3.1509 \(\int \frac{d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=38 \[ -\frac{b d-a e}{3 b^2 (a+b x)^3}-\frac{e}{2 b^2 (a+b x)^2} \]

[Out]

-(b*d - a*e)/(3*b^2*(a + b*x)^3) - e/(2*b^2*(a + b*x)^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0624565, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{b d-a e}{3 b^2 (a+b x)^3}-\frac{e}{2 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(b*d - a*e)/(3*b^2*(a + b*x)^3) - e/(2*b^2*(a + b*x)^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.7304, size = 31, normalized size = 0.82 \[ - \frac{e}{2 b^{2} \left (a + b x\right )^{2}} + \frac{a e - b d}{3 b^{2} \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e/(2*b**2*(a + b*x)**2) + (a*e - b*d)/(3*b**2*(a + b*x)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0159326, size = 27, normalized size = 0.71 \[ -\frac{a e+2 b d+3 b e x}{6 b^2 (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-(2*b*d + a*e + 3*b*e*x)/(6*b^2*(a + b*x)^3)

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 35, normalized size = 0.9 \[ -{\frac{e}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{-ae+bd}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*e/b^2/(b*x+a)^2-1/3*(-a*e+b*d)/b^2/(b*x+a)^3

_______________________________________________________________________________________

Maxima [A]  time = 0.681757, size = 68, normalized size = 1.79 \[ -\frac{3 \, b e x + 2 \, b d + a e}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(3*b*e*x + 2*b*d + a*e)/(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x + a^3*b^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.199822, size = 68, normalized size = 1.79 \[ -\frac{3 \, b e x + 2 \, b d + a e}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(3*b*e*x + 2*b*d + a*e)/(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x + a^3*b^2)

_______________________________________________________________________________________

Sympy [A]  time = 1.96857, size = 53, normalized size = 1.39 \[ - \frac{a e + 2 b d + 3 b e x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*e + 2*b*d + 3*b*e*x)/(6*a**3*b**2 + 18*a**2*b**3*x + 18*a*b**4*x**2 + 6*b**5
*x**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.208639, size = 36, normalized size = 0.95 \[ -\frac{3 \, b x e + 2 \, b d + a e}{6 \,{\left (b x + a\right )}^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(3*b*x*e + 2*b*d + a*e)/((b*x + a)^3*b^2)