Integrand size = 13, antiderivative size = 61 \[ \int x^m \left (a+b x^4\right )^3 \, dx=\frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{5+m}}{5+m}+\frac {3 a b^2 x^{9+m}}{9+m}+\frac {b^3 x^{13+m}}{13+m} \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \[ \int x^m \left (a+b x^4\right )^3 \, dx=\frac {a^3 x^{m+1}}{m+1}+\frac {3 a^2 b x^{m+5}}{m+5}+\frac {3 a b^2 x^{m+9}}{m+9}+\frac {b^3 x^{m+13}}{m+13} \]
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Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 x^m+3 a^2 b x^{4+m}+3 a b^2 x^{8+m}+b^3 x^{12+m}\right ) \, dx \\ & = \frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{5+m}}{5+m}+\frac {3 a b^2 x^{9+m}}{9+m}+\frac {b^3 x^{13+m}}{13+m} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int x^m \left (a+b x^4\right )^3 \, dx=x^{1+m} \left (\frac {a^3}{1+m}+\frac {3 a^2 b x^4}{5+m}+\frac {3 a b^2 x^8}{9+m}+\frac {b^3 x^{12}}{13+m}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(176\) vs. \(2(61)=122\).
Time = 4.06 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.90
method | result | size |
risch | \(\frac {x \left (b^{3} m^{3} x^{12}+15 b^{3} m^{2} x^{12}+59 m \,x^{12} b^{3}+45 b^{3} x^{12}+3 a \,b^{2} m^{3} x^{8}+57 a \,b^{2} m^{2} x^{8}+249 m \,x^{8} a \,b^{2}+195 a \,b^{2} x^{8}+3 a^{2} b \,m^{3} x^{4}+69 a^{2} b \,m^{2} x^{4}+417 m \,x^{4} a^{2} b +351 a^{2} b \,x^{4}+a^{3} m^{3}+27 a^{3} m^{2}+227 m \,a^{3}+585 a^{3}\right ) x^{m}}{\left (13+m \right ) \left (9+m \right ) \left (5+m \right ) \left (1+m \right )}\) | \(177\) |
gosper | \(\frac {x^{1+m} \left (b^{3} m^{3} x^{12}+15 b^{3} m^{2} x^{12}+59 m \,x^{12} b^{3}+45 b^{3} x^{12}+3 a \,b^{2} m^{3} x^{8}+57 a \,b^{2} m^{2} x^{8}+249 m \,x^{8} a \,b^{2}+195 a \,b^{2} x^{8}+3 a^{2} b \,m^{3} x^{4}+69 a^{2} b \,m^{2} x^{4}+417 m \,x^{4} a^{2} b +351 a^{2} b \,x^{4}+a^{3} m^{3}+27 a^{3} m^{2}+227 m \,a^{3}+585 a^{3}\right )}{\left (1+m \right ) \left (5+m \right ) \left (9+m \right ) \left (13+m \right )}\) | \(178\) |
parallelrisch | \(\frac {x^{13} x^{m} b^{3} m^{3}+15 x^{13} x^{m} b^{3} m^{2}+59 x^{13} x^{m} b^{3} m +45 x^{13} x^{m} b^{3}+3 x^{9} x^{m} a \,b^{2} m^{3}+57 x^{9} x^{m} a \,b^{2} m^{2}+249 x^{9} x^{m} a \,b^{2} m +195 x^{9} x^{m} a \,b^{2}+3 x^{5} x^{m} a^{2} b \,m^{3}+69 x^{5} x^{m} a^{2} b \,m^{2}+417 x^{5} x^{m} a^{2} b m +351 x^{5} x^{m} a^{2} b +x \,x^{m} a^{3} m^{3}+27 x \,x^{m} a^{3} m^{2}+227 x \,x^{m} a^{3} m +585 x \,x^{m} a^{3}}{\left (13+m \right ) \left (9+m \right ) \left (5+m \right ) \left (1+m \right )}\) | \(225\) |
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Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (61) = 122\).
Time = 0.29 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.57 \[ \int x^m \left (a+b x^4\right )^3 \, dx=\frac {{\left ({\left (b^{3} m^{3} + 15 \, b^{3} m^{2} + 59 \, b^{3} m + 45 \, b^{3}\right )} x^{13} + 3 \, {\left (a b^{2} m^{3} + 19 \, a b^{2} m^{2} + 83 \, a b^{2} m + 65 \, a b^{2}\right )} x^{9} + 3 \, {\left (a^{2} b m^{3} + 23 \, a^{2} b m^{2} + 139 \, a^{2} b m + 117 \, a^{2} b\right )} x^{5} + {\left (a^{3} m^{3} + 27 \, a^{3} m^{2} + 227 \, a^{3} m + 585 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 28 \, m^{3} + 254 \, m^{2} + 812 \, m + 585} \]
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Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (53) = 106\).
Time = 0.72 (sec) , antiderivative size = 683, normalized size of antiderivative = 11.20 \[ \int x^m \left (a+b x^4\right )^3 \, dx=\begin {cases} - \frac {a^{3}}{12 x^{12}} - \frac {3 a^{2} b}{8 x^{8}} - \frac {3 a b^{2}}{4 x^{4}} + b^{3} \log {\left (x \right )} & \text {for}\: m = -13 \\- \frac {a^{3}}{8 x^{8}} - \frac {3 a^{2} b}{4 x^{4}} + 3 a b^{2} \log {\left (x \right )} + \frac {b^{3} x^{4}}{4} & \text {for}\: m = -9 \\- \frac {a^{3}}{4 x^{4}} + 3 a^{2} b \log {\left (x \right )} + \frac {3 a b^{2} x^{4}}{4} + \frac {b^{3} x^{8}}{8} & \text {for}\: m = -5 \\a^{3} \log {\left (x \right )} + \frac {3 a^{2} b x^{4}}{4} + \frac {3 a b^{2} x^{8}}{8} + \frac {b^{3} x^{12}}{12} & \text {for}\: m = -1 \\\frac {a^{3} m^{3} x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {27 a^{3} m^{2} x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {227 a^{3} m x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {585 a^{3} x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {3 a^{2} b m^{3} x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {69 a^{2} b m^{2} x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {417 a^{2} b m x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {351 a^{2} b x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {3 a b^{2} m^{3} x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {57 a b^{2} m^{2} x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {249 a b^{2} m x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {195 a b^{2} x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {b^{3} m^{3} x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {15 b^{3} m^{2} x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {59 b^{3} m x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {45 b^{3} x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int x^m \left (a+b x^4\right )^3 \, dx=\frac {b^{3} x^{m + 13}}{m + 13} + \frac {3 \, a b^{2} x^{m + 9}}{m + 9} + \frac {3 \, a^{2} b x^{m + 5}}{m + 5} + \frac {a^{3} x^{m + 1}}{m + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (61) = 122\).
Time = 0.28 (sec) , antiderivative size = 224, normalized size of antiderivative = 3.67 \[ \int x^m \left (a+b x^4\right )^3 \, dx=\frac {b^{3} m^{3} x^{13} x^{m} + 15 \, b^{3} m^{2} x^{13} x^{m} + 59 \, b^{3} m x^{13} x^{m} + 45 \, b^{3} x^{13} x^{m} + 3 \, a b^{2} m^{3} x^{9} x^{m} + 57 \, a b^{2} m^{2} x^{9} x^{m} + 249 \, a b^{2} m x^{9} x^{m} + 195 \, a b^{2} x^{9} x^{m} + 3 \, a^{2} b m^{3} x^{5} x^{m} + 69 \, a^{2} b m^{2} x^{5} x^{m} + 417 \, a^{2} b m x^{5} x^{m} + 351 \, a^{2} b x^{5} x^{m} + a^{3} m^{3} x x^{m} + 27 \, a^{3} m^{2} x x^{m} + 227 \, a^{3} m x x^{m} + 585 \, a^{3} x x^{m}}{m^{4} + 28 \, m^{3} + 254 \, m^{2} + 812 \, m + 585} \]
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Time = 5.92 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.74 \[ \int x^m \left (a+b x^4\right )^3 \, dx=x^m\,\left (\frac {a^3\,x\,\left (m^3+27\,m^2+227\,m+585\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}+\frac {b^3\,x^{13}\,\left (m^3+15\,m^2+59\,m+45\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}+\frac {3\,a\,b^2\,x^9\,\left (m^3+19\,m^2+83\,m+65\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}+\frac {3\,a^2\,b\,x^5\,\left (m^3+23\,m^2+139\,m+117\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}\right ) \]
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