Integrand size = 21, antiderivative size = 17 \[ \int \frac {x \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {c}{2 b \left (a+b x^2\right )} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {21, 267} \[ \int \frac {x \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {c}{2 b \left (a+b x^2\right )} \]
[In]
[Out]
Rule 21
Rule 267
Rubi steps \begin{align*} \text {integral}& = c \int \frac {x}{\left (a+b x^2\right )^2} \, dx \\ & = -\frac {c}{2 b \left (a+b x^2\right )} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {c}{2 b \left (a+b x^2\right )} \]
[In]
[Out]
Time = 2.60 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(-\frac {c}{2 b \left (b \,x^{2}+a \right )}\) | \(16\) |
default | \(-\frac {c}{2 b \left (b \,x^{2}+a \right )}\) | \(16\) |
risch | \(-\frac {c}{2 b \left (b \,x^{2}+a \right )}\) | \(16\) |
parallelrisch | \(-\frac {c}{2 b \left (b \,x^{2}+a \right )}\) | \(16\) |
norman | \(\frac {-\frac {a c}{2 b}-\frac {c \,x^{2}}{2}}{\left (b \,x^{2}+a \right )^{2}}\) | \(25\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {x \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {c}{2 \, {\left (b^{2} x^{2} + a b\right )}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {x \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=- \frac {c}{2 a b + 2 b^{2} x^{2}} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {x \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {c}{2 \, {\left (b^{2} x^{2} + a b\right )}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {x \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {c}{2 \, {\left (b x^{2} + a\right )} b} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {x \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {c}{2\,b\,\left (b\,x^2+a\right )} \]
[In]
[Out]