Optimal. Leaf size=109 \[ \frac{4 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0594681, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac{4 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{2 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac{\left (2 \left (d^2-\frac{a e^2}{c}\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 d}\\ &=\frac{4 \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}\\ \end{align*}
Mathematica [A] time = 0.0443115, size = 54, normalized size = 0.5 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (c d (3 d+e x)-2 a e^2\right )}{3 c^2 d^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 69, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -cdex+2\,a{e}^{2}-3\,c{d}^{2} \right ) }{3\,{c}^{2}{d}^{2}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.42706, size = 88, normalized size = 0.81 \begin{align*} \frac{2 \,{\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} +{\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )}}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.55217, size = 158, normalized size = 1.45 \begin{align*} \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d e x + 3 \, c d^{2} - 2 \, a e^{2}\right )} \sqrt{e x + d}}{3 \,{\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]