3.751 \(\int \frac{(a+b x)^2}{\left (a^2-b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{b (a-b x)} \]

[Out]

1/(b*(a - b*x))

_______________________________________________________________________________________

Rubi [A]  time = 0.0177434, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{b (a-b x)} \]

Antiderivative was successfully verified.

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

[Out]

1/(b*(a - b*x))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.47588, size = 7, normalized size = 0.58 \[ \frac{1}{b \left (a - b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(b*(a - b*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.00386891, size = 12, normalized size = 1. \[ \frac{1}{b (a-b x)} \]

Antiderivative was successfully verified.

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

[Out]

1/(b*(a - b*x))

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 15, normalized size = 1.3 \[ -{\frac{1}{b \left ( bx-a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/b/(b*x-a)

_______________________________________________________________________________________

Maxima [A]  time = 0.684583, size = 19, normalized size = 1.58 \[ -\frac{1}{b^{2} x - a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(b^2*x - a*b)

_______________________________________________________________________________________

Fricas [A]  time = 0.203691, size = 19, normalized size = 1.58 \[ -\frac{1}{b^{2} x - a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(b^2*x - a*b)

_______________________________________________________________________________________

Sympy [A]  time = 1.21084, size = 10, normalized size = 0.83 \[ - \frac{1}{- a b + b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(-a*b + b**2*x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215095, size = 19, normalized size = 1.58 \[ -\frac{1}{{\left (b x - a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/((b*x - a)*b)