Integrand size = 22, antiderivative size = 67 \[ \int \frac {\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^2} \, dx=\frac {(b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {a^2 \log (x)}{c^2}-\frac {1}{2} \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \log \left (c+d x^2\right ) \]
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Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 90} \[ \int \frac {\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^2} \, dx=-\frac {1}{2} \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \log \left (c+d x^2\right )+\frac {a^2 \log (x)}{c^2}+\frac {(b c-a d)^2}{2 c d^2 \left (c+d x^2\right )} \]
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Rule 90
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2}{x (c+d x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2}{c^2 x}-\frac {(b c-a d)^2}{c d (c+d x)^2}+\frac {b^2 c^2-a^2 d^2}{c^2 d (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {(b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {a^2 \log (x)}{c^2}-\frac {1}{2} \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \log \left (c+d x^2\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^2} \, dx=\frac {2 a^2 \log (x)+\frac {(b c-a d) \left (c (b c-a d)+(b c+a d) \left (c+d x^2\right ) \log \left (c+d x^2\right )\right )}{d^2 \left (c+d x^2\right )}}{2 c^2} \]
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Time = 2.58 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {a^{2} \ln \left (x \right )}{c^{2}}-\frac {\left (a d -b c \right ) \left (\frac {\left (a d +b c \right ) \ln \left (d \,x^{2}+c \right )}{d^{2}}-\frac {\left (a d -b c \right ) c}{d^{2} \left (d \,x^{2}+c \right )}\right )}{2 c^{2}}\) | \(67\) |
norman | \(\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 c \,d^{2} \left (d \,x^{2}+c \right )}+\frac {a^{2} \ln \left (x \right )}{c^{2}}-\frac {\left (a^{2} d^{2}-b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 c^{2} d^{2}}\) | \(81\) |
risch | \(\frac {a^{2}}{2 c \left (d \,x^{2}+c \right )}-\frac {a b}{d \left (d \,x^{2}+c \right )}+\frac {c \,b^{2}}{2 d^{2} \left (d \,x^{2}+c \right )}+\frac {a^{2} \ln \left (x \right )}{c^{2}}-\frac {\ln \left (d \,x^{2}+c \right ) a^{2}}{2 c^{2}}+\frac {\ln \left (d \,x^{2}+c \right ) b^{2}}{2 d^{2}}\) | \(94\) |
parallelrisch | \(\frac {2 \ln \left (x \right ) x^{2} a^{2} d^{3}-\ln \left (d \,x^{2}+c \right ) x^{2} a^{2} d^{3}+\ln \left (d \,x^{2}+c \right ) x^{2} b^{2} c^{2} d +2 \ln \left (x \right ) a^{2} c \,d^{2}-\ln \left (d \,x^{2}+c \right ) a^{2} c \,d^{2}+\ln \left (d \,x^{2}+c \right ) b^{2} c^{3}+c \,a^{2} d^{2}-2 a b \,c^{2} d +b^{2} c^{3}}{2 c^{2} d^{2} \left (d \,x^{2}+c \right )}\) | \(136\) |
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Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.73 \[ \int \frac {\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^2} \, dx=\frac {b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{3} - a^{2} c d^{2} + {\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \, {\left (a^{2} d^{3} x^{2} + a^{2} c d^{2}\right )} \log \left (x\right )}{2 \, {\left (c^{2} d^{3} x^{2} + c^{3} d^{2}\right )}} \]
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Time = 0.74 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^2} \, dx=\frac {a^{2} \log {\left (x \right )}}{c^{2}} + \frac {a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{2 c^{2} d^{2} + 2 c d^{3} x^{2}} - \frac {\left (a d - b c\right ) \left (a d + b c\right ) \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c^{2} d^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^2} \, dx=\frac {a^{2} \log \left (x^{2}\right )}{2 \, c^{2}} + \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}} + \frac {{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{2} d^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^2} \, dx=\frac {a^{2} \log \left (x^{2}\right )}{2 \, c^{2}} + \frac {{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{2} d^{2}} - \frac {b^{2} c^{2} x^{2} - a^{2} d^{2} x^{2} + 2 \, a b c^{2} - 2 \, a^{2} c d}{2 \, {\left (d x^{2} + c\right )} c^{2} d} \]
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Time = 5.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^2} \, dx=\frac {a^2\,\ln \left (x\right )}{c^2}+\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{2\,c\,d^2\,\left (d\,x^2+c\right )}-\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^2-b^2\,c^2\right )}{2\,c^2\,d^2} \]
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