3.1503 \(\int \frac{1}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

Optimal. Leaf size=133 \[ -\frac{b^3}{(a+b x) (b d-a e)^4}-\frac{4 b^3 e \log (a+b x)}{(b d-a e)^5}+\frac{4 b^3 e \log (d+e x)}{(b d-a e)^5}-\frac{3 b^2 e}{(d+e x) (b d-a e)^4}-\frac{b e}{(d+e x)^2 (b d-a e)^3}-\frac{e}{3 (d+e x)^3 (b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.253951, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{b^3}{(a+b x) (b d-a e)^4}-\frac{4 b^3 e \log (a+b x)}{(b d-a e)^5}+\frac{4 b^3 e \log (d+e x)}{(b d-a e)^5}-\frac{3 b^2 e}{(d+e x) (b d-a e)^4}-\frac{b e}{(d+e x)^2 (b d-a e)^3}-\frac{e}{3 (d+e x)^3 (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 86.1945, size = 117, normalized size = 0.88 \[ \frac{4 b^{3} e \log{\left (a + b x \right )}}{\left (a e - b d\right )^{5}} - \frac{4 b^{3} e \log{\left (d + e x \right )}}{\left (a e - b d\right )^{5}} - \frac{b^{3}}{\left (a + b x\right ) \left (a e - b d\right )^{4}} - \frac{3 b^{2} e}{\left (d + e x\right ) \left (a e - b d\right )^{4}} + \frac{b e}{\left (d + e x\right )^{2} \left (a e - b d\right )^{3}} - \frac{e}{3 \left (d + e x\right )^{3} \left (a e - b d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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Mathematica [A]  time = 0.230602, size = 120, normalized size = 0.9 \[ \frac{-\frac{3 b^3 (b d-a e)}{a+b x}-12 b^3 e \log (a+b x)-\frac{9 b^2 e (b d-a e)}{d+e x}-\frac{3 b e (b d-a e)^2}{(d+e x)^2}+\frac{e (a e-b d)^3}{(d+e x)^3}+12 b^3 e \log (d+e x)}{3 (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

((-3*b^3*(b*d - a*e))/(a + b*x) + (e*(-(b*d) + a*e)^3)/(d + e*x)^3 - (3*b*e*(b*d
- a*e)^2)/(d + e*x)^2 - (9*b^2*e*(b*d - a*e))/(d + e*x) - 12*b^3*e*Log[a + b*x]
+ 12*b^3*e*Log[d + e*x])/(3*(b*d - a*e)^5)

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Maple [A]  time = 0.02, size = 131, normalized size = 1. \[ -{\frac{{b}^{3}}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}+4\,{\frac{{b}^{3}e\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{5}}}-{\frac{e}{3\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{3}}}-4\,{\frac{{b}^{3}e\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{5}}}-3\,{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}+{\frac{be}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

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

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Maxima [A]  time = 0.709142, size = 809, normalized size = 6.08 \[ -\frac{4 \, b^{3} e \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac{4 \, b^{3} e \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{12 \, b^{3} e^{3} x^{3} + 3 \, b^{3} d^{3} + 13 \, a b^{2} d^{2} e - 5 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (5 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (11 \, b^{3} d^{2} e + 8 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x}{3 \,{\left (a b^{4} d^{7} - 4 \, a^{2} b^{3} d^{6} e + 6 \, a^{3} b^{2} d^{5} e^{2} - 4 \, a^{4} b d^{4} e^{3} + a^{5} d^{3} e^{4} +{\left (b^{5} d^{4} e^{3} - 4 \, a b^{4} d^{3} e^{4} + 6 \, a^{2} b^{3} d^{2} e^{5} - 4 \, a^{3} b^{2} d e^{6} + a^{4} b e^{7}\right )} x^{4} +{\left (3 \, b^{5} d^{5} e^{2} - 11 \, a b^{4} d^{4} e^{3} + 14 \, a^{2} b^{3} d^{3} e^{4} - 6 \, a^{3} b^{2} d^{2} e^{5} - a^{4} b d e^{6} + a^{5} e^{7}\right )} x^{3} + 3 \,{\left (b^{5} d^{6} e - 3 \, a b^{4} d^{5} e^{2} + 2 \, a^{2} b^{3} d^{4} e^{3} + 2 \, a^{3} b^{2} d^{3} e^{4} - 3 \, a^{4} b d^{2} e^{5} + a^{5} d e^{6}\right )} x^{2} +{\left (b^{5} d^{7} - a b^{4} d^{6} e - 6 \, a^{2} b^{3} d^{5} e^{2} + 14 \, a^{3} b^{2} d^{4} e^{3} - 11 \, a^{4} b d^{3} e^{4} + 3 \, a^{5} d^{2} e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^4),x, algorithm="maxima")

[Out]

-4*b^3*e*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2
*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5) + 4*b^3*e*log(e*x + d)/(b^5*d^5 - 5*a*b^4*d^
4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5) - 1/3*(
12*b^3*e^3*x^3 + 3*b^3*d^3 + 13*a*b^2*d^2*e - 5*a^2*b*d*e^2 + a^3*e^3 + 6*(5*b^3
*d*e^2 + a*b^2*e^3)*x^2 + 2*(11*b^3*d^2*e + 8*a*b^2*d*e^2 - a^2*b*e^3)*x)/(a*b^4
*d^7 - 4*a^2*b^3*d^6*e + 6*a^3*b^2*d^5*e^2 - 4*a^4*b*d^4*e^3 + a^5*d^3*e^4 + (b^
5*d^4*e^3 - 4*a*b^4*d^3*e^4 + 6*a^2*b^3*d^2*e^5 - 4*a^3*b^2*d*e^6 + a^4*b*e^7)*x
^4 + (3*b^5*d^5*e^2 - 11*a*b^4*d^4*e^3 + 14*a^2*b^3*d^3*e^4 - 6*a^3*b^2*d^2*e^5
- a^4*b*d*e^6 + a^5*e^7)*x^3 + 3*(b^5*d^6*e - 3*a*b^4*d^5*e^2 + 2*a^2*b^3*d^4*e^
3 + 2*a^3*b^2*d^3*e^4 - 3*a^4*b*d^2*e^5 + a^5*d*e^6)*x^2 + (b^5*d^7 - a*b^4*d^6*
e - 6*a^2*b^3*d^5*e^2 + 14*a^3*b^2*d^4*e^3 - 11*a^4*b*d^3*e^4 + 3*a^5*d^2*e^5)*x
)

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Fricas [A]  time = 0.220492, size = 1017, normalized size = 7.65 \[ -\frac{3 \, b^{4} d^{4} + 10 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 6 \, a^{3} b d e^{3} - a^{4} e^{4} + 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (5 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (11 \, b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} e^{4} x^{4} + a b^{3} d^{3} e +{\left (3 \, b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3}\right )} x^{2} +{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (b x + a\right ) - 12 \,{\left (b^{4} e^{4} x^{4} + a b^{3} d^{3} e +{\left (3 \, b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3}\right )} x^{2} +{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (a b^{5} d^{8} - 5 \, a^{2} b^{4} d^{7} e + 10 \, a^{3} b^{3} d^{6} e^{2} - 10 \, a^{4} b^{2} d^{5} e^{3} + 5 \, a^{5} b d^{4} e^{4} - a^{6} d^{3} e^{5} +{\left (b^{6} d^{5} e^{3} - 5 \, a b^{5} d^{4} e^{4} + 10 \, a^{2} b^{4} d^{3} e^{5} - 10 \, a^{3} b^{3} d^{2} e^{6} + 5 \, a^{4} b^{2} d e^{7} - a^{5} b e^{8}\right )} x^{4} +{\left (3 \, b^{6} d^{6} e^{2} - 14 \, a b^{5} d^{5} e^{3} + 25 \, a^{2} b^{4} d^{4} e^{4} - 20 \, a^{3} b^{3} d^{3} e^{5} + 5 \, a^{4} b^{2} d^{2} e^{6} + 2 \, a^{5} b d e^{7} - a^{6} e^{8}\right )} x^{3} + 3 \,{\left (b^{6} d^{7} e - 4 \, a b^{5} d^{6} e^{2} + 5 \, a^{2} b^{4} d^{5} e^{3} - 5 \, a^{4} b^{2} d^{3} e^{5} + 4 \, a^{5} b d^{2} e^{6} - a^{6} d e^{7}\right )} x^{2} +{\left (b^{6} d^{8} - 2 \, a b^{5} d^{7} e - 5 \, a^{2} b^{4} d^{6} e^{2} + 20 \, a^{3} b^{3} d^{5} e^{3} - 25 \, a^{4} b^{2} d^{4} e^{4} + 14 \, a^{5} b d^{3} e^{5} - 3 \, a^{6} d^{2} e^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^4),x, algorithm="fricas")

[Out]

-1/3*(3*b^4*d^4 + 10*a*b^3*d^3*e - 18*a^2*b^2*d^2*e^2 + 6*a^3*b*d*e^3 - a^4*e^4
+ 12*(b^4*d*e^3 - a*b^3*e^4)*x^3 + 6*(5*b^4*d^2*e^2 - 4*a*b^3*d*e^3 - a^2*b^2*e^
4)*x^2 + 2*(11*b^4*d^3*e - 3*a*b^3*d^2*e^2 - 9*a^2*b^2*d*e^3 + a^3*b*e^4)*x + 12
*(b^4*e^4*x^4 + a*b^3*d^3*e + (3*b^4*d*e^3 + a*b^3*e^4)*x^3 + 3*(b^4*d^2*e^2 + a
*b^3*d*e^3)*x^2 + (b^4*d^3*e + 3*a*b^3*d^2*e^2)*x)*log(b*x + a) - 12*(b^4*e^4*x^
4 + a*b^3*d^3*e + (3*b^4*d*e^3 + a*b^3*e^4)*x^3 + 3*(b^4*d^2*e^2 + a*b^3*d*e^3)*
x^2 + (b^4*d^3*e + 3*a*b^3*d^2*e^2)*x)*log(e*x + d))/(a*b^5*d^8 - 5*a^2*b^4*d^7*
e + 10*a^3*b^3*d^6*e^2 - 10*a^4*b^2*d^5*e^3 + 5*a^5*b*d^4*e^4 - a^6*d^3*e^5 + (b
^6*d^5*e^3 - 5*a*b^5*d^4*e^4 + 10*a^2*b^4*d^3*e^5 - 10*a^3*b^3*d^2*e^6 + 5*a^4*b
^2*d*e^7 - a^5*b*e^8)*x^4 + (3*b^6*d^6*e^2 - 14*a*b^5*d^5*e^3 + 25*a^2*b^4*d^4*e
^4 - 20*a^3*b^3*d^3*e^5 + 5*a^4*b^2*d^2*e^6 + 2*a^5*b*d*e^7 - a^6*e^8)*x^3 + 3*(
b^6*d^7*e - 4*a*b^5*d^6*e^2 + 5*a^2*b^4*d^5*e^3 - 5*a^4*b^2*d^3*e^5 + 4*a^5*b*d^
2*e^6 - a^6*d*e^7)*x^2 + (b^6*d^8 - 2*a*b^5*d^7*e - 5*a^2*b^4*d^6*e^2 + 20*a^3*b
^3*d^5*e^3 - 25*a^4*b^2*d^4*e^4 + 14*a^5*b*d^3*e^5 - 3*a^6*d^2*e^6)*x)

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Sympy [A]  time = 11.766, size = 881, normalized size = 6.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

-4*b**3*e*log(x + (-4*a**6*b**3*e**7/(a*e - b*d)**5 + 24*a**5*b**4*d*e**6/(a*e -
b*d)**5 - 60*a**4*b**5*d**2*e**5/(a*e - b*d)**5 + 80*a**3*b**6*d**3*e**4/(a*e -
b*d)**5 - 60*a**2*b**7*d**4*e**3/(a*e - b*d)**5 + 24*a*b**8*d**5*e**2/(a*e - b*
d)**5 + 4*a*b**3*e**2 - 4*b**9*d**6*e/(a*e - b*d)**5 + 4*b**4*d*e)/(8*b**4*e**2)
)/(a*e - b*d)**5 + 4*b**3*e*log(x + (4*a**6*b**3*e**7/(a*e - b*d)**5 - 24*a**5*b
**4*d*e**6/(a*e - b*d)**5 + 60*a**4*b**5*d**2*e**5/(a*e - b*d)**5 - 80*a**3*b**6
*d**3*e**4/(a*e - b*d)**5 + 60*a**2*b**7*d**4*e**3/(a*e - b*d)**5 - 24*a*b**8*d*
*5*e**2/(a*e - b*d)**5 + 4*a*b**3*e**2 + 4*b**9*d**6*e/(a*e - b*d)**5 + 4*b**4*d
*e)/(8*b**4*e**2))/(a*e - b*d)**5 - (a**3*e**3 - 5*a**2*b*d*e**2 + 13*a*b**2*d**
2*e + 3*b**3*d**3 + 12*b**3*e**3*x**3 + x**2*(6*a*b**2*e**3 + 30*b**3*d*e**2) +
x*(-2*a**2*b*e**3 + 16*a*b**2*d*e**2 + 22*b**3*d**2*e))/(3*a**5*d**3*e**4 - 12*a
**4*b*d**4*e**3 + 18*a**3*b**2*d**5*e**2 - 12*a**2*b**3*d**6*e + 3*a*b**4*d**7 +
x**4*(3*a**4*b*e**7 - 12*a**3*b**2*d*e**6 + 18*a**2*b**3*d**2*e**5 - 12*a*b**4*
d**3*e**4 + 3*b**5*d**4*e**3) + x**3*(3*a**5*e**7 - 3*a**4*b*d*e**6 - 18*a**3*b*
*2*d**2*e**5 + 42*a**2*b**3*d**3*e**4 - 33*a*b**4*d**4*e**3 + 9*b**5*d**5*e**2)
+ x**2*(9*a**5*d*e**6 - 27*a**4*b*d**2*e**5 + 18*a**3*b**2*d**3*e**4 + 18*a**2*b
**3*d**4*e**3 - 27*a*b**4*d**5*e**2 + 9*b**5*d**6*e) + x*(9*a**5*d**2*e**5 - 33*
a**4*b*d**3*e**4 + 42*a**3*b**2*d**4*e**3 - 18*a**2*b**3*d**5*e**2 - 3*a*b**4*d*
*6*e + 3*b**5*d**7))

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GIAC/XCAS [A]  time = 0.216429, size = 455, normalized size = 3.42 \[ -\frac{4 \, b^{4} e{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} + \frac{4 \, b^{3} e^{2}{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac{3 \, b^{4} d^{4} + 10 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 6 \, a^{3} b d e^{3} - a^{4} e^{4} + 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (5 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (11 \, b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{3 \,{\left (b d - a e\right )}^{5}{\left (b x + a\right )}{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^4),x, algorithm="giac")

[Out]

-4*b^4*e*ln(abs(b*x + a))/(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3
*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5) + 4*b^3*e^2*ln(abs(x*e + d))/(b^5*d^
5*e - 5*a*b^4*d^4*e^2 + 10*a^2*b^3*d^3*e^3 - 10*a^3*b^2*d^2*e^4 + 5*a^4*b*d*e^5
- a^5*e^6) - 1/3*(3*b^4*d^4 + 10*a*b^3*d^3*e - 18*a^2*b^2*d^2*e^2 + 6*a^3*b*d*e^
3 - a^4*e^4 + 12*(b^4*d*e^3 - a*b^3*e^4)*x^3 + 6*(5*b^4*d^2*e^2 - 4*a*b^3*d*e^3
- a^2*b^2*e^4)*x^2 + 2*(11*b^4*d^3*e - 3*a*b^3*d^2*e^2 - 9*a^2*b^2*d*e^3 + a^3*b
*e^4)*x)/((b*d - a*e)^5*(b*x + a)*(x*e + d)^3)