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Time = 13.26 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.34 \[ \int (c x)^m \left (b x^2+c x^4\right )^3 \, dx={\left (c\,x\right )}^m\,\left (\frac {b^3\,x^7\,\left (m^3+33\,m^2+359\,m+1287\right )}{m^4+40\,m^3+590\,m^2+3800\,m+9009}+\frac {c^3\,x^{13}\,\left (m^3+27\,m^2+239\,m+693\right )}{m^4+40\,m^3+590\,m^2+3800\,m+9009}+\frac {3\,b\,c^2\,x^{11}\,\left (m^3+29\,m^2+271\,m+819\right )}{m^4+40\,m^3+590\,m^2+3800\,m+9009}+\frac {3\,b^2\,c\,x^9\,\left (m^3+31\,m^2+311\,m+1001\right )}{m^4+40\,m^3+590\,m^2+3800\,m+9009}\right ) \]
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