Integrand size = 18, antiderivative size = 45 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^5} \, dx=-\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b c x^2+\frac {c^2 x^4}{4}+\left (b^2+2 a c\right ) \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1128, 712} \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^5} \, dx=-\frac {a^2}{4 x^4}+\log (x) \left (2 a c+b^2\right )-\frac {a b}{x^2}+b c x^2+\frac {c^2 x^4}{4} \]
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Rule 712
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^2}{x^3} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (2 b c+\frac {a^2}{x^3}+\frac {2 a b}{x^2}+\frac {b^2+2 a c}{x}+c^2 x\right ) \, dx,x,x^2\right ) \\ & = -\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b c x^2+\frac {c^2 x^4}{4}+\left (b^2+2 a c\right ) \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^5} \, dx=\frac {\left (-a+c x^4\right ) \left (a+4 b x^2+c x^4\right )}{4 x^4}+\left (b^2+2 a c\right ) \log (x) \]
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Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {a^{2}}{4 x^{4}}-\frac {a b}{x^{2}}+b c \,x^{2}+\frac {c^{2} x^{4}}{4}+\left (2 a c +b^{2}\right ) \ln \left (x \right )\) | \(42\) |
norman | \(\frac {b c \,x^{6}-\frac {1}{4} a^{2}+\frac {1}{4} c^{2} x^{8}-a b \,x^{2}}{x^{4}}+\left (2 a c +b^{2}\right ) \ln \left (x \right )\) | \(44\) |
risch | \(\frac {c^{2} x^{4}}{4}+b c \,x^{2}+b^{2}+\frac {-\frac {1}{4} a^{2}-a b \,x^{2}}{x^{4}}+2 \ln \left (x \right ) a c +b^{2} \ln \left (x \right )\) | \(48\) |
parallelrisch | \(\frac {c^{2} x^{8}+4 b c \,x^{6}+8 \ln \left (x \right ) x^{4} a c +4 b^{2} \ln \left (x \right ) x^{4}-4 a b \,x^{2}-a^{2}}{4 x^{4}}\) | \(52\) |
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Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^5} \, dx=\frac {c^{2} x^{8} + 4 \, b c x^{6} + 4 \, {\left (b^{2} + 2 \, a c\right )} x^{4} \log \left (x\right ) - 4 \, a b x^{2} - a^{2}}{4 \, x^{4}} \]
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Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^5} \, dx=b c x^{2} + \frac {c^{2} x^{4}}{4} + \left (2 a c + b^{2}\right ) \log {\left (x \right )} + \frac {- a^{2} - 4 a b x^{2}}{4 x^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^5} \, dx=\frac {1}{4} \, c^{2} x^{4} + b c x^{2} + \frac {1}{2} \, {\left (b^{2} + 2 \, a c\right )} \log \left (x^{2}\right ) - \frac {4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^5} \, dx=\frac {1}{4} \, c^{2} x^{4} + b c x^{2} + \frac {1}{2} \, {\left (b^{2} + 2 \, a c\right )} \log \left (x^{2}\right ) - \frac {3 \, b^{2} x^{4} + 6 \, a c x^{4} + 4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \]
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Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{x^5} \, dx=\ln \left (x\right )\,\left (b^2+2\,a\,c\right )-\frac {\frac {a^2}{4}+b\,a\,x^2}{x^4}+\frac {c^2\,x^4}{4}+b\,c\,x^2 \]
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