Integrand size = 21, antiderivative size = 22 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log \left (a+b (c+d x)^3\right )}{3 b d} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 22, 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 = {379, 266} \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log \left (a+b (c+d x)^3\right )}{3 b d} \]
[In]
[Out]
Rule 266
Rule 379
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {\log \left (a+b (c+d x)^3\right )}{3 b d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log \left (a+b (c+d x)^3\right )}{3 b d} \]
[In]
[Out]
Time = 6.42 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\ln \left (a +b \left (d x +c \right )^{3}\right )}{3 b d}\) | \(21\) |
default | \(\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\) | \(43\) |
norman | \(\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\) | \(43\) |
risch | \(\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\) | \(43\) |
parallelrisch | \(\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\) | \(43\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, b d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).
Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log {\left (a + b c^{3} + 3 b c^{2} d x + 3 b c d^{2} x^{2} + b d^{3} x^{3} \right )}}{3 b d} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log \left ({\left (d x + c\right )}^{3} b + a\right )}{3 \, b d} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\log \left ({\left | {\left (d x + c\right )}^{3} b + a \right |}\right )}{3 \, b d} \]
[In]
[Out]
Time = 5.55 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx=\frac {\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,b\,d} \]
[In]
[Out]