3.2127 \(\int \frac{a+b x}{(d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{2 (a+b x)}{e \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.118313, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ -\frac{2 (a+b x)}{e \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.9834, size = 39, normalized size = 1. \[ - \frac{2 \left (a + b x\right )}{e \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(e*x+d)**(3/2)/((b*x+a)**2)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0241446, size = 30, normalized size = 0.77 \[ -\frac{2 (a+b x)}{e \sqrt{(a+b x)^2} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 27, normalized size = 0.7 \[ -2\,{\frac{bx+a}{e\sqrt{ex+d}\sqrt{ \left ( bx+a \right ) ^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.28485, size = 16, normalized size = 0.41 \[ -\frac{2}{\sqrt{e x + d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(sqrt(e*x + d)*e)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(e*x+d)**(3/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.285795, size = 24, normalized size = 0.62 \[ -\frac{2 \, e^{\left (-1\right )}{\rm sign}\left (b x + a\right )}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*e^(-1)*sign(b*x + a)/sqrt(x*e + d)