Integrand size = 20, antiderivative size = 50 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c}{2 a x^2}-\frac {(b c-a d) \log (x)}{a^2}+\frac {(b c-a d) \log \left (a+b x^2\right )}{2 a^2} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 50, 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} \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )} \, dx=\frac {(b c-a d) \log \left (a+b x^2\right )}{2 a^2}-\frac {\log (x) (b c-a d)}{a^2}-\frac {c}{2 a x^2} \]
[In]
[Out]
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)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {c}{a x^2}+\frac {-b c+a d}{a^2 x}-\frac {b (-b c+a d)}{a^2 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {c}{2 a x^2}-\frac {(b c-a d) \log (x)}{a^2}+\frac {(b c-a d) \log \left (a+b x^2\right )}{2 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c}{2 a x^2}+\frac {(-b c+a d) \log (x)}{a^2}+\frac {(b c-a d) \log \left (a+b x^2\right )}{2 a^2} \]
[In]
[Out]
Time = 2.64 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {c}{2 a \,x^{2}}+\frac {\left (a d -b c \right ) \ln \left (x \right )}{a^{2}}-\frac {\left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\) | \(46\) |
norman | \(-\frac {c}{2 a \,x^{2}}+\frac {\left (a d -b c \right ) \ln \left (x \right )}{a^{2}}-\frac {\left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\) | \(46\) |
risch | \(-\frac {c}{2 a \,x^{2}}+\frac {\ln \left (x \right ) d}{a}-\frac {b c \ln \left (x \right )}{a^{2}}-\frac {\ln \left (b \,x^{2}+a \right ) d}{2 a}+\frac {b c \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\) | \(56\) |
parallelrisch | \(\frac {2 \ln \left (x \right ) x^{2} a d -2 b c \ln \left (x \right ) x^{2}-\ln \left (b \,x^{2}+a \right ) x^{2} a d +b c \ln \left (b \,x^{2}+a \right ) x^{2}-a c}{2 a^{2} x^{2}}\) | \(61\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )} \, dx=\frac {{\left (b c - a d\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \, {\left (b c - a d\right )} x^{2} \log \left (x\right ) - a c}{2 \, a^{2} x^{2}} \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.82 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )} \, dx=- \frac {c}{2 a x^{2}} + \frac {\left (a d - b c\right ) \log {\left (x \right )}}{a^{2}} - \frac {\left (a d - b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2}} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )} \, dx=\frac {{\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2}} - \frac {{\left (b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {c}{2 \, a x^{2}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.44 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )} \, dx=-\frac {{\left (b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {{\left (b^{2} c - a b d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} + \frac {b c x^{2} - a d x^{2} - a c}{2 \, a^{2} x^{2}} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )} \, dx=\frac {\ln \left (x\right )\,\left (a\,d-b\,c\right )}{a^2}-\frac {\ln \left (b\,x^2+a\right )\,\left (a\,d-b\,c\right )}{2\,a^2}-\frac {c}{2\,a\,x^2} \]
[In]
[Out]