3.2022 \(\int \frac{a+b x}{(d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.113516, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a+b x}{2 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.9688, size = 36, normalized size = 0.92 \[ - \frac{a + b x}{2 e \left (d + e x\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0204092, size = 30, normalized size = 0.77 \[ -\frac{a+b x}{2 e \sqrt{(a+b x)^2} (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 27, normalized size = 0.7 \[ -{\frac{bx+a}{2\,e \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(e*x+d)^2/e*(b*x+a)/((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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.27426, size = 32, normalized size = 0.82 \[ -\frac{1}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(e^3*x^2 + 2*d*e^2*x + d^2*e)

_______________________________________________________________________________________

Sympy [A]  time = 1.3913, size = 26, normalized size = 0.67 \[ - \frac{1}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*d**2*e + 4*d*e**2*x + 2*e**3*x**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.290976, size = 24, normalized size = 0.62 \[ -\frac{e^{\left (-1\right )}{\rm sign}\left (b x + a\right )}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*e^(-1)*sign(b*x + a)/(x*e + d)^2