3.1511 \(\int \frac{1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=107 \[ -\frac{e^3 \log (a+b x)}{(b d-a e)^4}+\frac{e^3 \log (d+e x)}{(b d-a e)^4}-\frac{e^2}{(a+b x) (b d-a e)^3}+\frac{e}{2 (a+b x)^2 (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.176583, antiderivative size = 107, 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 \[ -\frac{e^3 \log (a+b x)}{(b d-a e)^4}+\frac{e^3 \log (d+e x)}{(b d-a e)^4}-\frac{e^2}{(a+b x) (b d-a e)^3}+\frac{e}{2 (a+b x)^2 (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 56.7139, size = 88, normalized size = 0.82 \[ - \frac{e^{3} \log{\left (a + b x \right )}}{\left (a e - b d\right )^{4}} + \frac{e^{3} \log{\left (d + e x \right )}}{\left (a e - b d\right )^{4}} + \frac{e^{2}}{\left (a + b x\right ) \left (a e - b d\right )^{3}} + \frac{e}{2 \left (a + b x\right )^{2} \left (a e - b d\right )^{2}} + \frac{1}{3 \left (a + b x\right )^{3} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0847369, size = 107, normalized size = 1. \[ -\frac{e^3 \log (a+b x)}{(b d-a e)^4}+\frac{e^3 \log (d+e x)}{(b d-a e)^4}-\frac{e^2}{(a+b x) (b d-a e)^3}+\frac{e}{2 (a+b x)^2 (b d-a e)^2}+\frac{1}{3 (a+b x)^3 (a e-b d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 103, normalized size = 1. \[{\frac{1}{ \left ( 3\,ae-3\,bd \right ) \left ( bx+a \right ) ^{3}}}+{\frac{e}{2\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{2}}}+{\frac{{e}^{2}}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}-{\frac{{e}^{3}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{4}}}+{\frac{{e}^{3}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.702517, size = 487, normalized size = 4.55 \[ -\frac{e^{3} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac{e^{3} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac{6 \, b^{2} e^{2} x^{2} + 2 \, b^{2} d^{2} - 7 \, a b d e + 11 \, a^{2} e^{2} - 3 \,{\left (b^{2} d e - 5 \, a b e^{2}\right )} x}{6 \,{\left (a^{3} b^{3} d^{3} - 3 \, a^{4} b^{2} d^{2} e + 3 \, a^{5} b d e^{2} - a^{6} e^{3} +{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{3} + 3 \,{\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.212251, size = 574, normalized size = 5.36 \[ -\frac{2 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 5 \, a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (a^{3} b^{4} d^{4} - 4 \, a^{4} b^{3} d^{3} e + 6 \, a^{5} b^{2} d^{2} e^{2} - 4 \, a^{6} b d e^{3} + a^{7} e^{4} +{\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{6} d^{4} - 4 \, a^{2} b^{5} d^{3} e + 6 \, a^{3} b^{4} d^{2} e^{2} - 4 \, a^{4} b^{3} d e^{3} + a^{5} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{5} d^{4} - 4 \, a^{3} b^{4} d^{3} e + 6 \, a^{4} b^{3} d^{2} e^{2} - 4 \, a^{5} b^{2} d e^{3} + a^{6} b e^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(2*b^3*d^3 - 9*a*b^2*d^2*e + 18*a^2*b*d*e^2 - 11*a^3*e^3 + 6*(b^3*d*e^2 - a
*b^2*e^3)*x^2 - 3*(b^3*d^2*e - 6*a*b^2*d*e^2 + 5*a^2*b*e^3)*x + 6*(b^3*e^3*x^3 +
3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*log(b*x + a) - 6*(b^3*e^3*x^3 + 3*a*
b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*log(e*x + d))/(a^3*b^4*d^4 - 4*a^4*b^3*d^
3*e + 6*a^5*b^2*d^2*e^2 - 4*a^6*b*d*e^3 + a^7*e^4 + (b^7*d^4 - 4*a*b^6*d^3*e + 6
*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)*x^3 + 3*(a*b^6*d^4 - 4*a^2*b^5
*d^3*e + 6*a^3*b^4*d^2*e^2 - 4*a^4*b^3*d*e^3 + a^5*b^2*e^4)*x^2 + 3*(a^2*b^5*d^4
- 4*a^3*b^4*d^3*e + 6*a^4*b^3*d^2*e^2 - 4*a^5*b^2*d*e^3 + a^6*b*e^4)*x)

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Sympy [A]  time = 6.84504, size = 570, normalized size = 5.33 \[ \frac{e^{3} \log{\left (x + \frac{- \frac{a^{5} e^{8}}{\left (a e - b d\right )^{4}} + \frac{5 a^{4} b d e^{7}}{\left (a e - b d\right )^{4}} - \frac{10 a^{3} b^{2} d^{2} e^{6}}{\left (a e - b d\right )^{4}} + \frac{10 a^{2} b^{3} d^{3} e^{5}}{\left (a e - b d\right )^{4}} - \frac{5 a b^{4} d^{4} e^{4}}{\left (a e - b d\right )^{4}} + a e^{4} + \frac{b^{5} d^{5} e^{3}}{\left (a e - b d\right )^{4}} + b d e^{3}}{2 b e^{4}} \right )}}{\left (a e - b d\right )^{4}} - \frac{e^{3} \log{\left (x + \frac{\frac{a^{5} e^{8}}{\left (a e - b d\right )^{4}} - \frac{5 a^{4} b d e^{7}}{\left (a e - b d\right )^{4}} + \frac{10 a^{3} b^{2} d^{2} e^{6}}{\left (a e - b d\right )^{4}} - \frac{10 a^{2} b^{3} d^{3} e^{5}}{\left (a e - b d\right )^{4}} + \frac{5 a b^{4} d^{4} e^{4}}{\left (a e - b d\right )^{4}} + a e^{4} - \frac{b^{5} d^{5} e^{3}}{\left (a e - b d\right )^{4}} + b d e^{3}}{2 b e^{4}} \right )}}{\left (a e - b d\right )^{4}} + \frac{11 a^{2} e^{2} - 7 a b d e + 2 b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (15 a b e^{2} - 3 b^{2} d e\right )}{6 a^{6} e^{3} - 18 a^{5} b d e^{2} + 18 a^{4} b^{2} d^{2} e - 6 a^{3} b^{3} d^{3} + x^{3} \left (6 a^{3} b^{3} e^{3} - 18 a^{2} b^{4} d e^{2} + 18 a b^{5} d^{2} e - 6 b^{6} d^{3}\right ) + x^{2} \left (18 a^{4} b^{2} e^{3} - 54 a^{3} b^{3} d e^{2} + 54 a^{2} b^{4} d^{2} e - 18 a b^{5} d^{3}\right ) + x \left (18 a^{5} b e^{3} - 54 a^{4} b^{2} d e^{2} + 54 a^{3} b^{3} d^{2} e - 18 a^{2} b^{4} d^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**3*log(x + (-a**5*e**8/(a*e - b*d)**4 + 5*a**4*b*d*e**7/(a*e - b*d)**4 - 10*a*
*3*b**2*d**2*e**6/(a*e - b*d)**4 + 10*a**2*b**3*d**3*e**5/(a*e - b*d)**4 - 5*a*b
**4*d**4*e**4/(a*e - b*d)**4 + a*e**4 + b**5*d**5*e**3/(a*e - b*d)**4 + b*d*e**3
)/(2*b*e**4))/(a*e - b*d)**4 - e**3*log(x + (a**5*e**8/(a*e - b*d)**4 - 5*a**4*b
*d*e**7/(a*e - b*d)**4 + 10*a**3*b**2*d**2*e**6/(a*e - b*d)**4 - 10*a**2*b**3*d*
*3*e**5/(a*e - b*d)**4 + 5*a*b**4*d**4*e**4/(a*e - b*d)**4 + a*e**4 - b**5*d**5*
e**3/(a*e - b*d)**4 + b*d*e**3)/(2*b*e**4))/(a*e - b*d)**4 + (11*a**2*e**2 - 7*a
*b*d*e + 2*b**2*d**2 + 6*b**2*e**2*x**2 + x*(15*a*b*e**2 - 3*b**2*d*e))/(6*a**6*
e**3 - 18*a**5*b*d*e**2 + 18*a**4*b**2*d**2*e - 6*a**3*b**3*d**3 + x**3*(6*a**3*
b**3*e**3 - 18*a**2*b**4*d*e**2 + 18*a*b**5*d**2*e - 6*b**6*d**3) + x**2*(18*a**
4*b**2*e**3 - 54*a**3*b**3*d*e**2 + 54*a**2*b**4*d**2*e - 18*a*b**5*d**3) + x*(1
8*a**5*b*e**3 - 54*a**4*b**2*d*e**2 + 54*a**3*b**3*d**2*e - 18*a**2*b**4*d**3))

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GIAC/XCAS [A]  time = 0.212459, size = 316, normalized size = 2.95 \[ -\frac{b e^{3}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac{e^{4}{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac{2 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 5 \, a^{2} b e^{3}\right )} x}{6 \,{\left (b d - a e\right )}^{4}{\left (b x + a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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