3.1692 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=206 \[ \frac{b^3 (-4 a B e-A b e+5 b B d)}{7 e^6 (d+e x)^7}-\frac{b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{4 e^6 (d+e x)^8}+\frac{2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6 (d+e x)^9}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{10 e^6 (d+e x)^{10}}+\frac{(b d-a e)^4 (B d-A e)}{11 e^6 (d+e x)^{11}}-\frac{b^4 B}{6 e^6 (d+e x)^6} \]

[Out]

((b*d - a*e)^4*(B*d - A*e))/(11*e^6*(d + e*x)^11) - ((b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e))/(10*e^6*(d + e
*x)^10) + (2*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e))/(9*e^6*(d + e*x)^9) - (b^2*(b*d - a*e)*(5*b*B*d -
2*A*b*e - 3*a*B*e))/(4*e^6*(d + e*x)^8) + (b^3*(5*b*B*d - A*b*e - 4*a*B*e))/(7*e^6*(d + e*x)^7) - (b^4*B)/(6*e
^6*(d + e*x)^6)

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Rubi [A]  time = 0.167545, antiderivative size = 206, 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} \[ \frac{b^3 (-4 a B e-A b e+5 b B d)}{7 e^6 (d+e x)^7}-\frac{b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{4 e^6 (d+e x)^8}+\frac{2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6 (d+e x)^9}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{10 e^6 (d+e x)^{10}}+\frac{(b d-a e)^4 (B d-A e)}{11 e^6 (d+e x)^{11}}-\frac{b^4 B}{6 e^6 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^12,x]

[Out]

((b*d - a*e)^4*(B*d - A*e))/(11*e^6*(d + e*x)^11) - ((b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e))/(10*e^6*(d + e
*x)^10) + (2*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e))/(9*e^6*(d + e*x)^9) - (b^2*(b*d - a*e)*(5*b*B*d -
2*A*b*e - 3*a*B*e))/(4*e^6*(d + e*x)^8) + (b^3*(5*b*B*d - A*b*e - 4*a*B*e))/(7*e^6*(d + e*x)^7) - (b^4*B)/(6*e
^6*(d + e*x)^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)^{12}} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{(d+e x)^{12}} \, dx\\ &=\int \left (\frac{(-b d+a e)^4 (-B d+A e)}{e^5 (d+e x)^{12}}+\frac{(-b d+a e)^3 (-5 b B d+4 A b e+a B e)}{e^5 (d+e x)^{11}}+\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)^{10}}-\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)^9}+\frac{b^3 (-5 b B d+A b e+4 a B e)}{e^5 (d+e x)^8}+\frac{b^4 B}{e^5 (d+e x)^7}\right ) \, dx\\ &=\frac{(b d-a e)^4 (B d-A e)}{11 e^6 (d+e x)^{11}}-\frac{(b d-a e)^3 (5 b B d-4 A b e-a B e)}{10 e^6 (d+e x)^{10}}+\frac{2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e)}{9 e^6 (d+e x)^9}-\frac{b^2 (b d-a e) (5 b B d-2 A b e-3 a B e)}{4 e^6 (d+e x)^8}+\frac{b^3 (5 b B d-A b e-4 a B e)}{7 e^6 (d+e x)^7}-\frac{b^4 B}{6 e^6 (d+e x)^6}\\ \end{align*}

Mathematica [A]  time = 0.150694, size = 323, normalized size = 1.57 \[ -\frac{21 a^2 b^2 e^2 \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )\right )+56 a^3 b e^3 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+126 a^4 e^4 (10 A e+B (d+11 e x))+6 a b^3 e \left (7 A e \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )\right )+b^4 \left (6 A e \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )+5 B \left (55 d^3 e^2 x^2+165 d^2 e^3 x^3+11 d^4 e x+d^5+330 d e^4 x^4+462 e^5 x^5\right )\right )}{13860 e^6 (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^12,x]

[Out]

-(126*a^4*e^4*(10*A*e + B*(d + 11*e*x)) + 56*a^3*b*e^3*(9*A*e*(d + 11*e*x) + 2*B*(d^2 + 11*d*e*x + 55*e^2*x^2)
) + 21*a^2*b^2*e^2*(8*A*e*(d^2 + 11*d*e*x + 55*e^2*x^2) + 3*B*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3))
+ 6*a*b^3*e*(7*A*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 4*B*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 +
165*d*e^3*x^3 + 330*e^4*x^4)) + b^4*(6*A*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4)
+ 5*B*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5)))/(13860*e^6*(d + e*
x)^11)

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Maple [B]  time = 0.009, size = 430, normalized size = 2.1 \begin{align*} -{\frac{4\,A{a}^{3}b{e}^{4}-12\,Ad{a}^{2}{b}^{2}{e}^{3}+12\,A{d}^{2}a{b}^{3}{e}^{2}-4\,A{d}^{3}{b}^{4}e+B{e}^{4}{a}^{4}-8\,Bd{a}^{3}b{e}^{3}+18\,B{d}^{2}{a}^{2}{b}^{2}{e}^{2}-16\,B{d}^{3}a{b}^{3}e+5\,{b}^{4}B{d}^{4}}{10\,{e}^{6} \left ( ex+d \right ) ^{10}}}-{\frac{{b}^{2} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+3\,{a}^{2}B{e}^{2}-8\,Bdabe+5\,B{b}^{2}{d}^{2} \right ) }{4\,{e}^{6} \left ( ex+d \right ) ^{8}}}-{\frac{{b}^{4}B}{6\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{A{a}^{4}{e}^{5}-4\,Ad{a}^{3}b{e}^{4}+6\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-4\,A{d}^{3}a{b}^{3}{e}^{2}+A{d}^{4}{b}^{4}e-Bd{a}^{4}{e}^{4}+4\,B{d}^{2}{a}^{3}b{e}^{3}-6\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+4\,B{d}^{4}a{b}^{3}e-{b}^{4}B{d}^{5}}{11\,{e}^{6} \left ( ex+d \right ) ^{11}}}-{\frac{2\,b \left ( 3\,A{a}^{2}b{e}^{3}-6\,Aa{b}^{2}d{e}^{2}+3\,A{b}^{3}{d}^{2}e+2\,B{e}^{3}{a}^{3}-9\,B{a}^{2}bd{e}^{2}+12\,Ba{b}^{2}{d}^{2}e-5\,B{b}^{3}{d}^{3} \right ) }{9\,{e}^{6} \left ( ex+d \right ) ^{9}}}-{\frac{{b}^{3} \left ( Abe+4\,aBe-5\,Bbd \right ) }{7\,{e}^{6} \left ( ex+d \right ) ^{7}}} \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)^12,x)

[Out]

-1/10*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+B*a^4*e^4-8*B*a^3*b*d*e^3+18*B*a^2*b^
2*d^2*e^2-16*B*a*b^3*d^3*e+5*B*b^4*d^4)/e^6/(e*x+d)^10-1/4*b^2*(2*A*a*b*e^2-2*A*b^2*d*e+3*B*a^2*e^2-8*B*a*b*d*
e+5*B*b^2*d^2)/e^6/(e*x+d)^8-1/6*b^4*B/e^6/(e*x+d)^6-1/11*(A*a^4*e^5-4*A*a^3*b*d*e^4+6*A*a^2*b^2*d^2*e^3-4*A*a
*b^3*d^3*e^2+A*b^4*d^4*e-B*a^4*d*e^4+4*B*a^3*b*d^2*e^3-6*B*a^2*b^2*d^3*e^2+4*B*a*b^3*d^4*e-B*b^4*d^5)/e^6/(e*x
+d)^11-2/9*b*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+2*B*a^3*e^3-9*B*a^2*b*d*e^2+12*B*a*b^2*d^2*e-5*B*b^3
*d^3)/e^6/(e*x+d)^9-1/7*b^3*(A*b*e+4*B*a*e-5*B*b*d)/e^6/(e*x+d)^7

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Maxima [B]  time = 1.24921, size = 705, normalized size = 3.42 \begin{align*} -\frac{2310 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 1260 \, A a^{4} e^{5} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 56 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 126 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 330 \,{\left (5 \, B b^{4} d e^{4} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 165 \,{\left (5 \, B b^{4} d^{2} e^{3} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 55 \,{\left (5 \, B b^{4} d^{3} e^{2} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 56 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 11 \,{\left (5 \, B b^{4} d^{4} e + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 56 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 126 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{13860 \,{\left (e^{17} x^{11} + 11 \, d e^{16} x^{10} + 55 \, d^{2} e^{15} x^{9} + 165 \, d^{3} e^{14} x^{8} + 330 \, d^{4} e^{13} x^{7} + 462 \, d^{5} e^{12} x^{6} + 462 \, d^{6} e^{11} x^{5} + 330 \, d^{7} e^{10} x^{4} + 165 \, d^{8} e^{9} x^{3} + 55 \, d^{9} e^{8} x^{2} + 11 \, d^{10} e^{7} x + d^{11} 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)^12,x, algorithm="maxima")

[Out]

-1/13860*(2310*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 1260*A*a^4*e^5 + 6*(4*B*a*b^3 + A*b^4)*d^4*e + 21*(3*B*a^2*b^2 +
2*A*a*b^3)*d^3*e^2 + 56*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 126*(B*a^4 + 4*A*a^3*b)*d*e^4 + 330*(5*B*b^4*d*e^4
+ 6*(4*B*a*b^3 + A*b^4)*e^5)*x^4 + 165*(5*B*b^4*d^2*e^3 + 6*(4*B*a*b^3 + A*b^4)*d*e^4 + 21*(3*B*a^2*b^2 + 2*A
*a*b^3)*e^5)*x^3 + 55*(5*B*b^4*d^3*e^2 + 6*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 21*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 +
56*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 11*(5*B*b^4*d^4*e + 6*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 21*(3*B*a^2*b^2 +
2*A*a*b^3)*d^2*e^3 + 56*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 126*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^17*x^11 + 11*d*e^
16*x^10 + 55*d^2*e^15*x^9 + 165*d^3*e^14*x^8 + 330*d^4*e^13*x^7 + 462*d^5*e^12*x^6 + 462*d^6*e^11*x^5 + 330*d^
7*e^10*x^4 + 165*d^8*e^9*x^3 + 55*d^9*e^8*x^2 + 11*d^10*e^7*x + d^11*e^6)

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Fricas [B]  time = 1.46746, size = 1141, normalized size = 5.54 \begin{align*} -\frac{2310 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 1260 \, A a^{4} e^{5} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 56 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 126 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 330 \,{\left (5 \, B b^{4} d e^{4} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 165 \,{\left (5 \, B b^{4} d^{2} e^{3} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 55 \,{\left (5 \, B b^{4} d^{3} e^{2} + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 56 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 11 \,{\left (5 \, B b^{4} d^{4} e + 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 21 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 56 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 126 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{13860 \,{\left (e^{17} x^{11} + 11 \, d e^{16} x^{10} + 55 \, d^{2} e^{15} x^{9} + 165 \, d^{3} e^{14} x^{8} + 330 \, d^{4} e^{13} x^{7} + 462 \, d^{5} e^{12} x^{6} + 462 \, d^{6} e^{11} x^{5} + 330 \, d^{7} e^{10} x^{4} + 165 \, d^{8} e^{9} x^{3} + 55 \, d^{9} e^{8} x^{2} + 11 \, d^{10} e^{7} x + d^{11} 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)^12,x, algorithm="fricas")

[Out]

-1/13860*(2310*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 1260*A*a^4*e^5 + 6*(4*B*a*b^3 + A*b^4)*d^4*e + 21*(3*B*a^2*b^2 +
2*A*a*b^3)*d^3*e^2 + 56*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 126*(B*a^4 + 4*A*a^3*b)*d*e^4 + 330*(5*B*b^4*d*e^4
+ 6*(4*B*a*b^3 + A*b^4)*e^5)*x^4 + 165*(5*B*b^4*d^2*e^3 + 6*(4*B*a*b^3 + A*b^4)*d*e^4 + 21*(3*B*a^2*b^2 + 2*A
*a*b^3)*e^5)*x^3 + 55*(5*B*b^4*d^3*e^2 + 6*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 21*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 +
56*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 11*(5*B*b^4*d^4*e + 6*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 21*(3*B*a^2*b^2 +
2*A*a*b^3)*d^2*e^3 + 56*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 126*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^17*x^11 + 11*d*e^
16*x^10 + 55*d^2*e^15*x^9 + 165*d^3*e^14*x^8 + 330*d^4*e^13*x^7 + 462*d^5*e^12*x^6 + 462*d^6*e^11*x^5 + 330*d^
7*e^10*x^4 + 165*d^8*e^9*x^3 + 55*d^9*e^8*x^2 + 11*d^10*e^7*x + d^11*e^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**12,x)

[Out]

Timed out

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Giac [B]  time = 1.16343, size = 594, normalized size = 2.88 \begin{align*} -\frac{{\left (2310 \, B b^{4} x^{5} e^{5} + 1650 \, B b^{4} d x^{4} e^{4} + 825 \, B b^{4} d^{2} x^{3} e^{3} + 275 \, B b^{4} d^{3} x^{2} e^{2} + 55 \, B b^{4} d^{4} x e + 5 \, B b^{4} d^{5} + 7920 \, B a b^{3} x^{4} e^{5} + 1980 \, A b^{4} x^{4} e^{5} + 3960 \, B a b^{3} d x^{3} e^{4} + 990 \, A b^{4} d x^{3} e^{4} + 1320 \, B a b^{3} d^{2} x^{2} e^{3} + 330 \, A b^{4} d^{2} x^{2} e^{3} + 264 \, B a b^{3} d^{3} x e^{2} + 66 \, A b^{4} d^{3} x e^{2} + 24 \, B a b^{3} d^{4} e + 6 \, A b^{4} d^{4} e + 10395 \, B a^{2} b^{2} x^{3} e^{5} + 6930 \, A a b^{3} x^{3} e^{5} + 3465 \, B a^{2} b^{2} d x^{2} e^{4} + 2310 \, A a b^{3} d x^{2} e^{4} + 693 \, B a^{2} b^{2} d^{2} x e^{3} + 462 \, A a b^{3} d^{2} x e^{3} + 63 \, B a^{2} b^{2} d^{3} e^{2} + 42 \, A a b^{3} d^{3} e^{2} + 6160 \, B a^{3} b x^{2} e^{5} + 9240 \, A a^{2} b^{2} x^{2} e^{5} + 1232 \, B a^{3} b d x e^{4} + 1848 \, A a^{2} b^{2} d x e^{4} + 112 \, B a^{3} b d^{2} e^{3} + 168 \, A a^{2} b^{2} d^{2} e^{3} + 1386 \, B a^{4} x e^{5} + 5544 \, A a^{3} b x e^{5} + 126 \, B a^{4} d e^{4} + 504 \, A a^{3} b d e^{4} + 1260 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{13860 \,{\left (x e + d\right )}^{11}} \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)^12,x, algorithm="giac")

[Out]

-1/13860*(2310*B*b^4*x^5*e^5 + 1650*B*b^4*d*x^4*e^4 + 825*B*b^4*d^2*x^3*e^3 + 275*B*b^4*d^3*x^2*e^2 + 55*B*b^4
*d^4*x*e + 5*B*b^4*d^5 + 7920*B*a*b^3*x^4*e^5 + 1980*A*b^4*x^4*e^5 + 3960*B*a*b^3*d*x^3*e^4 + 990*A*b^4*d*x^3*
e^4 + 1320*B*a*b^3*d^2*x^2*e^3 + 330*A*b^4*d^2*x^2*e^3 + 264*B*a*b^3*d^3*x*e^2 + 66*A*b^4*d^3*x*e^2 + 24*B*a*b
^3*d^4*e + 6*A*b^4*d^4*e + 10395*B*a^2*b^2*x^3*e^5 + 6930*A*a*b^3*x^3*e^5 + 3465*B*a^2*b^2*d*x^2*e^4 + 2310*A*
a*b^3*d*x^2*e^4 + 693*B*a^2*b^2*d^2*x*e^3 + 462*A*a*b^3*d^2*x*e^3 + 63*B*a^2*b^2*d^3*e^2 + 42*A*a*b^3*d^3*e^2
+ 6160*B*a^3*b*x^2*e^5 + 9240*A*a^2*b^2*x^2*e^5 + 1232*B*a^3*b*d*x*e^4 + 1848*A*a^2*b^2*d*x*e^4 + 112*B*a^3*b*
d^2*e^3 + 168*A*a^2*b^2*d^2*e^3 + 1386*B*a^4*x*e^5 + 5544*A*a^3*b*x*e^5 + 126*B*a^4*d*e^4 + 504*A*a^3*b*d*e^4
+ 1260*A*a^4*e^5)*e^(-6)/(x*e + d)^11