3.561 \(\int \frac{d+e x}{\left (a+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=28 \[ -\frac{a e-c d x}{a c \sqrt{a+c x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0318201, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{a e-c d x}{a c \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(a + c*x^2)^(3/2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.07493, size = 22, normalized size = 0.79 \[ - \frac{a e - c d x}{a c \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(c*x**2+a)**(3/2),x)

[Out]

-(a*e - c*d*x)/(a*c*sqrt(a + c*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0284599, size = 27, normalized size = 0.96 \[ \frac{c d x-a e}{a c \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(a + c*x^2)^(3/2),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 27, normalized size = 1. \[ -{\frac{-cdx+ae}{ac}{\frac{1}{\sqrt{c{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+a)^(3/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.716031, size = 42, normalized size = 1.5 \[ \frac{d x}{\sqrt{c x^{2} + a} a} - \frac{e}{\sqrt{c x^{2} + a} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.221072, size = 47, normalized size = 1.68 \[ \frac{{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{a c^{2} x^{2} + a^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 8.71059, size = 46, normalized size = 1.64 \[ e \left (\begin{cases} - \frac{1}{c \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{d x}{a^{\frac{3}{2}} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(c*x**2+a)**(3/2),x)

[Out]

e*Piecewise((-1/(c*sqrt(a + c*x**2)), Ne(c, 0)), (x**2/(2*a**(3/2)), True)) + d*
x/(a**(3/2)*sqrt(1 + c*x**2/a))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215611, size = 32, normalized size = 1.14 \[ \frac{\frac{d x}{a} - \frac{e}{c}}{\sqrt{c x^{2} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

(d*x/a - e/c)/sqrt(c*x^2 + a)