Integrand size = 22, antiderivative size = 56 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {653, 197} \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}} \]
[In]
[Out]
Rule 197
Rule 653
Rubi steps \begin{align*} \text {integral}& = \frac {d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d} \\ & = \frac {d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.04 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\left (d^2+2 d e x-2 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{3 d^3 e (d-e x)^2 (d+e x)} \]
[In]
[Out]
Time = 2.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{2} \left (-e x +d \right ) \left (-2 x^{2} e^{2}+2 d e x +d^{2}\right )}{3 d^{3} e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\) | \(53\) |
trager | \(\frac {\left (-2 x^{2} e^{2}+2 d e x +d^{2}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{3 d^{3} \left (-e x +d \right )^{2} e \left (e x +d \right )}\) | \(55\) |
default | \(d \left (\frac {x}{3 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}\right )+\frac {1}{3 e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}\) | \(64\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (48) = 96\).
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.82 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {e^{3} x^{3} - d e^{2} x^{2} - d^{2} e x + d^{3} - {\left (2 \, e^{2} x^{2} - 2 \, d e x - d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{3} e^{4} x^{3} - d^{4} e^{3} x^{2} - d^{5} e^{2} x + d^{6} e\right )}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 3.66 (sec) , antiderivative size = 296, normalized size of antiderivative = 5.29 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=d \left (\begin {cases} \frac {3 i d^{2} x}{- 3 d^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {2 i e^{2} x^{3}}{- 3 d^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {3 d^{2} x}{- 3 d^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {2 e^{2} x^{3}}{- 3 d^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} - \frac {1}{- 3 d^{2} e^{2} \sqrt {d^{2} - e^{2} x^{2}} + 3 e^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{2}}{2 \left (d^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {1}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} + \frac {2 \, x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}} \]
[In]
[Out]
\[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {e x + d}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Time = 9.81 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (d^2+2\,d\,e\,x-2\,e^2\,x^2\right )}{3\,d^3\,e\,\left (d+e\,x\right )\,{\left (d-e\,x\right )}^2} \]
[In]
[Out]