3.1939 \(\int \frac{(a+b x) (d+e x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=28 \[ -\frac{(d+e x)^2}{2 (a+b x)^2 (b d-a e)} \]

[Out]

-(d + e*x)^2/(2*(b*d - a*e)*(a + b*x)^2)

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Rubi [A]  time = 0.0191468, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{(d+e x)^2}{2 (a+b x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

-(d + e*x)^2/(2*(b*d - a*e)*(a + b*x)^2)

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Rubi in Sympy [A]  time = 15.2326, size = 20, normalized size = 0.71 \[ \frac{\left (d + e x\right )^{2}}{2 \left (a + b x\right )^{2} \left (a e - b d\right )} \]

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)**2,x)

[Out]

(d + e*x)**2/(2*(a + b*x)**2*(a*e - b*d))

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Mathematica [A]  time = 0.0170663, size = 26, normalized size = 0.93 \[ -\frac{a e+b (d+2 e x)}{2 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a*e + b*(d + 2*e*x))/(2*b^2*(a + b*x)^2)

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Maple [A]  time = 0.007, size = 35, normalized size = 1.3 \[ -{\frac{-ae+bd}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{e}{{b}^{2} \left ( bx+a \right ) }} \]

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)^2,x)

[Out]

-1/2*(-a*e+b*d)/b^2/(b*x+a)^2-e/b^2/(b*x+a)

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Maxima [A]  time = 0.703843, size = 51, normalized size = 1.82 \[ -\frac{2 \, b e x + b d + a e}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]

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)^2,x, algorithm="maxima")

[Out]

-1/2*(2*b*e*x + b*d + a*e)/(b^4*x^2 + 2*a*b^3*x + a^2*b^2)

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Fricas [A]  time = 0.27338, size = 51, normalized size = 1.82 \[ -\frac{2 \, b e x + b d + a e}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

-1/2*(2*b*e*x + b*d + a*e)/(b^4*x^2 + 2*a*b^3*x + a^2*b^2)

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Sympy [A]  time = 1.78584, size = 39, normalized size = 1.39 \[ - \frac{a e + b d + 2 b e x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \]

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)**2,x)

[Out]

-(a*e + b*d + 2*b*e*x)/(2*a**2*b**2 + 4*a*b**3*x + 2*b**4*x**2)

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GIAC/XCAS [A]  time = 0.283089, size = 35, normalized size = 1.25 \[ -\frac{2 \, b x e + b d + a e}{2 \,{\left (b x + a\right )}^{2} b^{2}} \]

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)^2,x, algorithm="giac")

[Out]

-1/2*(2*b*x*e + b*d + a*e)/((b*x + a)^2*b^2)