3.1947 \(\int \frac{(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0897226, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{2 e (b d-a e)}{3 b^3 (a+b x)^3}-\frac{(b d-a e)^2}{4 b^3 (a+b x)^4}-\frac{e^2}{2 b^3 (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.2779, size = 56, normalized size = 0.86 \[ - \frac{e^{2}}{2 b^{3} \left (a + b x\right )^{2}} + \frac{2 e \left (a e - b d\right )}{3 b^{3} \left (a + b x\right )^{3}} - \frac{\left (a e - b d\right )^{2}}{4 b^{3} \left (a + b x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0381164, size = 56, normalized size = 0.86 \[ -\frac{a^2 e^2+2 a b e (d+2 e x)+b^2 \left (3 d^2+8 d e x+6 e^2 x^2\right )}{12 b^3 (a+b x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2*e^2 + 2*a*b*e*(d + 2*e*x) + b^2*(3*d^2 + 8*d*e*x + 6*e^2*x^2))/(12*b^3*(a
+ b*x)^4)

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Maple [A]  time = 0.009, size = 71, normalized size = 1.1 \[ -{\frac{{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{4}}}+{\frac{2\,e \left ( ae-bd \right ) }{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}-{\frac{{e}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.718462, size = 132, normalized size = 2.03 \[ -\frac{6 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 2 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")

[Out]

-1/12*(6*b^2*e^2*x^2 + 3*b^2*d^2 + 2*a*b*d*e + a^2*e^2 + 4*(2*b^2*d*e + a*b*e^2)
*x)/(b^7*x^4 + 4*a*b^6*x^3 + 6*a^2*b^5*x^2 + 4*a^3*b^4*x + a^4*b^3)

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Fricas [A]  time = 0.278517, size = 132, normalized size = 2.03 \[ -\frac{6 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 2 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

-1/12*(6*b^2*e^2*x^2 + 3*b^2*d^2 + 2*a*b*d*e + a^2*e^2 + 4*(2*b^2*d*e + a*b*e^2)
*x)/(b^7*x^4 + 4*a*b^6*x^3 + 6*a^2*b^5*x^2 + 4*a^3*b^4*x + a^4*b^3)

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Sympy [A]  time = 3.7498, size = 104, normalized size = 1.6 \[ - \frac{a^{2} e^{2} + 2 a b d e + 3 b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (4 a b e^{2} + 8 b^{2} d e\right )}{12 a^{4} b^{3} + 48 a^{3} b^{4} x + 72 a^{2} b^{5} x^{2} + 48 a b^{6} x^{3} + 12 b^{7} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a**2*e**2 + 2*a*b*d*e + 3*b**2*d**2 + 6*b**2*e**2*x**2 + x*(4*a*b*e**2 + 8*b**
2*d*e))/(12*a**4*b**3 + 48*a**3*b**4*x + 72*a**2*b**5*x**2 + 48*a*b**6*x**3 + 12
*b**7*x**4)

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GIAC/XCAS [A]  time = 0.292116, size = 81, normalized size = 1.25 \[ -\frac{6 \, b^{2} x^{2} e^{2} + 8 \, b^{2} d x e + 3 \, b^{2} d^{2} + 4 \, a b x e^{2} + 2 \, a b d e + a^{2} e^{2}}{12 \,{\left (b x + a\right )}^{4} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")

[Out]

-1/12*(6*b^2*x^2*e^2 + 8*b^2*d*x*e + 3*b^2*d^2 + 4*a*b*x*e^2 + 2*a*b*d*e + a^2*e
^2)/((b*x + a)^4*b^3)