Optimal. Leaf size=28 \[ -\frac{(d+e x)^3}{3 (a+b x)^3 (b d-a e)} \]
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Rubi [A] time = 0.0229329, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{(d+e x)^3}{3 (a+b x)^3 (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 11.8525, size = 20, normalized size = 0.71 \[ \frac{\left (d + e x\right )^{3}}{3 \left (a + b x\right )^{3} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.0452853, size = 53, normalized size = 1.89 \[ -\frac{a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 b^3 (a+b x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Maple [B] time = 0.008, size = 70, normalized size = 2.5 \[ -{\frac{{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}-{\frac{{a}^{2}{e}^{2}-2\,aedb+{b}^{2}{d}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{e \left ( ae-bd \right ) }{{b}^{3} \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
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Maxima [A] time = 0.68296, size = 113, normalized size = 4.04 \[ -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.19681, size = 113, normalized size = 4.04 \[ -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.10007, size = 88, normalized size = 3.14 \[ - \frac{a^{2} e^{2} + a b d e + b^{2} d^{2} + 3 b^{2} e^{2} x^{2} + x \left (3 a b e^{2} + 3 b^{2} d e\right )}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.210047, size = 78, normalized size = 2.79 \[ -\frac{3 \, b^{2} x^{2} e^{2} + 3 \, b^{2} d x e + b^{2} d^{2} + 3 \, a b x e^{2} + a b d e + a^{2} e^{2}}{3 \,{\left (b x + a\right )}^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="giac")
[Out]