3.1521 \(\int \frac{d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=38 \[ -\frac{b d-a e}{5 b^2 (a+b x)^5}-\frac{e}{4 b^2 (a+b x)^4} \]

[Out]

-(b*d - a*e)/(5*b^2*(a + b*x)^5) - e/(4*b^2*(a + b*x)^4)

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Rubi [A]  time = 0.0617724, 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 \[ -\frac{b d-a e}{5 b^2 (a+b x)^5}-\frac{e}{4 b^2 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(b*d - a*e)/(5*b^2*(a + b*x)^5) - e/(4*b^2*(a + b*x)^4)

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Rubi in Sympy [A]  time = 19.9694, size = 31, normalized size = 0.82 \[ - \frac{e}{4 b^{2} \left (a + b x\right )^{4}} + \frac{a e - b d}{5 b^{2} \left (a + b x\right )^{5}} \]

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)**3,x)

[Out]

-e/(4*b**2*(a + b*x)**4) + (a*e - b*d)/(5*b**2*(a + b*x)**5)

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Mathematica [A]  time = 0.0171789, size = 27, normalized size = 0.71 \[ -\frac{a e+4 b d+5 b e x}{20 b^2 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(4*b*d + a*e + 5*b*e*x)/(20*b^2*(a + b*x)^5)

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Maple [A]  time = 0.008, size = 35, normalized size = 0.9 \[ -{\frac{-ae+bd}{5\,{b}^{2} \left ( bx+a \right ) ^{5}}}-{\frac{e}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}} \]

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)^3,x)

[Out]

-1/5*(-a*e+b*d)/b^2/(b*x+a)^5-1/4*e/b^2/(b*x+a)^4

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Maxima [A]  time = 0.690478, size = 97, normalized size = 2.55 \[ -\frac{5 \, b e x + 4 \, b d + a e}{20 \,{\left (b^{7} x^{5} + 5 \, a b^{6} x^{4} + 10 \, a^{2} b^{5} x^{3} + 10 \, a^{3} b^{4} x^{2} + 5 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \]

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)^3,x, algorithm="maxima")

[Out]

-1/20*(5*b*e*x + 4*b*d + a*e)/(b^7*x^5 + 5*a*b^6*x^4 + 10*a^2*b^5*x^3 + 10*a^3*b
^4*x^2 + 5*a^4*b^3*x + a^5*b^2)

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Fricas [A]  time = 0.196858, size = 97, normalized size = 2.55 \[ -\frac{5 \, b e x + 4 \, b d + a e}{20 \,{\left (b^{7} x^{5} + 5 \, a b^{6} x^{4} + 10 \, a^{2} b^{5} x^{3} + 10 \, a^{3} b^{4} x^{2} + 5 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

-1/20*(5*b*e*x + 4*b*d + a*e)/(b^7*x^5 + 5*a*b^6*x^4 + 10*a^2*b^5*x^3 + 10*a^3*b
^4*x^2 + 5*a^4*b^3*x + a^5*b^2)

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Sympy [A]  time = 2.69749, size = 76, normalized size = 2. \[ - \frac{a e + 4 b d + 5 b e x}{20 a^{5} b^{2} + 100 a^{4} b^{3} x + 200 a^{3} b^{4} x^{2} + 200 a^{2} b^{5} x^{3} + 100 a b^{6} x^{4} + 20 b^{7} x^{5}} \]

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)**3,x)

[Out]

-(a*e + 4*b*d + 5*b*e*x)/(20*a**5*b**2 + 100*a**4*b**3*x + 200*a**3*b**4*x**2 +
200*a**2*b**5*x**3 + 100*a*b**6*x**4 + 20*b**7*x**5)

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GIAC/XCAS [A]  time = 0.209069, size = 36, normalized size = 0.95 \[ -\frac{5 \, b x e + 4 \, b d + a e}{20 \,{\left (b x + a\right )}^{5} b^{2}} \]

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)^3,x, algorithm="giac")

[Out]

-1/20*(5*b*x*e + 4*b*d + a*e)/((b*x + a)^5*b^2)