3.1520 \(\int \frac{(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=65 \[ -\frac{e (b d-a e)}{2 b^3 (a+b x)^4}-\frac{(b d-a e)^2}{5 b^3 (a+b x)^5}-\frac{e^2}{3 b^3 (a+b x)^3} \]

[Out]

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

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Rubi [A]  time = 0.107336, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{e (b d-a e)}{2 b^3 (a+b x)^4}-\frac{(b d-a e)^2}{5 b^3 (a+b x)^5}-\frac{e^2}{3 b^3 (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.5751, size = 54, normalized size = 0.83 \[ - \frac{e^{2}}{3 b^{3} \left (a + b x\right )^{3}} + \frac{e \left (a e - b d\right )}{2 b^{3} \left (a + b x\right )^{4}} - \frac{\left (a e - b d\right )^{2}}{5 b^{3} \left (a + b x\right )^{5}} \]

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0467927, size = 57, normalized size = 0.88 \[ -\frac{a^2 e^2+a b e (3 d+5 e x)+b^2 \left (6 d^2+15 d e x+10 e^2 x^2\right )}{30 b^3 (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2*e^2 + a*b*e*(3*d + 5*e*x) + b^2*(6*d^2 + 15*d*e*x + 10*e^2*x^2))/(30*b^3*(
a + b*x)^5)

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Maple [A]  time = 0.008, size = 71, normalized size = 1.1 \[{\frac{e \left ( ae-bd \right ) }{2\,{b}^{3} \left ( bx+a \right ) ^{4}}}-{\frac{{a}^{2}{e}^{2}-2\,aedb+{b}^{2}{d}^{2}}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}-{\frac{{e}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}} \]

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)^3,x)

[Out]

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

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Maxima [A]  time = 0.690112, size = 147, normalized size = 2.26 \[ -\frac{10 \, b^{2} e^{2} x^{2} + 6 \, b^{2} d^{2} + 3 \, a b d e + a^{2} e^{2} + 5 \,{\left (3 \, b^{2} d e + a b e^{2}\right )} x}{30 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]

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)^3,x, algorithm="maxima")

[Out]

-1/30*(10*b^2*e^2*x^2 + 6*b^2*d^2 + 3*a*b*d*e + a^2*e^2 + 5*(3*b^2*d*e + a*b*e^2
)*x)/(b^8*x^5 + 5*a*b^7*x^4 + 10*a^2*b^6*x^3 + 10*a^3*b^5*x^2 + 5*a^4*b^4*x + a^
5*b^3)

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Fricas [A]  time = 0.19758, size = 147, normalized size = 2.26 \[ -\frac{10 \, b^{2} e^{2} x^{2} + 6 \, b^{2} d^{2} + 3 \, a b d e + a^{2} e^{2} + 5 \,{\left (3 \, b^{2} d e + a b e^{2}\right )} x}{30 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

-1/30*(10*b^2*e^2*x^2 + 6*b^2*d^2 + 3*a*b*d*e + a^2*e^2 + 5*(3*b^2*d*e + a*b*e^2
)*x)/(b^8*x^5 + 5*a*b^7*x^4 + 10*a^2*b^6*x^3 + 10*a^3*b^5*x^2 + 5*a^4*b^4*x + a^
5*b^3)

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Sympy [A]  time = 4.46008, size = 116, normalized size = 1.78 \[ - \frac{a^{2} e^{2} + 3 a b d e + 6 b^{2} d^{2} + 10 b^{2} e^{2} x^{2} + x \left (5 a b e^{2} + 15 b^{2} d e\right )}{30 a^{5} b^{3} + 150 a^{4} b^{4} x + 300 a^{3} b^{5} x^{2} + 300 a^{2} b^{6} x^{3} + 150 a b^{7} x^{4} + 30 b^{8} x^{5}} \]

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)**3,x)

[Out]

-(a**2*e**2 + 3*a*b*d*e + 6*b**2*d**2 + 10*b**2*e**2*x**2 + x*(5*a*b*e**2 + 15*b
**2*d*e))/(30*a**5*b**3 + 150*a**4*b**4*x + 300*a**3*b**5*x**2 + 300*a**2*b**6*x
**3 + 150*a*b**7*x**4 + 30*b**8*x**5)

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GIAC/XCAS [A]  time = 0.211503, size = 81, normalized size = 1.25 \[ -\frac{10 \, b^{2} x^{2} e^{2} + 15 \, b^{2} d x e + 6 \, b^{2} d^{2} + 5 \, a b x e^{2} + 3 \, a b d e + a^{2} e^{2}}{30 \,{\left (b x + a\right )}^{5} b^{3}} \]

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)^3,x, algorithm="giac")

[Out]

-1/30*(10*b^2*x^2*e^2 + 15*b^2*d*x*e + 6*b^2*d^2 + 5*a*b*x*e^2 + 3*a*b*d*e + a^2
*e^2)/((b*x + a)^5*b^3)