Optimal. Leaf size=53 \[ \frac{x}{3 d^2 \sqrt{d^2-e^2 x^2}}+\frac{2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0133077, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {653, 191} \[ \frac{x}{3 d^2 \sqrt{d^2-e^2 x^2}}+\frac{2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 653
Rule 191
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac{2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{3} \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x}{3 d^2 \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0281359, size = 47, normalized size = 0.89 \[ \frac{(2 d-e x) (d+e x)}{3 d^2 e (d-e x) \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.044, size = 44, normalized size = 0.8 \begin{align*}{\frac{ \left ( ex+d \right ) ^{3} \left ( -ex+d \right ) \left ( -ex+2\,d \right ) }{3\,{d}^{2}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 = 1.14806, size = 78, normalized size = 1.47 \begin{align*} \frac{2 \, x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{2 \, d}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} + \frac{x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.03883, size = 143, normalized size = 2.7 \begin{align*} \frac{2 \, e^{2} x^{2} - 4 \, d e x + 2 \, d^{2} - \sqrt{-e^{2} x^{2} + d^{2}}{\left (e x - 2 \, d\right )}}{3 \,{\left (d^{2} e^{3} x^{2} - 2 \, d^{3} e^{2} x + d^{4} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.45495, size = 65, normalized size = 1.23 \begin{align*} -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left (x{\left (\frac{x^{2} e^{2}}{d^{2}} - 3\right )} - 2 \, d 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]