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

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 = 0.224081, 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, Rules used = {27, 77} \[ -\frac{b^3 x (-4 a B e-A b e+4 b B d)}{e^5}+\frac{2 b^2 (b d-a e) \log (d+e x) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac{2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 (d+e x)}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{2 e^6 (d+e x)^2}+\frac{(b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^3}+\frac{b^4 B x^2}{2 e^4} \]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{(d+e x)^4} \, dx\\ &=\int \left (\frac{b^3 (-4 b B d+A b e+4 a B e)}{e^5}+\frac{b^4 B x}{e^4}+\frac{(-b d+a e)^4 (-B d+A e)}{e^5 (d+e x)^4}+\frac{(-b d+a e)^3 (-5 b B d+4 A b e+a B e)}{e^5 (d+e x)^3}+\frac{2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e)}{e^5 (d+e x)^2}-\frac{2 b^2 (b d-a e) (-5 b B d+2 A b e+3 a B e)}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac{b^3 (4 b B d-A b e-4 a B e) x}{e^5}+\frac{b^4 B x^2}{2 e^4}+\frac{(b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^3}-\frac{(b d-a e)^3 (5 b B d-4 A b e-a B e)}{2 e^6 (d+e x)^2}+\frac{2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e)}{e^6 (d+e x)}+\frac{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}\\ \end{align*}

Mathematica [A]  time = 0.179438, size = 351, normalized size = 1.86 \[ \frac{6 a^2 b^2 e^2 \left (B d \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 A e \left (d^2+3 d e x+3 e^2 x^2\right )\right )-4 a^3 b e^3 \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )-a^4 e^4 (2 A e+B (d+3 e x))+4 a b^3 e \left (A d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 B \left (9 d^2 e^2 x^2+27 d^3 e x+13 d^4-9 d e^3 x^3-3 e^4 x^4\right )\right )+12 b^2 (d+e x)^3 (b d-a e) \log (d+e x) (-3 a B e-2 A b e+5 b B d)+b^4 \left (2 A e \left (-9 d^2 e^2 x^2-27 d^3 e x-13 d^4+9 d e^3 x^3+3 e^4 x^4\right )+B \left (-9 d^3 e^2 x^2-63 d^2 e^3 x^3+81 d^4 e x+47 d^5-15 d e^4 x^4+3 e^5 x^5\right )\right )}{6 e^6 (d+e x)^3} \]

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 = 0.012, size = 626, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

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]

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

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Maxima [B]  time = 1.05905, size = 582, normalized size = 3.08 \begin{align*} \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}} \end{align*}

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, 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 [B]  time = 1.611, size = 1345, normalized size = 7.12 \begin{align*} \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 )}} \end{align*}

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, 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 [B]  time = 49.2681, size = 483, normalized size = 2.56 \begin{align*} \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}} \end{align*}

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 [B]  time = 1.14082, size = 560, normalized size = 2.96 \begin{align*} 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 )} \log \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}} \end{align*}

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, 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)*log(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 - 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)*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