Integrand size = 18, antiderivative size = 35 \[ \int \frac {x \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {d x^2}{2 b}+\frac {(b c-a d) \log \left (a+b x^2\right )}{2 b^2} \]
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Time = 0.03 (sec) , antiderivative size = 35, 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 = {455, 45} \[ \int \frac {x \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {(b c-a d) \log \left (a+b x^2\right )}{2 b^2}+\frac {d x^2}{2 b} \]
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Rule 45
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {c+d x}{a+b x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {d x^2}{2 b}+\frac {(b c-a d) \log \left (a+b x^2\right )}{2 b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {x \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {b d x^2+(b c-a d) \log \left (a+b x^2\right )}{2 b^2} \]
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Time = 2.58 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {d \,x^{2}}{2 b}+\frac {\left (-a d +b c \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(32\) |
norman | \(\frac {d \,x^{2}}{2 b}-\frac {\left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(32\) |
parallelrisch | \(-\frac {-b d \,x^{2}+\ln \left (b \,x^{2}+a \right ) a d -\ln \left (b \,x^{2}+a \right ) b c}{2 b^{2}}\) | \(37\) |
risch | \(\frac {d \,x^{2}}{2 b}-\frac {\ln \left (b \,x^{2}+a \right ) a d}{2 b^{2}}+\frac {c \ln \left (b \,x^{2}+a \right )}{2 b}\) | \(40\) |
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none
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {x \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {b d x^{2} + {\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]
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Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {x \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {d x^{2}}{2 b} - \frac {\left (a d - b c\right ) \log {\left (a + b x^{2} \right )}}{2 b^{2}} \]
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none
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {x \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {d x^{2}}{2 \, b} + \frac {{\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]
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none
Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {x \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {d x^{2}}{2 \, b} + \frac {{\left (b c - a d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} \]
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Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {x \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {d\,x^2}{2\,b}-\frac {\ln \left (b\,x^2+a\right )\,\left (a\,d-b\,c\right )}{2\,b^2} \]
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