3.1448 \(\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=28 \[ \frac{(a+b x)^3}{3 (d+e x)^3 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.0246163, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(a+b x)^3}{3 (d+e x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.9508, size = 22, normalized size = 0.79 \[ - \frac{\left (a + b x\right )^{3}}{3 \left (d + e x\right )^{3} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d)**4,x)

[Out]

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

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Mathematica [A]  time = 0.0469223, size = 53, normalized size = 1.89 \[ -\frac{a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^4,x]

[Out]

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

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Maple [B]  time = 0.008, size = 71, normalized size = 2.5 \[ -{\frac{{a}^{2}{e}^{2}-2\,deab+{b}^{2}{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{b \left ( ae-bd \right ) }{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2}}{{e}^{3} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^4,x)

[Out]

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

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Maxima [A]  time = 0.6862, size = 113, normalized size = 4.04 \[ -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)/(e*x + d)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.198369, size = 113, normalized size = 4.04 \[ -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)/(e*x + d)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.06849, size = 88, normalized size = 3.14 \[ - \frac{a^{2} e^{2} + a b d e + b^{2} d^{2} + 3 b^{2} e^{2} x^{2} + x \left (3 a b e^{2} + 3 b^{2} d e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d)**4,x)

[Out]

-(a**2*e**2 + a*b*d*e + b**2*d**2 + 3*b**2*e**2*x**2 + x*(3*a*b*e**2 + 3*b**2*d*
e))/(3*d**3*e**3 + 9*d**2*e**4*x + 9*d*e**5*x**2 + 3*e**6*x**3)

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GIAC/XCAS [A]  time = 0.210827, size = 78, normalized size = 2.79 \[ -\frac{{\left (3 \, b^{2} x^{2} e^{2} + 3 \, b^{2} d x e + b^{2} d^{2} + 3 \, a b x e^{2} + a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)/(e*x + d)^4,x, algorithm="giac")

[Out]

-1/3*(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)*e
^(-3)/(x*e + d)^3