Optimal. Leaf size=36 \[ \frac{2 \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0130347, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {649} \[ \frac{2 \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 649
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=\frac{2 \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0451778, size = 35, normalized size = 0.97 \[ \frac{2 \sqrt{d+e x}}{c e \sqrt{c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 36, normalized size = 1. \begin{align*} 2\,{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{3/2}}{e \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.1656, size = 22, normalized size = 0.61 \begin{align*} \frac{2}{\sqrt{-e x + d} c^{\frac{3}{2}} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.22249, size = 93, normalized size = 2.58 \begin{align*} -\frac{2 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{c^{2} e^{3} x^{2} - c^{2} d^{2} e} \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 )^{\frac{3}{2}}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]