Integrand size = 20, antiderivative size = 43 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x} \, dx=a b c x^2+\frac {1}{4} b^2 c x^4+\frac {d \left (a+b x^2\right )^3}{6 b}+a^2 c \log (x) \]
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Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {457, 81, 45} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x} \, dx=a^2 c \log (x)+a b c x^2+\frac {d \left (a+b x^2\right )^3}{6 b}+\frac {1}{4} b^2 c x^4 \]
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Rule 45
Rule 81
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2 (c+d x)}{x} \, dx,x,x^2\right ) \\ & = \frac {d \left (a+b x^2\right )^3}{6 b}+\frac {1}{2} c \text {Subst}\left (\int \frac {(a+b x)^2}{x} \, dx,x,x^2\right ) \\ & = \frac {d \left (a+b x^2\right )^3}{6 b}+\frac {1}{2} c \text {Subst}\left (\int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx,x,x^2\right ) \\ & = a b c x^2+\frac {1}{4} b^2 c x^4+\frac {d \left (a+b x^2\right )^3}{6 b}+a^2 c \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x} \, dx=\frac {1}{2} a (2 b c+a d) x^2+\frac {1}{4} b (b c+2 a d) x^4+\frac {1}{6} b^2 d x^6+a^2 c \log (x) \]
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Time = 2.58 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14
method | result | size |
norman | \(\left (\frac {1}{2} a^{2} d +a b c \right ) x^{2}+\left (\frac {1}{2} a b d +\frac {1}{4} b^{2} c \right ) x^{4}+\frac {b^{2} d \,x^{6}}{6}+a^{2} c \ln \left (x \right )\) | \(49\) |
default | \(\frac {b^{2} d \,x^{6}}{6}+\frac {a b d \,x^{4}}{2}+\frac {b^{2} c \,x^{4}}{4}+\frac {a^{2} d \,x^{2}}{2}+a b c \,x^{2}+a^{2} c \ln \left (x \right )\) | \(51\) |
risch | \(\frac {b^{2} d \,x^{6}}{6}+\frac {a b d \,x^{4}}{2}+\frac {b^{2} c \,x^{4}}{4}+\frac {a^{2} d \,x^{2}}{2}+a b c \,x^{2}+a^{2} c \ln \left (x \right )\) | \(51\) |
parallelrisch | \(\frac {b^{2} d \,x^{6}}{6}+\frac {a b d \,x^{4}}{2}+\frac {b^{2} c \,x^{4}}{4}+\frac {a^{2} d \,x^{2}}{2}+a b c \,x^{2}+a^{2} c \ln \left (x \right )\) | \(51\) |
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x} \, dx=\frac {1}{6} \, b^{2} d x^{6} + \frac {1}{4} \, {\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c \log \left (x\right ) + \frac {1}{2} \, {\left (2 \, a b c + a^{2} d\right )} x^{2} \]
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Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x} \, dx=a^{2} c \log {\left (x \right )} + \frac {b^{2} d x^{6}}{6} + x^{4} \left (\frac {a b d}{2} + \frac {b^{2} c}{4}\right ) + x^{2} \left (\frac {a^{2} d}{2} + a b c\right ) \]
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Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x} \, dx=\frac {1}{6} \, b^{2} d x^{6} + \frac {1}{4} \, {\left (b^{2} c + 2 \, a b d\right )} x^{4} + \frac {1}{2} \, a^{2} c \log \left (x^{2}\right ) + \frac {1}{2} \, {\left (2 \, a b c + a^{2} d\right )} x^{2} \]
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Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x} \, dx=\frac {1}{6} \, b^{2} d x^{6} + \frac {1}{4} \, b^{2} c x^{4} + \frac {1}{2} \, a b d x^{4} + a b c x^{2} + \frac {1}{2} \, a^{2} d x^{2} + \frac {1}{2} \, a^{2} c \log \left (x^{2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x} \, dx=x^2\,\left (\frac {d\,a^2}{2}+b\,c\,a\right )+x^4\,\left (\frac {c\,b^2}{4}+\frac {a\,d\,b}{2}\right )+\frac {b^2\,d\,x^6}{6}+a^2\,c\,\ln \left (x\right ) \]
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