3.1451 \(\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^7} \, dx\)

Optimal. Leaf size=65 \[ \frac{2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac{(b d-a e)^2}{6 e^3 (d+e x)^6}-\frac{b^2}{4 e^3 (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.09123, 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 \[ \frac{2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac{(b d-a e)^2}{6 e^3 (d+e x)^6}-\frac{b^2}{4 e^3 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 31.723, size = 58, normalized size = 0.89 \[ - \frac{b^{2}}{4 e^{3} \left (d + e x\right )^{4}} - \frac{2 b \left (a e - b d\right )}{5 e^{3} \left (d + e x\right )^{5}} - \frac{\left (a e - b d\right )^{2}}{6 e^{3} \left (d + e x\right )^{6}} \]

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)**7,x)

[Out]

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

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Mathematica [A]  time = 0.0395, size = 55, normalized size = 0.85 \[ -\frac{10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )}{60 e^3 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^7,x]

[Out]

-(10*a^2*e^2 + 4*a*b*e*(d + 6*e*x) + b^2*(d^2 + 6*d*e*x + 15*e^2*x^2))/(60*e^3*(
d + e*x)^6)

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Maple [A]  time = 0.008, size = 71, normalized size = 1.1 \[ -{\frac{2\,b \left ( ae-bd \right ) }{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}-{\frac{{a}^{2}{e}^{2}-2\,deab+{b}^{2}{d}^{2}}{6\,{e}^{3} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}} \]

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)^7,x)

[Out]

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

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Maxima [A]  time = 0.697357, size = 162, normalized size = 2.49 \[ -\frac{15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \,{\left (b^{2} d e + 4 \, a b e^{2}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]

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)^7,x, algorithm="maxima")

[Out]

-1/60*(15*b^2*e^2*x^2 + b^2*d^2 + 4*a*b*d*e + 10*a^2*e^2 + 6*(b^2*d*e + 4*a*b*e^
2)*x)/(e^9*x^6 + 6*d*e^8*x^5 + 15*d^2*e^7*x^4 + 20*d^3*e^6*x^3 + 15*d^4*e^5*x^2
+ 6*d^5*e^4*x + d^6*e^3)

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Fricas [A]  time = 0.196676, size = 162, normalized size = 2.49 \[ -\frac{15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \,{\left (b^{2} d e + 4 \, a b e^{2}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]

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)^7,x, algorithm="fricas")

[Out]

-1/60*(15*b^2*e^2*x^2 + b^2*d^2 + 4*a*b*d*e + 10*a^2*e^2 + 6*(b^2*d*e + 4*a*b*e^
2)*x)/(e^9*x^6 + 6*d*e^8*x^5 + 15*d^2*e^7*x^4 + 20*d^3*e^6*x^3 + 15*d^4*e^5*x^2
+ 6*d^5*e^4*x + d^6*e^3)

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Sympy [A]  time = 5.28379, size = 128, normalized size = 1.97 \[ - \frac{10 a^{2} e^{2} + 4 a b d e + b^{2} d^{2} + 15 b^{2} e^{2} x^{2} + x \left (24 a b e^{2} + 6 b^{2} d e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \]

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)**7,x)

[Out]

-(10*a**2*e**2 + 4*a*b*d*e + b**2*d**2 + 15*b**2*e**2*x**2 + x*(24*a*b*e**2 + 6*
b**2*d*e))/(60*d**6*e**3 + 360*d**5*e**4*x + 900*d**4*e**5*x**2 + 1200*d**3*e**6
*x**3 + 900*d**2*e**7*x**4 + 360*d*e**8*x**5 + 60*e**9*x**6)

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GIAC/XCAS [A]  time = 0.209703, size = 81, normalized size = 1.25 \[ -\frac{{\left (15 \, b^{2} x^{2} e^{2} + 6 \, b^{2} d x e + b^{2} d^{2} + 24 \, a b x e^{2} + 4 \, a b d e + 10 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{6}} \]

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)^7,x, algorithm="giac")

[Out]

-1/60*(15*b^2*x^2*e^2 + 6*b^2*d*x*e + b^2*d^2 + 24*a*b*x*e^2 + 4*a*b*d*e + 10*a^
2*e^2)*e^(-3)/(x*e + d)^6