Integrand size = 15, antiderivative size = 63 \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=-\frac {b^2 c^5 (c x)^{-5+m}}{5-m}-\frac {2 a b c^2 (c x)^{-2+m}}{2-m}+\frac {a^2 (c x)^{1+m}}{c (1+m)} \]
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Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {276} \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=\frac {a^2 (c x)^{m+1}}{c (m+1)}-\frac {2 a b c^2 (c x)^{m-2}}{2-m}-\frac {b^2 c^5 (c x)^{m-5}}{5-m} \]
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Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \left (b^2 c^6 (c x)^{-6+m}+2 a b c^3 (c x)^{-3+m}+a^2 (c x)^m\right ) \, dx \\ & = -\frac {b^2 c^5 (c x)^{-5+m}}{5-m}-\frac {2 a b c^2 (c x)^{-2+m}}{2-m}+\frac {a^2 (c x)^{1+m}}{c (1+m)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.06 \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=\frac {(c x)^m \left (b^2 \left (-2-m+m^2\right )+2 a b \left (-5-4 m+m^2\right ) x^3+a^2 \left (10-7 m+m^2\right ) x^6\right )}{(-5+m) (-2+m) (1+m) x^5} \]
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Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95
method | result | size |
norman | \(\frac {\frac {b^{2} {\mathrm e}^{m \ln \left (c x \right )}}{-5+m}+\frac {a^{2} x^{6} {\mathrm e}^{m \ln \left (c x \right )}}{1+m}+\frac {2 a b \,x^{3} {\mathrm e}^{m \ln \left (c x \right )}}{-2+m}}{x^{5}}\) | \(60\) |
gosper | \(\frac {\left (c x \right )^{m} \left (a^{2} m^{2} x^{6}-7 a^{2} m \,x^{6}+10 a^{2} x^{6}+2 a b \,m^{2} x^{3}-8 m \,x^{3} a b -10 a b \,x^{3}+b^{2} m^{2}-b^{2} m -2 b^{2}\right )}{x^{5} \left (1+m \right ) \left (-2+m \right ) \left (-5+m \right )}\) | \(96\) |
risch | \(\frac {\left (c x \right )^{m} \left (a^{2} m^{2} x^{6}-7 a^{2} m \,x^{6}+10 a^{2} x^{6}+2 a b \,m^{2} x^{3}-8 m \,x^{3} a b -10 a b \,x^{3}+b^{2} m^{2}-b^{2} m -2 b^{2}\right )}{x^{5} \left (1+m \right ) \left (-2+m \right ) \left (-5+m \right )}\) | \(96\) |
parallelrisch | \(\frac {x^{6} \left (c x \right )^{m} a^{2} m^{2}-7 x^{6} \left (c x \right )^{m} a^{2} m +10 x^{6} \left (c x \right )^{m} a^{2}+2 x^{3} \left (c x \right )^{m} a b \,m^{2}-8 x^{3} \left (c x \right )^{m} a b m -10 \left (c x \right )^{m} a b \,x^{3}+\left (c x \right )^{m} b^{2} m^{2}-\left (c x \right )^{m} b^{2} m -2 b^{2} \left (c x \right )^{m}}{x^{5} \left (1+m \right ) \left (-2+m \right ) \left (-5+m \right )}\) | \(136\) |
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Time = 0.43 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.38 \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=\frac {{\left ({\left (a^{2} m^{2} - 7 \, a^{2} m + 10 \, a^{2}\right )} x^{6} + b^{2} m^{2} + 2 \, {\left (a b m^{2} - 4 \, a b m - 5 \, a b\right )} x^{3} - b^{2} m - 2 \, b^{2}\right )} \left (c x\right )^{m}}{{\left (m^{3} - 6 \, m^{2} + 3 \, m + 10\right )} x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (49) = 98\).
Time = 0.39 (sec) , antiderivative size = 449, normalized size of antiderivative = 7.13 \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=\begin {cases} \frac {a^{2} \log {\left (x \right )} - \frac {2 a b}{3 x^{3}} - \frac {b^{2}}{6 x^{6}}}{c} & \text {for}\: m = -1 \\c^{2} \left (\frac {a^{2} x^{3}}{3} + 2 a b \log {\left (x \right )} - \frac {b^{2}}{3 x^{3}}\right ) & \text {for}\: m = 2 \\c^{5} \left (\frac {a^{2} x^{6}}{6} + \frac {2 a b x^{3}}{3} + b^{2} \log {\left (x \right )}\right ) & \text {for}\: m = 5 \\\frac {a^{2} m^{2} x^{6} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {7 a^{2} m x^{6} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} + \frac {10 a^{2} x^{6} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} + \frac {2 a b m^{2} x^{3} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {8 a b m x^{3} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {10 a b x^{3} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} + \frac {b^{2} m^{2} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {b^{2} m \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {2 b^{2} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=\frac {\left (c x\right )^{m + 1} a^{2}}{c {\left (m + 1\right )}} + \frac {2 \, a b c^{m} x^{m}}{{\left (m - 2\right )} x^{2}} + \frac {b^{2} c^{m} x^{m}}{{\left (m - 5\right )} x^{5}} \]
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\[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=\int { \left (c x\right )^{m} {\left (a + \frac {b}{x^{3}}\right )}^{2} \,d x } \]
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Time = 5.72 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.54 \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=-\frac {{\left (c\,x\right )}^m\,\left (-a^2\,m^2\,x^6+7\,a^2\,m\,x^6-10\,a^2\,x^6-2\,a\,b\,m^2\,x^3+8\,a\,b\,m\,x^3+10\,a\,b\,x^3-b^2\,m^2+b^2\,m+2\,b^2\right )}{x^5\,\left (m^3-6\,m^2+3\,m+10\right )} \]
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