Optimal. Leaf size=56 \[ \frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}}+\frac{d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0123644, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {639, 191} \[ \frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}}+\frac{d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 639
Rule 191
Rubi steps
\begin{align*} \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 \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*}
Mathematica [A] time = 0.0237811, size = 51, normalized size = 0.91 \[ \frac{d^2+2 d e x-2 e^2 x^2}{3 d^3 e (d-e x) \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.044, size = 53, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) ^{2} \left ( -ex+d \right ) \left ( -2\,{e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ) }{3\,{d}^{3}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.20168, size = 81, normalized size = 1.45 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.08311, size = 192, normalized size = 3.43 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 4.95444, size = 298, normalized size = 5.32 \begin{align*} 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}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.42761, size = 70, normalized size = 1.25 \begin{align*} -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left (x{\left (\frac{2 \, x^{2} e^{2}}{d^{3}} - \frac{3}{d}\right )} - e^{\left (-1\right )}\right )}}{3 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]