3.1519 \(\int \frac{(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=58 \[ \frac{e (d+e x)^4}{20 (a+b x)^4 (b d-a e)^2}-\frac{(d+e x)^4}{5 (a+b x)^5 (b d-a e)} \]

[Out]

-(d + e*x)^4/(5*(b*d - a*e)*(a + b*x)^5) + (e*(d + e*x)^4)/(20*(b*d - a*e)^2*(a
+ b*x)^4)

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Rubi [A]  time = 0.0507115, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{e (d+e x)^4}{20 (a+b x)^4 (b d-a e)^2}-\frac{(d+e x)^4}{5 (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

-(d + e*x)^4/(5*(b*d - a*e)*(a + b*x)^5) + (e*(d + e*x)^4)/(20*(b*d - a*e)^2*(a
+ b*x)^4)

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Rubi in Sympy [A]  time = 23.724, size = 46, normalized size = 0.79 \[ \frac{e \left (d + e x\right )^{4}}{20 \left (a + b x\right )^{4} \left (a e - b d\right )^{2}} + \frac{\left (d + e x\right )^{4}}{5 \left (a + b x\right )^{5} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(d + e*x)**4/(20*(a + b*x)**4*(a*e - b*d)**2) + (d + e*x)**4/(5*(a + b*x)**5*(
a*e - b*d))

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Mathematica [A]  time = 0.0665712, size = 97, normalized size = 1.67 \[ -\frac{a^3 e^3+a^2 b e^2 (2 d+5 e x)+a b^2 e \left (3 d^2+10 d e x+10 e^2 x^2\right )+b^3 \left (4 d^3+15 d^2 e x+20 d e^2 x^2+10 e^3 x^3\right )}{20 b^4 (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-(a^3*e^3 + a^2*b*e^2*(2*d + 5*e*x) + a*b^2*e*(3*d^2 + 10*d*e*x + 10*e^2*x^2) +
b^3*(4*d^3 + 15*d^2*e*x + 20*d*e^2*x^2 + 10*e^3*x^3))/(20*b^4*(a + b*x)^5)

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Maple [B]  time = 0.01, size = 121, normalized size = 2.1 \[ -{\frac{-{a}^{3}{e}^{3}+3\,{a}^{2}bd{e}^{2}-3\,a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}-{\frac{3\,e \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}+{\frac{{e}^{2} \left ( ae-bd \right ) }{{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{{e}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.690887, size = 216, normalized size = 3.72 \[ -\frac{10 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 2 \, a^{2} b d e^{2} + a^{3} e^{3} + 10 \,{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (3 \, b^{3} d^{2} e + 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{20 \,{\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.198352, size = 216, normalized size = 3.72 \[ -\frac{10 \, b^{3} e^{3} x^{3} + 4 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 2 \, a^{2} b d e^{2} + a^{3} e^{3} + 10 \,{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (3 \, b^{3} d^{2} e + 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{20 \,{\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.3567, size = 170, normalized size = 2.93 \[ - \frac{a^{3} e^{3} + 2 a^{2} b d e^{2} + 3 a b^{2} d^{2} e + 4 b^{3} d^{3} + 10 b^{3} e^{3} x^{3} + x^{2} \left (10 a b^{2} e^{3} + 20 b^{3} d e^{2}\right ) + x \left (5 a^{2} b e^{3} + 10 a b^{2} d e^{2} + 15 b^{3} d^{2} e\right )}{20 a^{5} b^{4} + 100 a^{4} b^{5} x + 200 a^{3} b^{6} x^{2} + 200 a^{2} b^{7} x^{3} + 100 a b^{8} x^{4} + 20 b^{9} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a**3*e**3 + 2*a**2*b*d*e**2 + 3*a*b**2*d**2*e + 4*b**3*d**3 + 10*b**3*e**3*x**
3 + x**2*(10*a*b**2*e**3 + 20*b**3*d*e**2) + x*(5*a**2*b*e**3 + 10*a*b**2*d*e**2
+ 15*b**3*d**2*e))/(20*a**5*b**4 + 100*a**4*b**5*x + 200*a**3*b**6*x**2 + 200*a
**2*b**7*x**3 + 100*a*b**8*x**4 + 20*b**9*x**5)

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GIAC/XCAS [A]  time = 0.212333, size = 147, normalized size = 2.53 \[ -\frac{10 \, b^{3} x^{3} e^{3} + 20 \, b^{3} d x^{2} e^{2} + 15 \, b^{3} d^{2} x e + 4 \, b^{3} d^{3} + 10 \, a b^{2} x^{2} e^{3} + 10 \, a b^{2} d x e^{2} + 3 \, a b^{2} d^{2} e + 5 \, a^{2} b x e^{3} + 2 \, a^{2} b d e^{2} + a^{3} e^{3}}{20 \,{\left (b x + a\right )}^{5} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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