3.1450 \(\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^6} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.094707, 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 \[ \frac{b (b d-a e)}{2 e^3 (d+e x)^4}-\frac{(b d-a e)^2}{5 e^3 (d+e x)^5}-\frac{b^2}{3 e^3 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 31.6695, size = 56, normalized size = 0.86 \[ - \frac{b^{2}}{3 e^{3} \left (d + e x\right )^{3}} - \frac{b \left (a e - b d\right )}{2 e^{3} \left (d + e x\right )^{4}} - \frac{\left (a e - b d\right )^{2}}{5 e^{3} \left (d + e x\right )^{5}} \]

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)**6,x)

[Out]

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

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Mathematica [A]  time = 0.0492028, size = 55, normalized size = 0.85 \[ -\frac{6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )}{30 e^3 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^6,x]

[Out]

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

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

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)^6,x)

[Out]

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

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Maxima [A]  time = 0.688273, size = 147, normalized size = 2.26 \[ -\frac{10 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 3 \, a b d e + 6 \, a^{2} e^{2} + 5 \,{\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{30 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]

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)^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.196158, size = 147, normalized size = 2.26 \[ -\frac{10 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 3 \, a b d e + 6 \, a^{2} e^{2} + 5 \,{\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{30 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.44098, size = 116, normalized size = 1.78 \[ - \frac{6 a^{2} e^{2} + 3 a b d e + b^{2} d^{2} + 10 b^{2} e^{2} x^{2} + x \left (15 a b e^{2} + 5 b^{2} d e\right )}{30 d^{5} e^{3} + 150 d^{4} e^{4} x + 300 d^{3} e^{5} x^{2} + 300 d^{2} e^{6} x^{3} + 150 d e^{7} x^{4} + 30 e^{8} x^{5}} \]

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)**6,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.210131, size = 81, normalized size = 1.25 \[ -\frac{{\left (10 \, b^{2} x^{2} e^{2} + 5 \, b^{2} d x e + b^{2} d^{2} + 15 \, a b x e^{2} + 3 \, a b d e + 6 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{30 \,{\left (x e + d\right )}^{5}} \]

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)^6,x, algorithm="giac")

[Out]

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