3.1449 \(\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^5} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0996398, antiderivative size = 65, 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 \[ \frac{2 b (b d-a e)}{3 e^3 (d+e x)^3}-\frac{(b d-a e)^2}{4 e^3 (d+e x)^4}-\frac{b^2}{2 e^3 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 31.3469, size = 58, normalized size = 0.89 \[ - \frac{b^{2}}{2 e^{3} \left (d + e x\right )^{2}} - \frac{2 b \left (a e - b d\right )}{3 e^{3} \left (d + e x\right )^{3}} - \frac{\left (a e - b d\right )^{2}}{4 e^{3} \left (d + e x\right )^{4}} \]

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)**5,x)

[Out]

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

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Mathematica [A]  time = 0.0397531, size = 55, normalized size = 0.85 \[ -\frac{3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^5,x]

[Out]

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

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

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)^5,x)

[Out]

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

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Maxima [A]  time = 0.695402, size = 132, normalized size = 2.03 \[ -\frac{6 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 2 \, a b d e + 3 \, a^{2} e^{2} + 4 \,{\left (b^{2} d e + 2 \, a b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

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)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.197234, size = 132, normalized size = 2.03 \[ -\frac{6 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 2 \, a b d e + 3 \, a^{2} e^{2} + 4 \,{\left (b^{2} d e + 2 \, a b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

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)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.76947, size = 104, normalized size = 1.6 \[ - \frac{3 a^{2} e^{2} + 2 a b d e + b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (8 a b e^{2} + 4 b^{2} d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]

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)**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.210628, size = 132, normalized size = 2.03 \[ -\frac{1}{12} \,{\left (\frac{6 \, b^{2} e}{{\left (x e + d\right )}^{2}} - \frac{8 \, b^{2} d e}{{\left (x e + d\right )}^{3}} + \frac{3 \, b^{2} d^{2} e}{{\left (x e + d\right )}^{4}} + \frac{8 \, a b e^{2}}{{\left (x e + d\right )}^{3}} - \frac{6 \, a b d e^{2}}{{\left (x e + d\right )}^{4}} + \frac{3 \, a^{2} e^{3}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-4\right )} \]

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)^5,x, algorithm="giac")

[Out]

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