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