Integrand size = 32, antiderivative size = 34 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {c^3}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {657, 643} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {c^3}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
[In]
[Out]
Rule 643
Rule 657
Rubi steps \begin{align*} \text {integral}& = c^4 \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx \\ & = -\frac {c^3}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {\left (c (d+e x)^2\right )^{3/2}}{3 e (d+e x)^6} \]
[In]
[Out]
Time = 2.78 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74
method | result | size |
risch | \(-\frac {c \sqrt {c \left (e x +d \right )^{2}}}{3 \left (e x +d \right )^{4} e}\) | \(25\) |
pseudoelliptic | \(-\frac {c \sqrt {c \left (e x +d \right )^{2}}}{3 \left (e x +d \right )^{4} e}\) | \(25\) |
gosper | \(-\frac {\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{3 \left (e x +d \right )^{6} e}\) | \(35\) |
default | \(-\frac {\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{3 \left (e x +d \right )^{6} e}\) | \(35\) |
trager | \(\frac {c \left (x^{2} e^{2}+3 d e x +3 d^{2}\right ) x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{3 d^{3} \left (e x +d \right )^{4}}\) | \(55\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (30) = 60\).
Time = 0.41 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.03 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c}{3 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} \]
[In]
[Out]
\[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {\left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{7}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.62 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {c^{\frac {3}{2}} \mathrm {sgn}\left (e x + d\right )}{3 \, {\left (e x + d\right )}^{3} e} \]
[In]
[Out]
Time = 9.73 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {c\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{3\,e\,{\left (d+e\,x\right )}^4} \]
[In]
[Out]