Integrand size = 35, antiderivative size = 20 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^3} \, dx=\frac {(a e+c d x)^4}{4 c d} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 32} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^3} \, dx=\frac {(a e+c d x)^4}{4 c d} \]
[In]
[Out]
Rule 32
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int (a e+c d x)^3 \, dx \\ & = \frac {(a e+c d x)^4}{4 c d} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^3} \, dx=\frac {(a e+c d x)^4}{4 c d} \]
[In]
[Out]
Time = 2.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\left (c d x +a e \right )^{4}}{4 c d}\) | \(19\) |
parallelrisch | \(\frac {1}{4} c^{3} d^{3} x^{4}+a \,c^{2} d^{2} e \,x^{3}+\frac {3}{2} a^{2} c d \,e^{2} x^{2}+a^{3} e^{3} x\) | \(46\) |
gosper | \(\frac {x \left (c^{3} d^{3} x^{3}+4 c^{2} d^{2} a e \,x^{2}+6 d \,e^{2} a^{2} c x +4 a^{3} e^{3}\right )}{4}\) | \(47\) |
risch | \(\frac {c^{3} d^{3} x^{4}}{4}+a \,c^{2} d^{2} e \,x^{3}+\frac {3 a^{2} c d \,e^{2} x^{2}}{2}+a^{3} e^{3} x +\frac {e^{4} a^{4}}{4 d c}\) | \(60\) |
norman | \(\frac {\left (e^{3} a \,c^{2} d^{2}+\frac {1}{2} d^{4} e \,c^{3}\right ) x^{5}+\left (\frac {3}{2} d \,e^{4} a^{2} c +2 d^{3} e^{2} c^{2} a +\frac {1}{4} d^{5} c^{3}\right ) x^{4}+\left (a^{3} e^{5}+3 d^{2} e^{3} a^{2} c +d^{4} a \,c^{2} e \right ) x^{3}-\frac {d^{2} \left (4 d \,e^{4} a^{3}+3 d^{3} e^{2} a^{2} c \right )}{2 e^{2}}-\frac {d \left (3 d \,e^{4} a^{3}+3 d^{3} e^{2} a^{2} c \right ) x}{e}+\frac {e^{2} c^{3} d^{3} x^{6}}{4}}{\left (e x +d \right )^{2}}\) | \(176\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (18) = 36\).
Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.25 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^3} \, dx=\frac {1}{4} \, c^{3} d^{3} x^{4} + a c^{2} d^{2} e x^{3} + \frac {3}{2} \, a^{2} c d e^{2} x^{2} + a^{3} e^{3} x \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (14) = 28\).
Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^3} \, dx=a^{3} e^{3} x + \frac {3 a^{2} c d e^{2} x^{2}}{2} + a c^{2} d^{2} e x^{3} + \frac {c^{3} d^{3} x^{4}}{4} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (18) = 36\).
Time = 0.20 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.25 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^3} \, dx=\frac {1}{4} \, c^{3} d^{3} x^{4} + a c^{2} d^{2} e x^{3} + \frac {3}{2} \, a^{2} c d e^{2} x^{2} + a^{3} e^{3} x \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (18) = 36\).
Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.25 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^3} \, dx=\frac {1}{4} \, c^{3} d^{3} x^{4} + a c^{2} d^{2} e x^{3} + \frac {3}{2} \, a^{2} c d e^{2} x^{2} + a^{3} e^{3} x \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.25 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^3} \, dx=a^3\,e^3\,x+\frac {3\,a^2\,c\,d\,e^2\,x^2}{2}+a\,c^2\,d^2\,e\,x^3+\frac {c^3\,d^3\,x^4}{4} \]
[In]
[Out]