3.638 \(\int \frac{A+B x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

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

[Out]

-(A*b - a*B)/(3*b^2*(a + b*x)^3) - B/(2*b^2*(a + b*x)^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0522174, 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 \[ -\frac{A b-a B}{3 b^2 (a+b x)^3}-\frac{B}{2 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

-(A*b - a*B)/(3*b^2*(a + b*x)^3) - B/(2*b^2*(a + b*x)^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.6748, size = 32, normalized size = 0.84 \[ - \frac{B}{2 b^{2} \left (a + b x\right )^{2}} - \frac{A b - B a}{3 b^{2} \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

-B/(2*b**2*(a + b*x)**2) - (A*b - B*a)/(3*b**2*(a + b*x)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0168167, size = 27, normalized size = 0.71 \[ -\frac{B (a+3 b x)+2 A b}{6 b^2 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

-(2*A*b + B*(a + 3*b*x))/(6*b^2*(a + b*x)^3)

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 35, normalized size = 0.9 \[ -{\frac{Ab-Ba}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{B}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

-1/3*(A*b-B*a)/b^2/(b*x+a)^3-1/2*B/b^2/(b*x+a)^2

_______________________________________________________________________________________

Maxima [A]  time = 0.697634, size = 68, normalized size = 1.79 \[ -\frac{3 \, B b x + B a + 2 \, A b}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.267353, size = 68, normalized size = 1.79 \[ -\frac{3 \, B b x + B a + 2 \, A b}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.03968, size = 53, normalized size = 1.39 \[ - \frac{2 A b + B a + 3 B b x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

-(2*A*b + B*a + 3*B*b*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 = 0.266002, size = 34, normalized size = 0.89 \[ -\frac{3 \, B b x + B a + 2 \, A b}{6 \,{\left (b x + a\right )}^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(3*B*b*x + B*a + 2*A*b)/((b*x + a)^3*b^2)