Integrand size = 17, antiderivative size = 40 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{11}} \, dx=-\frac {b^3}{4 x^4}-\frac {3 b^2 c}{2 x^2}+\frac {c^3 x^2}{2}+3 b c^2 \log (x) \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1598, 272, 45} \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{11}} \, dx=-\frac {b^3}{4 x^4}-\frac {3 b^2 c}{2 x^2}+3 b c^2 \log (x)+\frac {c^3 x^2}{2} \]
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Rule 45
Rule 272
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (b+c x^2\right )^3}{x^5} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {(b+c x)^3}{x^3} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (c^3+\frac {b^3}{x^3}+\frac {3 b^2 c}{x^2}+\frac {3 b c^2}{x}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b^3}{4 x^4}-\frac {3 b^2 c}{2 x^2}+\frac {c^3 x^2}{2}+3 b c^2 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{11}} \, dx=-\frac {b^3}{4 x^4}-\frac {3 b^2 c}{2 x^2}+\frac {c^3 x^2}{2}+3 b c^2 \log (x) \]
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Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {b^{3}}{4 x^{4}}-\frac {3 b^{2} c}{2 x^{2}}+\frac {c^{3} x^{2}}{2}+3 b \,c^{2} \ln \left (x \right )\) | \(35\) |
risch | \(\frac {c^{3} x^{2}}{2}+\frac {-\frac {3}{2} b^{2} c \,x^{2}-\frac {1}{4} b^{3}}{x^{4}}+3 b \,c^{2} \ln \left (x \right )\) | \(37\) |
norman | \(\frac {-\frac {1}{4} b^{3} x^{6}+\frac {1}{2} c^{3} x^{12}-\frac {3}{2} b^{2} c \,x^{8}}{x^{10}}+3 b \,c^{2} \ln \left (x \right )\) | \(40\) |
parallelrisch | \(\frac {2 c^{3} x^{6}+12 b \,c^{2} \ln \left (x \right ) x^{4}-6 b^{2} c \,x^{2}-b^{3}}{4 x^{4}}\) | \(40\) |
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Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{11}} \, dx=\frac {2 \, c^{3} x^{6} + 12 \, b c^{2} x^{4} \log \left (x\right ) - 6 \, b^{2} c x^{2} - b^{3}}{4 \, x^{4}} \]
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Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{11}} \, dx=3 b c^{2} \log {\left (x \right )} + \frac {c^{3} x^{2}}{2} + \frac {- b^{3} - 6 b^{2} c x^{2}}{4 x^{4}} \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{11}} \, dx=\frac {1}{2} \, c^{3} x^{2} + \frac {3}{2} \, b c^{2} \log \left (x^{2}\right ) - \frac {6 \, b^{2} c x^{2} + b^{3}}{4 \, x^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{11}} \, dx=\frac {1}{2} \, c^{3} x^{2} + \frac {3}{2} \, b c^{2} \log \left (x^{2}\right ) - \frac {9 \, b c^{2} x^{4} + 6 \, b^{2} c x^{2} + b^{3}}{4 \, x^{4}} \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{11}} \, dx=\frac {c^3\,x^2}{2}-\frac {\frac {b^3}{4}+\frac {3\,c\,b^2\,x^2}{2}}{x^4}+3\,b\,c^2\,\ln \left (x\right ) \]
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