Integrand size = 24, antiderivative size = 91 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d^2 e (d+e x) \sqrt {d^2-e^2 x^2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 197} \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{5 d^2 e (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}} \]
[In]
[Out]
Rule 197
Rule 673
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {3 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 d} \\ & = -\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d^2 e (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 d^2} \\ & = \frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {1}{5 d^2 e (d+e x) \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^3+d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )}{5 d^4 e (d-e x) (d+e x)^3} \]
[In]
[Out]
Time = 0.38 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (-2 e^{3} x^{3}-4 d \,e^{2} x^{2}-d^{2} e x +2 d^{3}\right )}{5 \left (e x +d \right ) d^{4} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\) | \(66\) |
trager | \(-\frac {\left (-2 e^{3} x^{3}-4 d \,e^{2} x^{2}-d^{2} e x +2 d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{5 d^{4} \left (e x +d \right )^{3} e \left (-e x +d \right )}\) | \(68\) |
default | \(\frac {-\frac {1}{5 d e \left (x +\frac {d}{e}\right )^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}+\frac {3 e \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}}{e^{2}}\) | \(156\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, e^{4} x^{4} + 4 \, d e^{3} x^{3} - 4 \, d^{3} e x - 2 \, d^{4} + {\left (2 \, e^{3} x^{3} + 4 \, d e^{2} x^{2} + d^{2} e x - 2 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{4} e^{5} x^{4} + 2 \, d^{5} e^{4} x^{3} - 2 \, d^{7} e^{2} x - d^{8} e\right )}} \]
[In]
[Out]
\[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{5 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} d e^{3} x^{2} + 2 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{2} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e\right )}} - \frac {1}{5 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{2} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e\right )}} + \frac {2 \, x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.12 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {e^{3} {\left (\frac {5}{d^{4} e^{3} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {d^{16} e^{12} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{4} \mathrm {sgn}\left (e\right )^{4} + 5 \, d^{16} e^{12} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{4} \mathrm {sgn}\left (e\right )^{4} + 15 \, d^{16} e^{12} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{4} \mathrm {sgn}\left (e\right )^{4}}{d^{20} e^{15} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{5} \mathrm {sgn}\left (e\right )^{5}}\right )} + \frac {16 i \, \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{4}}}{40 \, {\left | e \right |}} \]
[In]
[Out]
Time = 11.85 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (-2\,d^3+d^2\,e\,x+4\,d\,e^2\,x^2+2\,e^3\,x^3\right )}{5\,d^4\,e\,{\left (d+e\,x\right )}^3\,\left (d-e\,x\right )} \]
[In]
[Out]