Integrand size = 17, antiderivative size = 34 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^8} \, dx=-\frac {b^3}{x}+3 b^2 c x+b c^2 x^3+\frac {c^3 x^5}{5} \]
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Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 276} \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^8} \, dx=-\frac {b^3}{x}+3 b^2 c x+b c^2 x^3+\frac {c^3 x^5}{5} \]
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Rule 276
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (b+c x^2\right )^3}{x^2} \, dx \\ & = \int \left (3 b^2 c+\frac {b^3}{x^2}+3 b c^2 x^2+c^3 x^4\right ) \, dx \\ & = -\frac {b^3}{x}+3 b^2 c x+b c^2 x^3+\frac {c^3 x^5}{5} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^8} \, dx=-\frac {b^3}{x}+3 b^2 c x+b c^2 x^3+\frac {c^3 x^5}{5} \]
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Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {b^{3}}{x}+3 b^{2} c x +b \,c^{2} x^{3}+\frac {c^{3} x^{5}}{5}\) | \(33\) |
risch | \(-\frac {b^{3}}{x}+3 b^{2} c x +b \,c^{2} x^{3}+\frac {c^{3} x^{5}}{5}\) | \(33\) |
parallelrisch | \(\frac {c^{3} x^{6}+5 b \,c^{2} x^{4}+15 b^{2} c \,x^{2}-5 b^{3}}{5 x}\) | \(37\) |
gosper | \(-\frac {-c^{3} x^{6}-5 b \,c^{2} x^{4}-15 b^{2} c \,x^{2}+5 b^{3}}{5 x}\) | \(38\) |
norman | \(\frac {b \,c^{2} x^{10}-b^{3} x^{6}+\frac {1}{5} c^{3} x^{12}+3 b^{2} c \,x^{8}}{x^{7}}\) | \(39\) |
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Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^8} \, dx=\frac {c^{3} x^{6} + 5 \, b c^{2} x^{4} + 15 \, b^{2} c x^{2} - 5 \, b^{3}}{5 \, x} \]
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Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^8} \, dx=- \frac {b^{3}}{x} + 3 b^{2} c x + b c^{2} x^{3} + \frac {c^{3} x^{5}}{5} \]
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Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^8} \, dx=\frac {1}{5} \, c^{3} x^{5} + b c^{2} x^{3} + 3 \, b^{2} c x - \frac {b^{3}}{x} \]
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Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^8} \, dx=\frac {1}{5} \, c^{3} x^{5} + b c^{2} x^{3} + 3 \, b^{2} c x - \frac {b^{3}}{x} \]
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Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^8} \, dx=\frac {c^3\,x^5}{5}-\frac {b^3}{x}+b\,c^2\,x^3+3\,b^2\,c\,x \]
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