3.15.71 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{11}} \, dx\)

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

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Rubi [A]  time = 0.16, antiderivative size = 206, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {b^3 (-4 a B e-A b e+5 b B d)}{6 e^6 (d+e x)^6}-\frac {2 b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{7 e^6 (d+e x)^7}+\frac {b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{4 e^6 (d+e x)^8}-\frac {(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{9 e^6 (d+e x)^9}+\frac {(b d-a e)^4 (B d-A e)}{10 e^6 (d+e x)^{10}}-\frac {b^4 B}{5 e^6 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^11,x]

[Out]

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

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

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Mathematica [A]  time = 0.15, size = 320, normalized size = 1.55 \begin {gather*} -\frac {14 a^4 e^4 (9 A e+B (d+10 e x))+14 a^3 b e^3 \left (4 A e (d+10 e x)+B \left (d^2+10 d e x+45 e^2 x^2\right )\right )+3 a^2 b^2 e^2 \left (7 A e \left (d^2+10 d e x+45 e^2 x^2\right )+3 B \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )\right )+2 a b^3 e \left (3 A e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+2 B \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )+b^4 \left (A e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+B \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )}{1260 e^6 (d+e x)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^11,x]

[Out]

-1/1260*(14*a^4*e^4*(9*A*e + B*(d + 10*e*x)) + 14*a^3*b*e^3*(4*A*e*(d + 10*e*x) + B*(d^2 + 10*d*e*x + 45*e^2*x
^2)) + 3*a^2*b^2*e^2*(7*A*e*(d^2 + 10*d*e*x + 45*e^2*x^2) + 3*B*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3
)) + 2*a*b^3*e*(3*A*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 2*B*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2
+ 120*d*e^3*x^3 + 210*e^4*x^4)) + b^4*(A*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4)
+ B*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5)))/(e^6*(d + e*x)^10)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^11,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^11, x]

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fricas [B]  time = 0.40, size = 501, normalized size = 2.43 \begin {gather*} -\frac {252 \, B b^{4} e^{5} x^{5} + B b^{4} d^{5} + 126 \, A a^{4} e^{5} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 7 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 14 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 210 \, {\left (B b^{4} d e^{4} + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 120 \, {\left (B b^{4} d^{2} e^{3} + {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 45 \, {\left (B b^{4} d^{3} e^{2} + {\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} + 7 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 10 \, {\left (B b^{4} d^{4} e + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 7 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 14 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{1260 \, {\left (e^{16} x^{10} + 10 \, d e^{15} x^{9} + 45 \, d^{2} e^{14} x^{8} + 120 \, d^{3} e^{13} x^{7} + 210 \, d^{4} e^{12} x^{6} + 252 \, d^{5} e^{11} x^{5} + 210 \, d^{6} e^{10} x^{4} + 120 \, d^{7} e^{9} x^{3} + 45 \, d^{8} e^{8} x^{2} + 10 \, d^{9} e^{7} x + d^{10} e^{6}\right )}} \end {gather*}

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)^11,x, algorithm="fricas")

[Out]

-1/1260*(252*B*b^4*e^5*x^5 + B*b^4*d^5 + 126*A*a^4*e^5 + (4*B*a*b^3 + A*b^4)*d^4*e + 3*(3*B*a^2*b^2 + 2*A*a*b^
3)*d^3*e^2 + 7*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 14*(B*a^4 + 4*A*a^3*b)*d*e^4 + 210*(B*b^4*d*e^4 + (4*B*a*b^
3 + A*b^4)*e^5)*x^4 + 120*(B*b^4*d^2*e^3 + (4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 +
45*(B*b^4*d^3*e^2 + (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 + 7*(2*B*a^3*b + 3*A*a^2*b
^2)*e^5)*x^2 + 10*(B*b^4*d^4*e + (4*B*a*b^3 + A*b^4)*d^3*e^2 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 7*(2*B*a^
3*b + 3*A*a^2*b^2)*d*e^4 + 14*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^16*x^10 + 10*d*e^15*x^9 + 45*d^2*e^14*x^8 + 120*d
^3*e^13*x^7 + 210*d^4*e^12*x^6 + 252*d^5*e^11*x^5 + 210*d^6*e^10*x^4 + 120*d^7*e^9*x^3 + 45*d^8*e^8*x^2 + 10*d
^9*e^7*x + d^10*e^6)

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giac [B]  time = 0.17, size = 438, normalized size = 2.13 \begin {gather*} -\frac {{\left (252 \, B b^{4} x^{5} e^{5} + 210 \, B b^{4} d x^{4} e^{4} + 120 \, B b^{4} d^{2} x^{3} e^{3} + 45 \, B b^{4} d^{3} x^{2} e^{2} + 10 \, B b^{4} d^{4} x e + B b^{4} d^{5} + 840 \, B a b^{3} x^{4} e^{5} + 210 \, A b^{4} x^{4} e^{5} + 480 \, B a b^{3} d x^{3} e^{4} + 120 \, A b^{4} d x^{3} e^{4} + 180 \, B a b^{3} d^{2} x^{2} e^{3} + 45 \, A b^{4} d^{2} x^{2} e^{3} + 40 \, B a b^{3} d^{3} x e^{2} + 10 \, A b^{4} d^{3} x e^{2} + 4 \, B a b^{3} d^{4} e + A b^{4} d^{4} e + 1080 \, B a^{2} b^{2} x^{3} e^{5} + 720 \, A a b^{3} x^{3} e^{5} + 405 \, B a^{2} b^{2} d x^{2} e^{4} + 270 \, A a b^{3} d x^{2} e^{4} + 90 \, B a^{2} b^{2} d^{2} x e^{3} + 60 \, A a b^{3} d^{2} x e^{3} + 9 \, B a^{2} b^{2} d^{3} e^{2} + 6 \, A a b^{3} d^{3} e^{2} + 630 \, B a^{3} b x^{2} e^{5} + 945 \, A a^{2} b^{2} x^{2} e^{5} + 140 \, B a^{3} b d x e^{4} + 210 \, A a^{2} b^{2} d x e^{4} + 14 \, B a^{3} b d^{2} e^{3} + 21 \, A a^{2} b^{2} d^{2} e^{3} + 140 \, B a^{4} x e^{5} + 560 \, A a^{3} b x e^{5} + 14 \, B a^{4} d e^{4} + 56 \, A a^{3} b d e^{4} + 126 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{1260 \, {\left (x e + d\right )}^{10}} \end {gather*}

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)^11,x, algorithm="giac")

[Out]

-1/1260*(252*B*b^4*x^5*e^5 + 210*B*b^4*d*x^4*e^4 + 120*B*b^4*d^2*x^3*e^3 + 45*B*b^4*d^3*x^2*e^2 + 10*B*b^4*d^4
*x*e + B*b^4*d^5 + 840*B*a*b^3*x^4*e^5 + 210*A*b^4*x^4*e^5 + 480*B*a*b^3*d*x^3*e^4 + 120*A*b^4*d*x^3*e^4 + 180
*B*a*b^3*d^2*x^2*e^3 + 45*A*b^4*d^2*x^2*e^3 + 40*B*a*b^3*d^3*x*e^2 + 10*A*b^4*d^3*x*e^2 + 4*B*a*b^3*d^4*e + A*
b^4*d^4*e + 1080*B*a^2*b^2*x^3*e^5 + 720*A*a*b^3*x^3*e^5 + 405*B*a^2*b^2*d*x^2*e^4 + 270*A*a*b^3*d*x^2*e^4 + 9
0*B*a^2*b^2*d^2*x*e^3 + 60*A*a*b^3*d^2*x*e^3 + 9*B*a^2*b^2*d^3*e^2 + 6*A*a*b^3*d^3*e^2 + 630*B*a^3*b*x^2*e^5 +
945*A*a^2*b^2*x^2*e^5 + 140*B*a^3*b*d*x*e^4 + 210*A*a^2*b^2*d*x*e^4 + 14*B*a^3*b*d^2*e^3 + 21*A*a^2*b^2*d^2*e
^3 + 140*B*a^4*x*e^5 + 560*A*a^3*b*x*e^5 + 14*B*a^4*d*e^4 + 56*A*a^3*b*d*e^4 + 126*A*a^4*e^5)*e^(-6)/(x*e + d)
^10

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maple [B]  time = 0.05, size = 430, normalized size = 2.09 \begin {gather*} -\frac {B \,b^{4}}{5 \left (e x +d \right )^{5} e^{6}}-\frac {\left (A b e +4 a B e -5 B b d \right ) b^{3}}{6 \left (e x +d \right )^{6} e^{6}}-\frac {2 \left (2 A a b \,e^{2}-2 A \,b^{2} d e +3 B \,a^{2} e^{2}-8 B d a b e +5 B \,b^{2} d^{2}\right ) b^{2}}{7 \left (e x +d \right )^{7} e^{6}}-\frac {\left (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}\right ) b}{4 \left (e x +d \right )^{8} e^{6}}-\frac {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 d \,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}}{10 \left (e x +d \right )^{10} e^{6}}-\frac {4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A \,d^{2} a \,b^{3} e^{2}-4 A \,b^{4} d^{3} e +B \,a^{4} e^{4}-8 B d \,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}}{9 \left (e x +d \right )^{9} e^{6}} \end {gather*}

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)^11,x)

[Out]

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

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maxima [B]  time = 0.64, size = 501, normalized size = 2.43 \begin {gather*} -\frac {252 \, B b^{4} e^{5} x^{5} + B b^{4} d^{5} + 126 \, A a^{4} e^{5} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 7 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 14 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 210 \, {\left (B b^{4} d e^{4} + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 120 \, {\left (B b^{4} d^{2} e^{3} + {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 45 \, {\left (B b^{4} d^{3} e^{2} + {\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} + 7 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 10 \, {\left (B b^{4} d^{4} e + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 7 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 14 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{1260 \, {\left (e^{16} x^{10} + 10 \, d e^{15} x^{9} + 45 \, d^{2} e^{14} x^{8} + 120 \, d^{3} e^{13} x^{7} + 210 \, d^{4} e^{12} x^{6} + 252 \, d^{5} e^{11} x^{5} + 210 \, d^{6} e^{10} x^{4} + 120 \, d^{7} e^{9} x^{3} + 45 \, d^{8} e^{8} x^{2} + 10 \, d^{9} e^{7} x + d^{10} e^{6}\right )}} \end {gather*}

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)^11,x, algorithm="maxima")

[Out]

-1/1260*(252*B*b^4*e^5*x^5 + B*b^4*d^5 + 126*A*a^4*e^5 + (4*B*a*b^3 + A*b^4)*d^4*e + 3*(3*B*a^2*b^2 + 2*A*a*b^
3)*d^3*e^2 + 7*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 14*(B*a^4 + 4*A*a^3*b)*d*e^4 + 210*(B*b^4*d*e^4 + (4*B*a*b^
3 + A*b^4)*e^5)*x^4 + 120*(B*b^4*d^2*e^3 + (4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 +
45*(B*b^4*d^3*e^2 + (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 + 7*(2*B*a^3*b + 3*A*a^2*b
^2)*e^5)*x^2 + 10*(B*b^4*d^4*e + (4*B*a*b^3 + A*b^4)*d^3*e^2 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 7*(2*B*a^
3*b + 3*A*a^2*b^2)*d*e^4 + 14*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^16*x^10 + 10*d*e^15*x^9 + 45*d^2*e^14*x^8 + 120*d
^3*e^13*x^7 + 210*d^4*e^12*x^6 + 252*d^5*e^11*x^5 + 210*d^6*e^10*x^4 + 120*d^7*e^9*x^3 + 45*d^8*e^8*x^2 + 10*d
^9*e^7*x + d^10*e^6)

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mupad [B]  time = 0.31, size = 502, normalized size = 2.44 \begin {gather*} -\frac {\frac {14\,B\,a^4\,d\,e^4+126\,A\,a^4\,e^5+14\,B\,a^3\,b\,d^2\,e^3+56\,A\,a^3\,b\,d\,e^4+9\,B\,a^2\,b^2\,d^3\,e^2+21\,A\,a^2\,b^2\,d^2\,e^3+4\,B\,a\,b^3\,d^4\,e+6\,A\,a\,b^3\,d^3\,e^2+B\,b^4\,d^5+A\,b^4\,d^4\,e}{1260\,e^6}+\frac {x\,\left (14\,B\,a^4\,e^4+14\,B\,a^3\,b\,d\,e^3+56\,A\,a^3\,b\,e^4+9\,B\,a^2\,b^2\,d^2\,e^2+21\,A\,a^2\,b^2\,d\,e^3+4\,B\,a\,b^3\,d^3\,e+6\,A\,a\,b^3\,d^2\,e^2+B\,b^4\,d^4+A\,b^4\,d^3\,e\right )}{126\,e^5}+\frac {b^3\,x^4\,\left (A\,b\,e+4\,B\,a\,e+B\,b\,d\right )}{6\,e^2}+\frac {b\,x^2\,\left (14\,B\,a^3\,e^3+9\,B\,a^2\,b\,d\,e^2+21\,A\,a^2\,b\,e^3+4\,B\,a\,b^2\,d^2\,e+6\,A\,a\,b^2\,d\,e^2+B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{28\,e^4}+\frac {2\,b^2\,x^3\,\left (9\,B\,a^2\,e^2+4\,B\,a\,b\,d\,e+6\,A\,a\,b\,e^2+B\,b^2\,d^2+A\,b^2\,d\,e\right )}{21\,e^3}+\frac {B\,b^4\,x^5}{5\,e}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^2)/(d + e*x)^11,x)

[Out]

-((126*A*a^4*e^5 + B*b^4*d^5 + A*b^4*d^4*e + 14*B*a^4*d*e^4 + 6*A*a*b^3*d^3*e^2 + 14*B*a^3*b*d^2*e^3 + 21*A*a^
2*b^2*d^2*e^3 + 9*B*a^2*b^2*d^3*e^2 + 56*A*a^3*b*d*e^4 + 4*B*a*b^3*d^4*e)/(1260*e^6) + (x*(14*B*a^4*e^4 + B*b^
4*d^4 + 56*A*a^3*b*e^4 + A*b^4*d^3*e + 6*A*a*b^3*d^2*e^2 + 21*A*a^2*b^2*d*e^3 + 9*B*a^2*b^2*d^2*e^2 + 4*B*a*b^
3*d^3*e + 14*B*a^3*b*d*e^3))/(126*e^5) + (b^3*x^4*(A*b*e + 4*B*a*e + B*b*d))/(6*e^2) + (b*x^2*(14*B*a^3*e^3 +
B*b^3*d^3 + 21*A*a^2*b*e^3 + A*b^3*d^2*e + 6*A*a*b^2*d*e^2 + 4*B*a*b^2*d^2*e + 9*B*a^2*b*d*e^2))/(28*e^4) + (2
*b^2*x^3*(9*B*a^2*e^2 + B*b^2*d^2 + 6*A*a*b*e^2 + A*b^2*d*e + 4*B*a*b*d*e))/(21*e^3) + (B*b^4*x^5)/(5*e))/(d^1
0 + e^10*x^10 + 10*d*e^9*x^9 + 45*d^8*e^2*x^2 + 120*d^7*e^3*x^3 + 210*d^6*e^4*x^4 + 252*d^5*e^5*x^5 + 210*d^4*
e^6*x^6 + 120*d^3*e^7*x^7 + 45*d^2*e^8*x^8 + 10*d^9*e*x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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)**11,x)

[Out]

Timed out

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