Integrand size = 17, antiderivative size = 39 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^7} \, dx=\frac {3}{2} b^2 c x^2+\frac {3}{4} b c^2 x^4+\frac {c^3 x^6}{6}+b^3 \log (x) \]
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Time = 0.02 (sec) , antiderivative size = 39, 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} \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^7} \, dx=b^3 \log (x)+\frac {3}{2} b^2 c x^2+\frac {3}{4} b c^2 x^4+\frac {c^3 x^6}{6} \]
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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} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {(b+c x)^3}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (3 b^2 c+\frac {b^3}{x}+3 b c^2 x+c^3 x^2\right ) \, dx,x,x^2\right ) \\ & = \frac {3}{2} b^2 c x^2+\frac {3}{4} b c^2 x^4+\frac {c^3 x^6}{6}+b^3 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^7} \, dx=\frac {3}{2} b^2 c x^2+\frac {3}{4} b c^2 x^4+\frac {c^3 x^6}{6}+b^3 \log (x) \]
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Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {3 b^{2} c \,x^{2}}{2}+\frac {3 b \,c^{2} x^{4}}{4}+\frac {c^{3} x^{6}}{6}+b^{3} \ln \left (x \right )\) | \(34\) |
risch | \(\frac {3 b^{2} c \,x^{2}}{2}+\frac {3 b \,c^{2} x^{4}}{4}+\frac {c^{3} x^{6}}{6}+b^{3} \ln \left (x \right )\) | \(34\) |
parallelrisch | \(\frac {3 b^{2} c \,x^{2}}{2}+\frac {3 b \,c^{2} x^{4}}{4}+\frac {c^{3} x^{6}}{6}+b^{3} \ln \left (x \right )\) | \(34\) |
norman | \(\frac {\frac {1}{6} c^{3} x^{12}+\frac {3}{4} b \,c^{2} x^{10}+\frac {3}{2} b^{2} c \,x^{8}}{x^{6}}+b^{3} \ln \left (x \right )\) | \(39\) |
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Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^7} \, dx=\frac {1}{6} \, c^{3} x^{6} + \frac {3}{4} \, b c^{2} x^{4} + \frac {3}{2} \, b^{2} c x^{2} + b^{3} \log \left (x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^7} \, dx=b^{3} \log {\left (x \right )} + \frac {3 b^{2} c x^{2}}{2} + \frac {3 b c^{2} x^{4}}{4} + \frac {c^{3} x^{6}}{6} \]
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Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^7} \, dx=\frac {1}{6} \, c^{3} x^{6} + \frac {3}{4} \, b c^{2} x^{4} + \frac {3}{2} \, b^{2} c x^{2} + \frac {1}{2} \, b^{3} \log \left (x^{2}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^7} \, dx=\frac {1}{6} \, c^{3} x^{6} + \frac {3}{4} \, b c^{2} x^{4} + \frac {3}{2} \, b^{2} c x^{2} + \frac {1}{2} \, b^{3} \log \left (x^{2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^7} \, dx=b^3\,\ln \left (x\right )+\frac {c^3\,x^6}{6}+\frac {3\,b^2\,c\,x^2}{2}+\frac {3\,b\,c^2\,x^4}{4} \]
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