Integrand size = 20, antiderivative size = 76 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {c}{2 a^2 x^2}-\frac {b c-a d}{2 a^2 \left (a+b x^2\right )}-\frac {(2 b c-a d) \log (x)}{a^3}+\frac {(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3} \]
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Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 78} \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3}-\frac {\log (x) (2 b c-a d)}{a^3}-\frac {b c-a d}{2 a^2 \left (a+b x^2\right )}-\frac {c}{2 a^2 x^2} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {c+d x}{x^2 (a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {c}{a^2 x^2}+\frac {-2 b c+a d}{a^3 x}-\frac {b (-b c+a d)}{a^2 (a+b x)^2}-\frac {b (-2 b c+a d)}{a^3 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {c}{2 a^2 x^2}-\frac {b c-a d}{2 a^2 \left (a+b x^2\right )}-\frac {(2 b c-a d) \log (x)}{a^3}+\frac {(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.84 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {-\frac {a c}{x^2}+\frac {a (-b c+a d)}{a+b x^2}+2 (-2 b c+a d) \log (x)+(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3} \]
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Time = 2.64 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {c}{2 a^{2} x^{2}}+\frac {\left (a d -2 b c \right ) \ln \left (x \right )}{a^{3}}-\frac {b \left (\frac {\left (a d -2 b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{3}}\) | \(75\) |
norman | \(\frac {-\frac {c}{2 a}+\frac {b \left (-a d +2 b c \right ) x^{4}}{2 a^{3}}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {\left (a d -2 b c \right ) \ln \left (x \right )}{a^{3}}-\frac {\left (a d -2 b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{3}}\) | \(75\) |
risch | \(\frac {\frac {\left (a d -2 b c \right ) x^{2}}{2 a^{2}}-\frac {c}{2 a}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {\ln \left (x \right ) d}{a^{2}}-\frac {2 b c \ln \left (x \right )}{a^{3}}-\frac {\ln \left (b \,x^{2}+a \right ) d}{2 a^{2}}+\frac {b c \ln \left (b \,x^{2}+a \right )}{a^{3}}\) | \(82\) |
parallelrisch | \(\frac {2 \ln \left (x \right ) x^{4} a b d -4 \ln \left (x \right ) x^{4} b^{2} c -\ln \left (b \,x^{2}+a \right ) x^{4} a b d +2 \ln \left (b \,x^{2}+a \right ) x^{4} b^{2} c -a b d \,x^{4}+2 b^{2} c \,x^{4}+2 \ln \left (x \right ) x^{2} a^{2} d -4 \ln \left (x \right ) x^{2} a b c -\ln \left (b \,x^{2}+a \right ) x^{2} a^{2} d +2 \ln \left (b \,x^{2}+a \right ) x^{2} a b c -a^{2} c}{2 a^{3} x^{2} \left (b \,x^{2}+a \right )}\) | \(150\) |
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Time = 0.24 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.61 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x^{2} - {\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} + {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} + {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}} \]
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Time = 0.50 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {- a c + x^{2} \left (a d - 2 b c\right )}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} + \frac {\left (a d - 2 b c\right ) \log {\left (x \right )}}{a^{3}} - \frac {\left (a d - 2 b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (2 \, b c - a d\right )} x^{2} + a c}{2 \, {\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac {{\left (2 \, b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3}} - \frac {{\left (2 \, b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.11 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (2 \, b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac {2 \, b c x^{2} - a d x^{2} + a c}{2 \, {\left (b x^{4} + a x^{2}\right )} a^{2}} + \frac {{\left (2 \, b^{2} c - a b d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} b} \]
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Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\ln \left (x\right )\,\left (a\,d-2\,b\,c\right )}{a^3}-\frac {\ln \left (b\,x^2+a\right )\,\left (a\,d-2\,b\,c\right )}{2\,a^3}-\frac {\frac {c}{2\,a}-\frac {x^2\,\left (a\,d-2\,b\,c\right )}{2\,a^2}}{b\,x^4+a\,x^2} \]
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