3.2049 \(\int \frac{\sqrt{d+e x}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=46 \[ \frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt{d+e x}} \]

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d*Sqrt[d + e*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.0767348, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026 \[ \frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d*Sqrt[d + e*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.5455, size = 41, normalized size = 0.89 \[ \frac{2 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{c d \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(c*d*sqrt(d + e*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0328645, size = 35, normalized size = 0.76 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)}}{c d \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)])/(c*d*Sqrt[d + e*x])

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 50, normalized size = 1.1 \[ 2\,{\frac{ \left ( cdx+ae \right ) \sqrt{ex+d}}{cd\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

2*(c*d*x+a*e)*(e*x+d)^(1/2)/d/c/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.813545, size = 24, normalized size = 0.52 \[ \frac{2 \, \sqrt{c d x + a e}}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")

[Out]

2*sqrt(c*d*x + a*e)/(c*d)

_______________________________________________________________________________________

Fricas [A]  time = 0.211239, size = 57, normalized size = 1.24 \[ \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{\sqrt{e x + d} c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="fricas")

[Out]

2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*c*d)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x}}{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Integral(sqrt(d + e*x)/sqrt((d + e*x)*(a*e + c*d*x)), x)

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")

[Out]

Timed out