3.1684 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=189 \[ -\frac{b^3 x (-4 a B e-A b e+4 b B d)}{e^5}+\frac{2 b^2 (b d-a e) \log (d+e x) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac{2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 (d+e x)}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{2 e^6 (d+e x)^2}+\frac{(b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^3}+\frac{b^4 B x^2}{2 e^4} \]

[Out]

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

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Rubi [A]  time = 0.529179, antiderivative size = 189, 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 \[ -\frac{b^3 x (-4 a B e-A b e+4 b B d)}{e^5}+\frac{2 b^2 (b d-a e) \log (d+e x) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac{2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 (d+e x)}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{2 e^6 (d+e x)^2}+\frac{(b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^3}+\frac{b^4 B x^2}{2 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B b^{4} \int x\, dx}{e^{4}} + \frac{b^{3} x \left (A b e + 4 B a e - 4 B b d\right )}{e^{5}} + \frac{2 b^{2} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{2 b \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{e^{6} \left (d + e x\right )} - \frac{\left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{2 e^{6} \left (d + e x\right )^{2}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{4}}{3 e^{6} \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b**4*Integral(x, x)/e**4 + b**3*x*(A*b*e + 4*B*a*e - 4*B*b*d)/e**5 + 2*b**2*(a
*e - b*d)*(2*A*b*e + 3*B*a*e - 5*B*b*d)*log(d + e*x)/e**6 - 2*b*(a*e - b*d)**2*(
3*A*b*e + 2*B*a*e - 5*B*b*d)/(e**6*(d + e*x)) - (a*e - b*d)**3*(4*A*b*e + B*a*e
- 5*B*b*d)/(2*e**6*(d + e*x)**2) - (A*e - B*d)*(a*e - b*d)**4/(3*e**6*(d + e*x)*
*3)

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Mathematica [A]  time = 0.30098, size = 351, normalized size = 1.86 \[ \frac{-a^4 e^4 (2 A e+B (d+3 e x))-4 a^3 b e^3 \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )+6 a^2 b^2 e^2 \left (B d \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 A e \left (d^2+3 d e x+3 e^2 x^2\right )\right )+4 a b^3 e \left (A d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 B \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )\right )+12 b^2 (d+e x)^3 (b d-a e) \log (d+e x) (-3 a B e-2 A b e+5 b B d)+b^4 \left (2 A e \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+B \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )\right )}{6 e^6 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-(a^4*e^4*(2*A*e + B*(d + 3*e*x))) - 4*a^3*b*e^3*(A*e*(d + 3*e*x) + 2*B*(d^2 +
3*d*e*x + 3*e^2*x^2)) + 6*a^2*b^2*e^2*(-2*A*e*(d^2 + 3*d*e*x + 3*e^2*x^2) + B*d*
(11*d^2 + 27*d*e*x + 18*e^2*x^2)) + 4*a*b^3*e*(A*d*e*(11*d^2 + 27*d*e*x + 18*e^2
*x^2) - 2*B*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3 - 3*e^4*x^4)) + b
^4*(2*A*e*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + B*(
47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5)
) + 12*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*e)*(d + e*x)^3*Log[d + e*x])/(
6*e^6*(d + e*x)^3)

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Maple [B]  time = 0.018, size = 626, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/e/(e*x+d)^3*A*a^4-1/2/e^2/(e*x+d)^2*B*a^4+b^4/e^4*A*x+10*b^4/e^6/(e*x+d)*B*
d^3+4*b^3/e^4*a*B*x-4*b^4/e^5*B*d*x+4*b^3/e^4*ln(e*x+d)*A*a-4*b^4/e^5*ln(e*x+d)*
A*d+6*b^2/e^4*ln(e*x+d)*a^2*B+10*b^4/e^6*ln(e*x+d)*B*d^2-6*b^2/e^3/(e*x+d)*A*a^2
-1/3/e^5/(e*x+d)^3*A*b^4*d^4-2/e^3/(e*x+d)^3*A*d^2*a^2*b^2+6/e^3/(e*x+d)^2*A*a^2
*b^2*d-6/e^4/(e*x+d)^2*A*a*b^3*d^2+4/e^3/(e*x+d)^2*B*a^3*b*d-9/e^4/(e*x+d)^2*B*a
^2*b^2*d^2+8/e^5/(e*x+d)^2*B*a*b^3*d^3+12*b^3/e^4/(e*x+d)*A*a*d+4/3/e^2/(e*x+d)^
3*A*d*a^3*b-16*b^3/e^5*ln(e*x+d)*B*d*a+1/2*b^4*B*x^2/e^4-4/3/e^3/(e*x+d)^3*B*d^2
*a^3*b+2/e^4/(e*x+d)^3*B*d^3*a^2*b^2-4/3/e^5/(e*x+d)^3*B*a*b^3*d^4+18*b^2/e^4/(e
*x+d)*B*a^2*d-24*b^3/e^5/(e*x+d)*B*a*d^2+4/3/e^4/(e*x+d)^3*A*d^3*a*b^3-6*b^4/e^5
/(e*x+d)*A*d^2-4*b/e^3/(e*x+d)*B*a^3+1/3/e^2/(e*x+d)^3*B*d*a^4+1/3/e^6/(e*x+d)^3
*B*b^4*d^5-2/e^2/(e*x+d)^2*A*a^3*b+2/e^5/(e*x+d)^2*A*b^4*d^3-5/2/e^6/(e*x+d)^2*B
*b^4*d^4

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Maxima [A]  time = 0.707303, size = 582, normalized size = 3.08 \[ \frac{47 \, B b^{4} d^{5} - 2 \, A a^{4} e^{5} - 26 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 22 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} -{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 12 \,{\left (5 \, B b^{4} d^{3} e^{2} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 3 \,{\left (35 \, B b^{4} d^{4} e - 20 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 18 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} -{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac{B b^{4} e x^{2} - 2 \,{\left (4 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} x}{2 \, e^{5}} + \frac{2 \,{\left (5 \, B b^{4} d^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(47*B*b^4*d^5 - 2*A*a^4*e^5 - 26*(4*B*a*b^3 + A*b^4)*d^4*e + 22*(3*B*a^2*b^2
+ 2*A*a*b^3)*d^3*e^2 - 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 - (B*a^4 + 4*A*a^3*b
)*d*e^4 + 12*(5*B*b^4*d^3*e^2 - 3*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 +
2*A*a*b^3)*d*e^4 - (2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 3*(35*B*b^4*d^4*e - 20*
(4*B*a*b^3 + A*b^4)*d^3*e^2 + 18*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 - 4*(2*B*a^3*
b + 3*A*a^2*b^2)*d*e^4 - (B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^9*x^3 + 3*d*e^8*x^2 + 3*
d^2*e^7*x + d^3*e^6) + 1/2*(B*b^4*e*x^2 - 2*(4*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*
x)/e^5 + 2*(5*B*b^4*d^2 - 2*(4*B*a*b^3 + A*b^4)*d*e + (3*B*a^2*b^2 + 2*A*a*b^3)*
e^2)*log(e*x + d)/e^6

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Fricas [A]  time = 0.290983, size = 878, normalized size = 4.65 \[ \frac{3 \, B b^{4} e^{5} x^{5} + 47 \, B b^{4} d^{5} - 2 \, A a^{4} e^{5} - 26 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 22 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} -{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 3 \,{\left (5 \, B b^{4} d e^{4} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} - 9 \,{\left (7 \, B b^{4} d^{2} e^{3} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4}\right )} x^{3} - 3 \,{\left (3 \, B b^{4} d^{3} e^{2} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} - 12 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 3 \,{\left (27 \, B b^{4} d^{4} e - 18 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 18 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} -{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x + 12 \,{\left (5 \, B b^{4} d^{5} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} +{\left (5 \, B b^{4} d^{2} e^{3} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 3 \,{\left (5 \, B b^{4} d^{3} e^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4}\right )} x^{2} + 3 \,{\left (5 \, B b^{4} d^{4} e - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(3*B*b^4*e^5*x^5 + 47*B*b^4*d^5 - 2*A*a^4*e^5 - 26*(4*B*a*b^3 + A*b^4)*d^4*e
+ 22*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 -
(B*a^4 + 4*A*a^3*b)*d*e^4 - 3*(5*B*b^4*d*e^4 - 2*(4*B*a*b^3 + A*b^4)*e^5)*x^4 -
9*(7*B*b^4*d^2*e^3 - 2*(4*B*a*b^3 + A*b^4)*d*e^4)*x^3 - 3*(3*B*b^4*d^3*e^2 + 6*(
4*B*a*b^3 + A*b^4)*d^2*e^3 - 12*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 4*(2*B*a^3*b +
3*A*a^2*b^2)*e^5)*x^2 + 3*(27*B*b^4*d^4*e - 18*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 18
*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 - 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 - (B*a^4
+ 4*A*a^3*b)*e^5)*x + 12*(5*B*b^4*d^5 - 2*(4*B*a*b^3 + A*b^4)*d^4*e + (3*B*a^2*b
^2 + 2*A*a*b^3)*d^3*e^2 + (5*B*b^4*d^2*e^3 - 2*(4*B*a*b^3 + A*b^4)*d*e^4 + (3*B*
a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 3*(5*B*b^4*d^3*e^2 - 2*(4*B*a*b^3 + A*b^4)*d^2*e
^3 + (3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4)*x^2 + 3*(5*B*b^4*d^4*e - 2*(4*B*a*b^3 + A*
b^4)*d^3*e^2 + (3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3)*x)*log(e*x + d))/(e^9*x^3 + 3*
d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)

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Sympy [A]  time = 115.793, size = 483, normalized size = 2.56 \[ \frac{B b^{4} x^{2}}{2 e^{4}} + \frac{2 b^{2} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{2 A a^{4} e^{5} + 4 A a^{3} b d e^{4} + 12 A a^{2} b^{2} d^{2} e^{3} - 44 A a b^{3} d^{3} e^{2} + 26 A b^{4} d^{4} e + B a^{4} d e^{4} + 8 B a^{3} b d^{2} e^{3} - 66 B a^{2} b^{2} d^{3} e^{2} + 104 B a b^{3} d^{4} e - 47 B b^{4} d^{5} + x^{2} \left (36 A a^{2} b^{2} e^{5} - 72 A a b^{3} d e^{4} + 36 A b^{4} d^{2} e^{3} + 24 B a^{3} b e^{5} - 108 B a^{2} b^{2} d e^{4} + 144 B a b^{3} d^{2} e^{3} - 60 B b^{4} d^{3} e^{2}\right ) + x \left (12 A a^{3} b e^{5} + 36 A a^{2} b^{2} d e^{4} - 108 A a b^{3} d^{2} e^{3} + 60 A b^{4} d^{3} e^{2} + 3 B a^{4} e^{5} + 24 B a^{3} b d e^{4} - 162 B a^{2} b^{2} d^{2} e^{3} + 240 B a b^{3} d^{3} e^{2} - 105 B b^{4} d^{4} e\right )}{6 d^{3} e^{6} + 18 d^{2} e^{7} x + 18 d e^{8} x^{2} + 6 e^{9} x^{3}} + \frac{x \left (A b^{4} e + 4 B a b^{3} e - 4 B b^{4} d\right )}{e^{5}} \]

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)**4,x)

[Out]

B*b**4*x**2/(2*e**4) + 2*b**2*(a*e - b*d)*(2*A*b*e + 3*B*a*e - 5*B*b*d)*log(d +
e*x)/e**6 - (2*A*a**4*e**5 + 4*A*a**3*b*d*e**4 + 12*A*a**2*b**2*d**2*e**3 - 44*A
*a*b**3*d**3*e**2 + 26*A*b**4*d**4*e + B*a**4*d*e**4 + 8*B*a**3*b*d**2*e**3 - 66
*B*a**2*b**2*d**3*e**2 + 104*B*a*b**3*d**4*e - 47*B*b**4*d**5 + x**2*(36*A*a**2*
b**2*e**5 - 72*A*a*b**3*d*e**4 + 36*A*b**4*d**2*e**3 + 24*B*a**3*b*e**5 - 108*B*
a**2*b**2*d*e**4 + 144*B*a*b**3*d**2*e**3 - 60*B*b**4*d**3*e**2) + x*(12*A*a**3*
b*e**5 + 36*A*a**2*b**2*d*e**4 - 108*A*a*b**3*d**2*e**3 + 60*A*b**4*d**3*e**2 +
3*B*a**4*e**5 + 24*B*a**3*b*d*e**4 - 162*B*a**2*b**2*d**2*e**3 + 240*B*a*b**3*d*
*3*e**2 - 105*B*b**4*d**4*e))/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6
*e**9*x**3) + x*(A*b**4*e + 4*B*a*b**3*e - 4*B*b**4*d)/e**5

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GIAC/XCAS [A]  time = 0.286035, size = 560, normalized size = 2.96 \[ 2 \,{\left (5 \, B b^{4} d^{2} - 8 \, B a b^{3} d e - 2 \, A b^{4} d e + 3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B b^{4} x^{2} e^{4} - 8 \, B b^{4} d x e^{3} + 8 \, B a b^{3} x e^{4} + 2 \, A b^{4} x e^{4}\right )} e^{\left (-8\right )} + \frac{{\left (47 \, B b^{4} d^{5} - 104 \, B a b^{3} d^{4} e - 26 \, A b^{4} d^{4} e + 66 \, B a^{2} b^{2} d^{3} e^{2} + 44 \, A a b^{3} d^{3} e^{2} - 8 \, B a^{3} b d^{2} e^{3} - 12 \, A a^{2} b^{2} d^{2} e^{3} - B a^{4} d e^{4} - 4 \, A a^{3} b d e^{4} - 2 \, A a^{4} e^{5} + 12 \,{\left (5 \, B b^{4} d^{3} e^{2} - 12 \, B a b^{3} d^{2} e^{3} - 3 \, A b^{4} d^{2} e^{3} + 9 \, B a^{2} b^{2} d e^{4} + 6 \, A a b^{3} d e^{4} - 2 \, B a^{3} b e^{5} - 3 \, A a^{2} b^{2} e^{5}\right )} x^{2} + 3 \,{\left (35 \, B b^{4} d^{4} e - 80 \, B a b^{3} d^{3} e^{2} - 20 \, A b^{4} d^{3} e^{2} + 54 \, B a^{2} b^{2} d^{2} e^{3} + 36 \, A a b^{3} d^{2} e^{3} - 8 \, B a^{3} b d e^{4} - 12 \, A a^{2} b^{2} d e^{4} - B a^{4} e^{5} - 4 \, A a^{3} b e^{5}\right )} x\right )} e^{\left (-6\right )}}{6 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(5*B*b^4*d^2 - 8*B*a*b^3*d*e - 2*A*b^4*d*e + 3*B*a^2*b^2*e^2 + 2*A*a*b^3*e^2)*
e^(-6)*ln(abs(x*e + d)) + 1/2*(B*b^4*x^2*e^4 - 8*B*b^4*d*x*e^3 + 8*B*a*b^3*x*e^4
+ 2*A*b^4*x*e^4)*e^(-8) + 1/6*(47*B*b^4*d^5 - 104*B*a*b^3*d^4*e - 26*A*b^4*d^4*
e + 66*B*a^2*b^2*d^3*e^2 + 44*A*a*b^3*d^3*e^2 - 8*B*a^3*b*d^2*e^3 - 12*A*a^2*b^2
*d^2*e^3 - B*a^4*d*e^4 - 4*A*a^3*b*d*e^4 - 2*A*a^4*e^5 + 12*(5*B*b^4*d^3*e^2 - 1
2*B*a*b^3*d^2*e^3 - 3*A*b^4*d^2*e^3 + 9*B*a^2*b^2*d*e^4 + 6*A*a*b^3*d*e^4 - 2*B*
a^3*b*e^5 - 3*A*a^2*b^2*e^5)*x^2 + 3*(35*B*b^4*d^4*e - 80*B*a*b^3*d^3*e^2 - 20*A
*b^4*d^3*e^2 + 54*B*a^2*b^2*d^2*e^3 + 36*A*a*b^3*d^2*e^3 - 8*B*a^3*b*d*e^4 - 12*
A*a^2*b^2*d*e^4 - B*a^4*e^5 - 4*A*a^3*b*e^5)*x)*e^(-6)/(x*e + d)^3