Optimal. Leaf size=63 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.064531, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.021, Rules used = {860} \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 860
Rubi steps
\begin{align*} \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^{5/2}} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0343866, size = 52, normalized size = 0.83 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2}}{3 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.05, size = 63, normalized size = 1. \begin{align*} -{\frac{2\,cdx+2\,ae}{3\,aeg-3\,cdf}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( gx+f \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{\sqrt{e x + d}{\left (g x + f\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.91476, size = 360, normalized size = 5.71 \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 x + a e\right )} \sqrt{e x + d} \sqrt{g x + f}}{3 \,{\left (c d^{2} f^{3} - a d e f^{2} g +{\left (c d e f g^{2} - a e^{2} g^{3}\right )} x^{3} +{\left (2 \, c d e f^{2} g - a d e g^{3} +{\left (c d^{2} - 2 \, a e^{2}\right )} f g^{2}\right )} x^{2} +{\left (c d e f^{3} - 2 \, a d e f g^{2} +{\left (2 \, c d^{2} - a e^{2}\right )} f^{2} g\right )} x\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{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{\sqrt{e x + d}{\left (g x + f\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]