3.1701 \(\int \frac{(A+B x) (d+e x)^4}{(a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=186 \[ \frac{e^3 x (-4 a B e+A b e+4 b B d)}{b^5}+\frac{2 e^2 (b d-a e) \log (a+b x) (-5 a B e+2 A b e+3 b B d)}{b^6}-\frac{2 e (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^6 (a+b x)}-\frac{(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{2 b^6 (a+b x)^2}-\frac{(A b-a B) (b d-a e)^4}{3 b^6 (a+b x)^3}+\frac{B e^4 x^2}{2 b^4} \]

[Out]

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

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Rubi [A]  time = 0.227663, antiderivative size = 186, 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} \[ \frac{e^3 x (-4 a B e+A b e+4 b B d)}{b^5}+\frac{2 e^2 (b d-a e) \log (a+b x) (-5 a B e+2 A b e+3 b B d)}{b^6}-\frac{2 e (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^6 (a+b x)}-\frac{(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{2 b^6 (a+b x)^2}-\frac{(A b-a B) (b d-a e)^4}{3 b^6 (a+b x)^3}+\frac{B e^4 x^2}{2 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.194464, size = 362, normalized size = 1.95 \[ \frac{-2 A b \left (3 a^2 b^2 e^2 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a^3 b e^3 (27 e x-22 d)+13 a^4 e^4+a b^3 e \left (18 d^2 e x+2 d^3-36 d e^2 x^2-9 e^3 x^3\right )+b^4 \left (18 d^2 e^2 x^2+6 d^3 e x+d^4-3 e^4 x^4\right )\right )+B \left (-3 a^3 b^2 e^2 \left (-22 d^2+72 d e x+3 e^2 x^2\right )-a^2 b^3 e \left (-162 d^2 e x+8 d^3+72 d e^2 x^2+63 e^3 x^3\right )+a^4 b e^3 (81 e x-104 d)+47 a^5 e^4-a b^4 \left (-108 d^2 e^2 x^2+24 d^3 e x+d^4-72 d e^3 x^3+15 e^4 x^4\right )+3 b^5 x \left (-8 d^3 e x-d^4+8 d e^3 x^3+e^4 x^4\right )\right )+12 e^2 (a+b x)^3 (b d-a e) \log (a+b x) (-5 a B e+2 A b e+3 b B d)}{6 b^6 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*b*(13*a^4*e^4 + a^3*b*e^3*(-22*d + 27*e*x) + 3*a^2*b^2*e^2*(2*d^2 - 18*d*e*x + 3*e^2*x^2) + a*b^3*e*(2*d
^3 + 18*d^2*e*x - 36*d*e^2*x^2 - 9*e^3*x^3) + b^4*(d^4 + 6*d^3*e*x + 18*d^2*e^2*x^2 - 3*e^4*x^4)) + B*(47*a^5*
e^4 + a^4*b*e^3*(-104*d + 81*e*x) - 3*a^3*b^2*e^2*(-22*d^2 + 72*d*e*x + 3*e^2*x^2) - a^2*b^3*e*(8*d^3 - 162*d^
2*e*x + 72*d*e^2*x^2 + 63*e^3*x^3) + 3*b^5*x*(-d^4 - 8*d^3*e*x + 8*d*e^3*x^3 + e^4*x^4) - a*b^4*(d^4 + 24*d^3*
e*x - 108*d^2*e^2*x^2 - 72*d*e^3*x^3 + 15*e^4*x^4)) + 12*e^2*(b*d - a*e)*(3*b*B*d + 2*A*b*e - 5*a*B*e)*(a + b*
x)^3*Log[a + b*x])/(6*b^6*(a + b*x)^3)

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Maple [B]  time = 0.012, size = 626, normalized size = 3.4 \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)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

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Maxima [B]  time = 1.21847, size = 586, normalized size = 3.15 \begin{align*} -\frac{{\left (B a b^{4} + 2 \, A b^{5}\right )} d^{4} + 4 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e - 6 \,{\left (11 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2} e^{2} + 4 \,{\left (26 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} d e^{3} -{\left (47 \, B a^{5} - 26 \, A a^{4} b\right )} e^{4} + 12 \,{\left (2 \, B b^{5} d^{3} e - 3 \,{\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 6 \,{\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} -{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 3 \,{\left (B b^{5} d^{4} + 4 \,{\left (2 \, B a b^{4} + A b^{5}\right )} d^{3} e - 6 \,{\left (9 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} + 4 \,{\left (20 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} d e^{3} - 5 \,{\left (7 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} e^{4}\right )} x}{6 \,{\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} + \frac{B b e^{4} x^{2} + 2 \,{\left (4 \, B b d e^{3} -{\left (4 \, B a - A b\right )} e^{4}\right )} x}{2 \, b^{5}} + \frac{2 \,{\left (3 \, B b^{2} d^{2} e^{2} - 2 \,{\left (4 \, B a b - A b^{2}\right )} d e^{3} +{\left (5 \, B a^{2} - 2 \, A a b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*((B*a*b^4 + 2*A*b^5)*d^4 + 4*(2*B*a^2*b^3 + A*a*b^4)*d^3*e - 6*(11*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e^2 + 4*(
26*B*a^4*b - 11*A*a^3*b^2)*d*e^3 - (47*B*a^5 - 26*A*a^4*b)*e^4 + 12*(2*B*b^5*d^3*e - 3*(3*B*a*b^4 - A*b^5)*d^2
*e^2 + 6*(2*B*a^2*b^3 - A*a*b^4)*d*e^3 - (5*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*x^2 + 3*(B*b^5*d^4 + 4*(2*B*a*b^4 +
A*b^5)*d^3*e - 6*(9*B*a^2*b^3 - 2*A*a*b^4)*d^2*e^2 + 4*(20*B*a^3*b^2 - 9*A*a^2*b^3)*d*e^3 - 5*(7*B*a^4*b - 4*A
*a^3*b^2)*e^4)*x)/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6) + 1/2*(B*b*e^4*x^2 + 2*(4*B*b*d*e^3 - (4*B*a
- A*b)*e^4)*x)/b^5 + 2*(3*B*b^2*d^2*e^2 - 2*(4*B*a*b - A*b^2)*d*e^3 + (5*B*a^2 - 2*A*a*b)*e^4)*log(b*x + a)/b
^6

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Fricas [B]  time = 1.52117, size = 1365, normalized size = 7.34 \begin{align*} \frac{3 \, B b^{5} e^{4} x^{5} -{\left (B a b^{4} + 2 \, A b^{5}\right )} d^{4} - 4 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e + 6 \,{\left (11 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \,{\left (26 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} d e^{3} +{\left (47 \, B a^{5} - 26 \, A a^{4} b\right )} e^{4} + 3 \,{\left (8 \, B b^{5} d e^{3} -{\left (5 \, B a b^{4} - 2 \, A b^{5}\right )} e^{4}\right )} x^{4} + 9 \,{\left (8 \, B a b^{4} d e^{3} -{\left (7 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} e^{4}\right )} x^{3} - 3 \,{\left (8 \, B b^{5} d^{3} e - 12 \,{\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 24 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d e^{3} + 3 \,{\left (B a^{3} b^{2} + 2 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 3 \,{\left (B b^{5} d^{4} + 4 \,{\left (2 \, B a b^{4} + A b^{5}\right )} d^{3} e - 6 \,{\left (9 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} + 36 \,{\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} - 9 \,{\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \,{\left (3 \, B a^{3} b^{2} d^{2} e^{2} - 2 \,{\left (4 \, B a^{4} b - A a^{3} b^{2}\right )} d e^{3} +{\left (5 \, B a^{5} - 2 \, A a^{4} b\right )} e^{4} +{\left (3 \, B b^{5} d^{2} e^{2} - 2 \,{\left (4 \, B a b^{4} - A b^{5}\right )} d e^{3} +{\left (5 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 3 \,{\left (3 \, B a b^{4} d^{2} e^{2} - 2 \,{\left (4 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} +{\left (5 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 3 \,{\left (3 \, B a^{2} b^{3} d^{2} e^{2} - 2 \,{\left (4 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} +{\left (5 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*B*b^5*e^4*x^5 - (B*a*b^4 + 2*A*b^5)*d^4 - 4*(2*B*a^2*b^3 + A*a*b^4)*d^3*e + 6*(11*B*a^3*b^2 - 2*A*a^2*b
^3)*d^2*e^2 - 4*(26*B*a^4*b - 11*A*a^3*b^2)*d*e^3 + (47*B*a^5 - 26*A*a^4*b)*e^4 + 3*(8*B*b^5*d*e^3 - (5*B*a*b^
4 - 2*A*b^5)*e^4)*x^4 + 9*(8*B*a*b^4*d*e^3 - (7*B*a^2*b^3 - 2*A*a*b^4)*e^4)*x^3 - 3*(8*B*b^5*d^3*e - 12*(3*B*a
*b^4 - A*b^5)*d^2*e^2 + 24*(B*a^2*b^3 - A*a*b^4)*d*e^3 + 3*(B*a^3*b^2 + 2*A*a^2*b^3)*e^4)*x^2 - 3*(B*b^5*d^4 +
4*(2*B*a*b^4 + A*b^5)*d^3*e - 6*(9*B*a^2*b^3 - 2*A*a*b^4)*d^2*e^2 + 36*(2*B*a^3*b^2 - A*a^2*b^3)*d*e^3 - 9*(3
*B*a^4*b - 2*A*a^3*b^2)*e^4)*x + 12*(3*B*a^3*b^2*d^2*e^2 - 2*(4*B*a^4*b - A*a^3*b^2)*d*e^3 + (5*B*a^5 - 2*A*a^
4*b)*e^4 + (3*B*b^5*d^2*e^2 - 2*(4*B*a*b^4 - A*b^5)*d*e^3 + (5*B*a^2*b^3 - 2*A*a*b^4)*e^4)*x^3 + 3*(3*B*a*b^4*
d^2*e^2 - 2*(4*B*a^2*b^3 - A*a*b^4)*d*e^3 + (5*B*a^3*b^2 - 2*A*a^2*b^3)*e^4)*x^2 + 3*(3*B*a^2*b^3*d^2*e^2 - 2*
(4*B*a^3*b^2 - A*a^2*b^3)*d*e^3 + (5*B*a^4*b - 2*A*a^3*b^2)*e^4)*x)*log(b*x + a))/(b^9*x^3 + 3*a*b^8*x^2 + 3*a
^2*b^7*x + a^3*b^6)

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Sympy [B]  time = 45.4421, size = 483, normalized size = 2.6 \begin{align*} \frac{B e^{4} x^{2}}{2 b^{4}} + \frac{- 26 A a^{4} b e^{4} + 44 A a^{3} b^{2} d e^{3} - 12 A a^{2} b^{3} d^{2} e^{2} - 4 A a b^{4} d^{3} e - 2 A b^{5} d^{4} + 47 B a^{5} e^{4} - 104 B a^{4} b d e^{3} + 66 B a^{3} b^{2} d^{2} e^{2} - 8 B a^{2} b^{3} d^{3} e - B a b^{4} d^{4} + x^{2} \left (- 36 A a^{2} b^{3} e^{4} + 72 A a b^{4} d e^{3} - 36 A b^{5} d^{2} e^{2} + 60 B a^{3} b^{2} e^{4} - 144 B a^{2} b^{3} d e^{3} + 108 B a b^{4} d^{2} e^{2} - 24 B b^{5} d^{3} e\right ) + x \left (- 60 A a^{3} b^{2} e^{4} + 108 A a^{2} b^{3} d e^{3} - 36 A a b^{4} d^{2} e^{2} - 12 A b^{5} d^{3} e + 105 B a^{4} b e^{4} - 240 B a^{3} b^{2} d e^{3} + 162 B a^{2} b^{3} d^{2} e^{2} - 24 B a b^{4} d^{3} e - 3 B b^{5} d^{4}\right )}{6 a^{3} b^{6} + 18 a^{2} b^{7} x + 18 a b^{8} x^{2} + 6 b^{9} x^{3}} - \frac{x \left (- A b e^{4} + 4 B a e^{4} - 4 B b d e^{3}\right )}{b^{5}} + \frac{2 e^{2} \left (a e - b d\right ) \left (- 2 A b e + 5 B a e - 3 B b d\right ) \log{\left (a + b x \right )}}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

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

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Giac [B]  time = 1.15325, size = 560, normalized size = 3.01 \begin{align*} \frac{2 \,{\left (3 \, B b^{2} d^{2} e^{2} - 8 \, B a b d e^{3} + 2 \, A b^{2} d e^{3} + 5 \, B a^{2} e^{4} - 2 \, A a b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{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}}{2 \, b^{8}} - \frac{B a b^{4} d^{4} + 2 \, A b^{5} d^{4} + 8 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e - 66 \, B a^{3} b^{2} d^{2} e^{2} + 12 \, A a^{2} b^{3} d^{2} e^{2} + 104 \, B a^{4} b d e^{3} - 44 \, A a^{3} b^{2} d e^{3} - 47 \, B a^{5} e^{4} + 26 \, A a^{4} b e^{4} + 12 \,{\left (2 \, B b^{5} d^{3} e - 9 \, B a b^{4} d^{2} e^{2} + 3 \, A b^{5} d^{2} e^{2} + 12 \, B a^{2} b^{3} d e^{3} - 6 \, A a b^{4} d e^{3} - 5 \, B a^{3} b^{2} e^{4} + 3 \, A a^{2} b^{3} e^{4}\right )} x^{2} + 3 \,{\left (B b^{5} d^{4} + 8 \, B a b^{4} d^{3} e + 4 \, A b^{5} d^{3} e - 54 \, B a^{2} b^{3} d^{2} e^{2} + 12 \, A a b^{4} d^{2} e^{2} + 80 \, B a^{3} b^{2} d e^{3} - 36 \, A a^{2} b^{3} d e^{3} - 35 \, B a^{4} b e^{4} + 20 \, A a^{3} b^{2} e^{4}\right )} x}{6 \,{\left (b x + a\right )}^{3} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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