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