3.1707 \(\int \frac{A+B x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=199 \[ \frac{e^2 (B d-A e)}{(d+e x) (b d-a e)^4}+\frac{e^2 \log (a+b x) (a B e-4 A b e+3 b B d)}{(b d-a e)^5}-\frac{e^2 \log (d+e x) (a B e-4 A b e+3 b B d)}{(b d-a e)^5}+\frac{e (a B e-3 A b e+2 b B d)}{(a+b x) (b d-a e)^4}-\frac{a B e-2 A b e+b B d}{2 (a+b x)^2 (b d-a e)^3}-\frac{A b-a B}{3 (a+b x)^3 (b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.525541, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{e^2 (B d-A e)}{(d+e x) (b d-a e)^4}+\frac{e^2 \log (a+b x) (a B e-4 A b e+3 b B d)}{(b d-a e)^5}-\frac{e^2 \log (d+e x) (a B e-4 A b e+3 b B d)}{(b d-a e)^5}+\frac{e (a B e-3 A b e+2 b B d)}{(a+b x) (b d-a e)^4}-\frac{a B e-2 A b e+b B d}{2 (a+b x)^2 (b d-a e)^3}-\frac{A b-a B}{3 (a+b x)^3 (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 164.785, size = 189, normalized size = 0.95 \[ \frac{e^{2} \left (4 A b e - B a e - 3 B b d\right ) \log{\left (a + b x \right )}}{\left (a e - b d\right )^{5}} - \frac{e^{2} \left (4 A b e - B a e - 3 B b d\right ) \log{\left (d + e x \right )}}{\left (a e - b d\right )^{5}} - \frac{e^{2} \left (A e - B d\right )}{\left (d + e x\right ) \left (a e - b d\right )^{4}} - \frac{e \left (3 A b e - B a e - 2 B b d\right )}{\left (a + b x\right ) \left (a e - b d\right )^{4}} - \frac{2 A b e - B a e - B b d}{2 \left (a + b x\right )^{2} \left (a e - b d\right )^{3}} - \frac{A b - B a}{3 \left (a + b x\right )^{3} \left (a e - b d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.217704, size = 189, normalized size = 0.95 \[ \frac{\frac{6 e^2 (a e-b d) (A e-B d)}{d+e x}+6 e^2 \log (a+b x) (a B e-4 A b e+3 b B d)-6 e^2 \log (d+e x) (a B e-4 A b e+3 b B d)+\frac{2 (a B-A b) (b d-a e)^3}{(a+b x)^3}-\frac{3 (b d-a e)^2 (a B e-2 A b e+b B d)}{(a+b x)^2}+\frac{6 e (a e-b d) (-a B e+3 A b e-2 b B d)}{a+b x}}{6 (b d-a e)^5} \]

Antiderivative was successfully verified.

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

[Out]

((2*(-(A*b) + a*B)*(b*d - a*e)^3)/(a + b*x)^3 - (3*(b*d - a*e)^2*(b*B*d - 2*A*b*
e + a*B*e))/(a + b*x)^2 + (6*e*(-(b*d) + a*e)*(-2*b*B*d + 3*A*b*e - a*B*e))/(a +
b*x) + (6*e^2*(-(b*d) + a*e)*(-(B*d) + A*e))/(d + e*x) + 6*e^2*(3*b*B*d - 4*A*b
*e + a*B*e)*Log[a + b*x] - 6*e^2*(3*b*B*d - 4*A*b*e + a*B*e)*Log[d + e*x])/(6*(b
*d - a*e)^5)

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Maple [A]  time = 0.026, size = 365, normalized size = 1.8 \[ -{\frac{Abe}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{2}}}+{\frac{aBe}{2\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{2}}}+{\frac{Bbd}{2\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{2}}}-{\frac{Ab}{3\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{3}}}+{\frac{Ba}{3\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{3}}}-3\,{\frac{Ab{e}^{2}}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}+{\frac{aB{e}^{2}}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}+2\,{\frac{bBed}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}+4\,{\frac{{e}^{3}\ln \left ( bx+a \right ) Ab}{ \left ( ae-bd \right ) ^{5}}}-{\frac{{e}^{3}\ln \left ( bx+a \right ) aB}{ \left ( ae-bd \right ) ^{5}}}-3\,{\frac{{e}^{2}\ln \left ( bx+a \right ) Bbd}{ \left ( ae-bd \right ) ^{5}}}-{\frac{{e}^{3}A}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}+{\frac{Bd{e}^{2}}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}-4\,{\frac{{e}^{3}\ln \left ( ex+d \right ) Ab}{ \left ( ae-bd \right ) ^{5}}}+{\frac{{e}^{3}\ln \left ( ex+d \right ) aB}{ \left ( ae-bd \right ) ^{5}}}+3\,{\frac{{e}^{2}\ln \left ( ex+d \right ) Bbd}{ \left ( ae-bd \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.734678, size = 1022, normalized size = 5.14 \[ \frac{{\left (3 \, B b d e^{2} +{\left (B a - 4 \, A b\right )} e^{3}\right )} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{{\left (3 \, B b d e^{2} +{\left (B a - 4 \, A b\right )} e^{3}\right )} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{6 \, A a^{3} e^{3} +{\left (B a b^{2} + 2 \, A b^{3}\right )} d^{3} - 2 \,{\left (4 \, B a^{2} b + 5 \, A a b^{2}\right )} d^{2} e -{\left (17 \, B a^{3} - 26 \, A a^{2} b\right )} d e^{2} - 6 \,{\left (3 \, B b^{3} d e^{2} +{\left (B a b^{2} - 4 \, A b^{3}\right )} e^{3}\right )} x^{3} - 3 \,{\left (3 \, B b^{3} d^{2} e + 4 \,{\left (4 \, B a b^{2} - A b^{3}\right )} d e^{2} + 5 \,{\left (B a^{2} b - 4 \, A a b^{2}\right )} e^{3}\right )} x^{2} +{\left (3 \, B b^{3} d^{3} -{\left (23 \, B a b^{2} + 4 \, A b^{3}\right )} d^{2} e -{\left (41 \, B a^{2} b - 32 \, A a b^{2}\right )} d e^{2} - 11 \,{\left (B a^{3} - 4 \, A a^{2} b\right )} e^{3}\right )} x}{6 \,{\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} +{\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} +{\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \,{\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} +{\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*B*b*d*e^2 + (B*a - 4*A*b)*e^3)*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) - (3*B*b*d*e^2 + (B
*a - 4*A*b)*e^3)*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) - 1/6*(6*A*a^3*e^3 + (B*a*b^2 + 2*A*
b^3)*d^3 - 2*(4*B*a^2*b + 5*A*a*b^2)*d^2*e - (17*B*a^3 - 26*A*a^2*b)*d*e^2 - 6*(
3*B*b^3*d*e^2 + (B*a*b^2 - 4*A*b^3)*e^3)*x^3 - 3*(3*B*b^3*d^2*e + 4*(4*B*a*b^2 -
A*b^3)*d*e^2 + 5*(B*a^2*b - 4*A*a*b^2)*e^3)*x^2 + (3*B*b^3*d^3 - (23*B*a*b^2 +
4*A*b^3)*d^2*e - (41*B*a^2*b - 32*A*a*b^2)*d*e^2 - 11*(B*a^3 - 4*A*a^2*b)*e^3)*x
)/(a^3*b^4*d^5 - 4*a^4*b^3*d^4*e + 6*a^5*b^2*d^3*e^2 - 4*a^6*b*d^2*e^3 + a^7*d*e
^4 + (b^7*d^4*e - 4*a*b^6*d^3*e^2 + 6*a^2*b^5*d^2*e^3 - 4*a^3*b^4*d*e^4 + a^4*b^
3*e^5)*x^4 + (b^7*d^5 - a*b^6*d^4*e - 6*a^2*b^5*d^3*e^2 + 14*a^3*b^4*d^2*e^3 - 1
1*a^4*b^3*d*e^4 + 3*a^5*b^2*e^5)*x^3 + 3*(a*b^6*d^5 - 3*a^2*b^5*d^4*e + 2*a^3*b^
4*d^3*e^2 + 2*a^4*b^3*d^2*e^3 - 3*a^5*b^2*d*e^4 + a^6*b*e^5)*x^2 + (3*a^2*b^5*d^
5 - 11*a^3*b^4*d^4*e + 14*a^4*b^3*d^3*e^2 - 6*a^5*b^2*d^2*e^3 - a^6*b*d*e^4 + a^
7*e^5)*x)

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Fricas [A]  time = 0.29019, size = 1658, normalized size = 8.33 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(6*A*a^4*e^4 - (B*a*b^3 + 2*A*b^4)*d^4 + 3*(3*B*a^2*b^2 + 4*A*a*b^3)*d^3*e +
9*(B*a^3*b - 4*A*a^2*b^2)*d^2*e^2 - (17*B*a^4 - 20*A*a^3*b)*d*e^3 + 6*(3*B*b^4*
d^2*e^2 - 2*(B*a*b^3 + 2*A*b^4)*d*e^3 - (B*a^2*b^2 - 4*A*a*b^3)*e^4)*x^3 + 3*(3*
B*b^4*d^3*e + (13*B*a*b^3 - 4*A*b^4)*d^2*e^2 - (11*B*a^2*b^2 + 16*A*a*b^3)*d*e^3
- 5*(B*a^3*b - 4*A*a^2*b^2)*e^4)*x^2 - (3*B*b^4*d^4 - 2*(13*B*a*b^3 + 2*A*b^4)*
d^3*e - 18*(B*a^2*b^2 - 2*A*a*b^3)*d^2*e^2 + 6*(5*B*a^3*b + 2*A*a^2*b^2)*d*e^3 +
11*(B*a^4 - 4*A*a^3*b)*e^4)*x + 6*(3*B*a^3*b*d^2*e^2 + (B*a^4 - 4*A*a^3*b)*d*e^
3 + (3*B*b^4*d*e^3 + (B*a*b^3 - 4*A*b^4)*e^4)*x^4 + (3*B*b^4*d^2*e^2 + 2*(5*B*a*
b^3 - 2*A*b^4)*d*e^3 + 3*(B*a^2*b^2 - 4*A*a*b^3)*e^4)*x^3 + 3*(3*B*a*b^3*d^2*e^2
+ 4*(B*a^2*b^2 - A*a*b^3)*d*e^3 + (B*a^3*b - 4*A*a^2*b^2)*e^4)*x^2 + (9*B*a^2*b
^2*d^2*e^2 + 6*(B*a^3*b - 2*A*a^2*b^2)*d*e^3 + (B*a^4 - 4*A*a^3*b)*e^4)*x)*log(b
*x + a) - 6*(3*B*a^3*b*d^2*e^2 + (B*a^4 - 4*A*a^3*b)*d*e^3 + (3*B*b^4*d*e^3 + (B
*a*b^3 - 4*A*b^4)*e^4)*x^4 + (3*B*b^4*d^2*e^2 + 2*(5*B*a*b^3 - 2*A*b^4)*d*e^3 +
3*(B*a^2*b^2 - 4*A*a*b^3)*e^4)*x^3 + 3*(3*B*a*b^3*d^2*e^2 + 4*(B*a^2*b^2 - A*a*b
^3)*d*e^3 + (B*a^3*b - 4*A*a^2*b^2)*e^4)*x^2 + (9*B*a^2*b^2*d^2*e^2 + 6*(B*a^3*b
- 2*A*a^2*b^2)*d*e^3 + (B*a^4 - 4*A*a^3*b)*e^4)*x)*log(e*x + d))/(a^3*b^5*d^6 -
5*a^4*b^4*d^5*e + 10*a^5*b^3*d^4*e^2 - 10*a^6*b^2*d^3*e^3 + 5*a^7*b*d^2*e^4 - a
^8*d*e^5 + (b^8*d^5*e - 5*a*b^7*d^4*e^2 + 10*a^2*b^6*d^3*e^3 - 10*a^3*b^5*d^2*e^
4 + 5*a^4*b^4*d*e^5 - a^5*b^3*e^6)*x^4 + (b^8*d^6 - 2*a*b^7*d^5*e - 5*a^2*b^6*d^
4*e^2 + 20*a^3*b^5*d^3*e^3 - 25*a^4*b^4*d^2*e^4 + 14*a^5*b^3*d*e^5 - 3*a^6*b^2*e
^6)*x^3 + 3*(a*b^7*d^6 - 4*a^2*b^6*d^5*e + 5*a^3*b^5*d^4*e^2 - 5*a^5*b^3*d^2*e^4
+ 4*a^6*b^2*d*e^5 - a^7*b*e^6)*x^2 + (3*a^2*b^6*d^6 - 14*a^3*b^5*d^5*e + 25*a^4
*b^4*d^4*e^2 - 20*a^5*b^3*d^3*e^3 + 5*a^6*b^2*d^2*e^4 + 2*a^7*b*d*e^5 - a^8*e^6)
*x)

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Sympy [A]  time = 21.8537, size = 1445, normalized size = 7.26 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**2*(-4*A*b*e + B*a*e + 3*B*b*d)*log(x + (-4*A*a*b*e**4 - 4*A*b**2*d*e**3 + B*a
**2*e**4 + 4*B*a*b*d*e**3 + 3*B*b**2*d**2*e**2 - a**6*e**8*(-4*A*b*e + B*a*e + 3
*B*b*d)/(a*e - b*d)**5 + 6*a**5*b*d*e**7*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d
)**5 - 15*a**4*b**2*d**2*e**6*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 + 20*a
**3*b**3*d**3*e**5*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 - 15*a**2*b**4*d*
*4*e**4*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 + 6*a*b**5*d**5*e**3*(-4*A*b
*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 - b**6*d**6*e**2*(-4*A*b*e + B*a*e + 3*B*b*
d)/(a*e - b*d)**5)/(-8*A*b**2*e**4 + 2*B*a*b*e**4 + 6*B*b**2*d*e**3))/(a*e - b*d
)**5 - e**2*(-4*A*b*e + B*a*e + 3*B*b*d)*log(x + (-4*A*a*b*e**4 - 4*A*b**2*d*e**
3 + B*a**2*e**4 + 4*B*a*b*d*e**3 + 3*B*b**2*d**2*e**2 + a**6*e**8*(-4*A*b*e + B*
a*e + 3*B*b*d)/(a*e - b*d)**5 - 6*a**5*b*d*e**7*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*
e - b*d)**5 + 15*a**4*b**2*d**2*e**6*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5
- 20*a**3*b**3*d**3*e**5*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 + 15*a**2*
b**4*d**4*e**4*(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 - 6*a*b**5*d**5*e**3*
(-4*A*b*e + B*a*e + 3*B*b*d)/(a*e - b*d)**5 + b**6*d**6*e**2*(-4*A*b*e + B*a*e +
3*B*b*d)/(a*e - b*d)**5)/(-8*A*b**2*e**4 + 2*B*a*b*e**4 + 6*B*b**2*d*e**3))/(a*
e - b*d)**5 + (-6*A*a**3*e**3 - 26*A*a**2*b*d*e**2 + 10*A*a*b**2*d**2*e - 2*A*b*
*3*d**3 + 17*B*a**3*d*e**2 + 8*B*a**2*b*d**2*e - B*a*b**2*d**3 + x**3*(-24*A*b**
3*e**3 + 6*B*a*b**2*e**3 + 18*B*b**3*d*e**2) + x**2*(-60*A*a*b**2*e**3 - 12*A*b*
*3*d*e**2 + 15*B*a**2*b*e**3 + 48*B*a*b**2*d*e**2 + 9*B*b**3*d**2*e) + x*(-44*A*
a**2*b*e**3 - 32*A*a*b**2*d*e**2 + 4*A*b**3*d**2*e + 11*B*a**3*e**3 + 41*B*a**2*
b*d*e**2 + 23*B*a*b**2*d**2*e - 3*B*b**3*d**3))/(6*a**7*d*e**4 - 24*a**6*b*d**2*
e**3 + 36*a**5*b**2*d**3*e**2 - 24*a**4*b**3*d**4*e + 6*a**3*b**4*d**5 + x**4*(6
*a**4*b**3*e**5 - 24*a**3*b**4*d*e**4 + 36*a**2*b**5*d**2*e**3 - 24*a*b**6*d**3*
e**2 + 6*b**7*d**4*e) + x**3*(18*a**5*b**2*e**5 - 66*a**4*b**3*d*e**4 + 84*a**3*
b**4*d**2*e**3 - 36*a**2*b**5*d**3*e**2 - 6*a*b**6*d**4*e + 6*b**7*d**5) + x**2*
(18*a**6*b*e**5 - 54*a**5*b**2*d*e**4 + 36*a**4*b**3*d**2*e**3 + 36*a**3*b**4*d*
*3*e**2 - 54*a**2*b**5*d**4*e + 18*a*b**6*d**5) + x*(6*a**7*e**5 - 6*a**6*b*d*e*
*4 - 36*a**5*b**2*d**2*e**3 + 84*a**4*b**3*d**3*e**2 - 66*a**3*b**4*d**4*e + 18*
a**2*b**5*d**5))

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GIAC/XCAS [A]  time = 0.300409, size = 555, normalized size = 2.79 \[ \frac{{\left (3 \, B b d e^{3} + B a e^{4} - 4 \, A b e^{4}\right )}{\rm ln}\left ({\left | b - \frac{b d}{x e + d} + \frac{a e}{x e + d} \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} + \frac{\frac{B d e^{6}}{x e + d} - \frac{A e^{7}}{x e + d}}{b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}} + \frac{15 \, B b^{4} d e^{2} + 11 \, B a b^{3} e^{3} - 26 \, A b^{4} e^{3} - \frac{3 \,{\left (11 \, B b^{4} d^{2} e^{3} - 2 \, B a b^{3} d e^{4} - 20 \, A b^{4} d e^{4} - 9 \, B a^{2} b^{2} e^{5} + 20 \, A a b^{3} e^{5}\right )} e^{\left (-1\right )}}{x e + d} + \frac{18 \,{\left (B b^{4} d^{3} e^{4} - B a b^{3} d^{2} e^{5} - 2 \, A b^{4} d^{2} e^{5} - B a^{2} b^{2} d e^{6} + 4 \, A a b^{3} d e^{6} + B a^{3} b e^{7} - 2 \, A a^{2} b^{2} e^{7}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}}{6 \,{\left (b d - a e\right )}^{5}{\left (b - \frac{b d}{x e + d} + \frac{a e}{x e + d}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*B*b*d*e^3 + B*a*e^4 - 4*A*b*e^4)*ln(abs(b - b*d/(x*e + d) + a*e/(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) + (B*d*e^6/(x*e + d) - A*e^7/(x*e + d))/(b^4*d^4*e^4 - 4*a*b^3*
d^3*e^5 + 6*a^2*b^2*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8) + 1/6*(15*B*b^4*d*e^2 + 1
1*B*a*b^3*e^3 - 26*A*b^4*e^3 - 3*(11*B*b^4*d^2*e^3 - 2*B*a*b^3*d*e^4 - 20*A*b^4*
d*e^4 - 9*B*a^2*b^2*e^5 + 20*A*a*b^3*e^5)*e^(-1)/(x*e + d) + 18*(B*b^4*d^3*e^4 -
B*a*b^3*d^2*e^5 - 2*A*b^4*d^2*e^5 - B*a^2*b^2*d*e^6 + 4*A*a*b^3*d*e^6 + B*a^3*b
*e^7 - 2*A*a^2*b^2*e^7)*e^(-2)/(x*e + d)^2)/((b*d - a*e)^5*(b - b*d/(x*e + d) +
a*e/(x*e + d))^3)