Optimal. Leaf size=38 \[ -\frac{b d-a e}{4 b^2 (a+b x)^4}-\frac{e}{3 b^2 (a+b x)^3} \]
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Rubi [A] time = 0.0543469, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{b d-a e}{4 b^2 (a+b x)^4}-\frac{e}{3 b^2 (a+b x)^3} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(d + e*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
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Rubi in Sympy [A] time = 21.168, size = 31, normalized size = 0.82 \[ - \frac{e}{3 b^{2} \left (a + b x\right )^{3}} + \frac{a e - b d}{4 b^{2} \left (a + b x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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Mathematica [A] time = 0.0154056, size = 27, normalized size = 0.71 \[ -\frac{a e+3 b d+4 b e x}{12 b^2 (a+b x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(d + e*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Maple [A] time = 0.009, size = 35, normalized size = 0.9 \[ -{\frac{-ae+bd}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}}-{\frac{e}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
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Maxima [A] time = 0.705829, size = 82, normalized size = 2.16 \[ -\frac{4 \, b e x + 3 \, b d + a e}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.290385, size = 82, normalized size = 2.16 \[ -\frac{4 \, b e x + 3 \, b d + a e}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.39521, size = 65, normalized size = 1.71 \[ - \frac{a e + 3 b d + 4 b e x}{12 a^{4} b^{2} + 48 a^{3} b^{3} x + 72 a^{2} b^{4} x^{2} + 48 a b^{5} x^{3} + 12 b^{6} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.276846, size = 36, normalized size = 0.95 \[ -\frac{4 \, b x e + 3 \, b d + a e}{12 \,{\left (b x + a\right )}^{4} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")
[Out]