Integrand size = 22, antiderivative size = 58 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c^2}{2 a x^2}-\frac {c (b c-2 a d) \log (x)}{a^2}+\frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 a^2 b} \]
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Time = 0.04 (sec) , antiderivative size = 58, 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} \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )} \, dx=\frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 a^2 b}-\frac {c \log (x) (b c-2 a d)}{a^2}-\frac {c^2}{2 a x^2} \]
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Rule 90
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^2}{x^2 (a+b x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {c^2}{a x^2}+\frac {c (-b c+2 a d)}{a^2 x}+\frac {(-b c+a d)^2}{a^2 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {c^2}{2 a x^2}-\frac {c (b c-2 a d) \log (x)}{a^2}+\frac {(b c-a d)^2 \log \left (a+b x^2\right )}{2 a^2 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )} \, dx=\frac {-a b c^2-2 b c (b c-2 a d) x^2 \log (x)+(b c-a d)^2 x^2 \log \left (a+b x^2\right )}{2 a^2 b x^2} \]
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Time = 2.65 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.14
method | result | size |
default | \(-\frac {c^{2}}{2 a \,x^{2}}+\frac {c \left (2 a d -b c \right ) \ln \left (x \right )}{a^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2} b}\) | \(66\) |
norman | \(-\frac {c^{2}}{2 a \,x^{2}}+\frac {c \left (2 a d -b c \right ) \ln \left (x \right )}{a^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2} b}\) | \(66\) |
risch | \(-\frac {c^{2}}{2 a \,x^{2}}+\frac {2 c \ln \left (x \right ) d}{a}-\frac {c^{2} \ln \left (x \right ) b}{a^{2}}+\frac {\ln \left (-b \,x^{2}-a \right ) d^{2}}{2 b}-\frac {\ln \left (-b \,x^{2}-a \right ) c d}{a}+\frac {b \ln \left (-b \,x^{2}-a \right ) c^{2}}{2 a^{2}}\) | \(90\) |
parallelrisch | \(\frac {4 \ln \left (x \right ) x^{2} a b c d -2 \ln \left (x \right ) x^{2} b^{2} c^{2}+\ln \left (b \,x^{2}+a \right ) x^{2} a^{2} d^{2}-2 \ln \left (b \,x^{2}+a \right ) x^{2} a b c d +\ln \left (b \,x^{2}+a \right ) x^{2} b^{2} c^{2}-b \,c^{2} a}{2 a^{2} x^{2} b}\) | \(97\) |
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Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {a b c^{2} - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \log \left (b x^{2} + a\right ) + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d\right )} x^{2} \log \left (x\right )}{2 \, a^{2} b x^{2}} \]
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Time = 0.82 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )} \, dx=- \frac {c^{2}}{2 a x^{2}} + \frac {c \left (2 a d - b c\right ) \log {\left (x \right )}}{a^{2}} + \frac {\left (a d - b c\right )^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2} b} \]
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Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.19 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {{\left (b c^{2} - 2 \, a c d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {c^{2}}{2 \, a x^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2} b} \]
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Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.55 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {{\left (b c^{2} - 2 \, a c d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} + \frac {b c^{2} x^{2} - 2 \, a c d x^{2} - a c^{2}}{2 \, a^{2} x^{2}} \]
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Time = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.16 \[ \int \frac {\left (c+d x^2\right )^2}{x^3 \left (a+b x^2\right )} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,a^2\,b}-\frac {c^2}{2\,a\,x^2}-\frac {\ln \left (x\right )\,\left (b\,c^2-2\,a\,c\,d\right )}{a^2} \]
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