3.15.53 \(\int (A+B x) (d+e x)^7 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=206 \[ -\frac {b^3 (d+e x)^{12} (-4 a B e-A b e+5 b B d)}{12 e^6}+\frac {2 b^2 (d+e x)^{11} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{11 e^6}-\frac {b (d+e x)^{10} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{5 e^6}+\frac {(d+e x)^9 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{9 e^6}-\frac {(d+e x)^8 (b d-a e)^4 (B d-A e)}{8 e^6}+\frac {b^4 B (d+e x)^{13}}{13 e^6} \]

________________________________________________________________________________________

Rubi [A]  time = 0.80, 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 (d+e x)^{12} (-4 a B e-A b e+5 b B d)}{12 e^6}+\frac {2 b^2 (d+e x)^{11} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{11 e^6}-\frac {b (d+e x)^{10} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{5 e^6}+\frac {(d+e x)^9 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{9 e^6}-\frac {(d+e x)^8 (b d-a e)^4 (B d-A e)}{8 e^6}+\frac {b^4 B (d+e x)^{13}}{13 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

-((b*d - a*e)^4*(B*d - A*e)*(d + e*x)^8)/(8*e^6) + ((b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x)^9)/(9*
e^6) - (b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^10)/(5*e^6) + (2*b^2*(b*d - a*e)*(5*b*B*d - 2*
A*b*e - 3*a*B*e)*(d + e*x)^11)/(11*e^6) - (b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^12)/(12*e^6) + (b^4*B*(d
+ e*x)^13)/(13*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 (A+B x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (A+B x) (d+e x)^7 \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (-B d+A e) (d+e x)^7}{e^5}+\frac {(-b d+a e)^3 (-5 b B d+4 A b e+a B e) (d+e x)^8}{e^5}+\frac {2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e) (d+e x)^9}{e^5}-\frac {2 b^2 (b d-a e) (-5 b B d+2 A b e+3 a B e) (d+e x)^{10}}{e^5}+\frac {b^3 (-5 b B d+A b e+4 a B e) (d+e x)^{11}}{e^5}+\frac {b^4 B (d+e x)^{12}}{e^5}\right ) \, dx\\ &=-\frac {(b d-a e)^4 (B d-A e) (d+e x)^8}{8 e^6}+\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^9}{9 e^6}-\frac {b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^{10}}{5 e^6}+\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^{11}}{11 e^6}-\frac {b^3 (5 b B d-A b e-4 a B e) (d+e x)^{12}}{12 e^6}+\frac {b^4 B (d+e x)^{13}}{13 e^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.28, size = 823, normalized size = 4.00 \begin {gather*} \frac {1}{13} b^4 B e^7 x^{13}+\frac {1}{12} b^3 e^6 (7 b B d+A b e+4 a B e) x^{12}+\frac {1}{11} b^2 e^5 \left (7 d (3 B d+A e) b^2+4 a e (7 B d+A e) b+6 a^2 B e^2\right ) x^{11}+\frac {1}{10} b e^4 \left (7 d^2 (5 B d+3 A e) b^3+28 a d e (3 B d+A e) b^2+6 a^2 e^2 (7 B d+A e) b+4 a^3 B e^3\right ) x^{10}+\frac {1}{9} e^3 \left (35 d^3 (B d+A e) b^4+28 a d^2 e (5 B d+3 A e) b^3+42 a^2 d e^2 (3 B d+A e) b^2+4 a^3 e^3 (7 B d+A e) b+a^4 B e^4\right ) x^9+\frac {1}{8} e^2 \left (7 b^4 (3 B d+5 A e) d^4+140 a b^3 e (B d+A e) d^3+42 a^2 b^2 e^2 (5 B d+3 A e) d^2+28 a^3 b e^3 (3 B d+A e) d+a^4 e^4 (7 B d+A e)\right ) x^8+d e \left (b^4 (B d+3 A e) d^4+4 a b^3 e (3 B d+5 A e) d^3+30 a^2 b^2 e^2 (B d+A e) d^2+4 a^3 b e^3 (5 B d+3 A e) d+a^4 e^4 (3 B d+A e)\right ) x^7+\frac {1}{6} d^2 \left (b^4 (B d+7 A e) d^4+28 a b^3 e (B d+3 A e) d^3+42 a^2 b^2 e^2 (3 B d+5 A e) d^2+140 a^3 b e^3 (B d+A e) d+7 a^4 e^4 (5 B d+3 A e)\right ) x^6+\frac {1}{5} d^3 \left (a B d \left (4 b^3 d^3+42 a b^2 e d^2+84 a^2 b e^2 d+35 a^3 e^3\right )+A \left (b^4 d^4+28 a b^3 e d^3+126 a^2 b^2 e^2 d^2+140 a^3 b e^3 d+35 a^4 e^4\right )\right ) x^5+\frac {1}{4} a d^4 \left (a B d \left (6 b^2 d^2+28 a b e d+21 a^2 e^2\right )+A \left (4 b^3 d^3+42 a b^2 e d^2+84 a^2 b e^2 d+35 a^3 e^3\right )\right ) x^4+\frac {1}{3} a^2 d^5 \left (a B d (4 b d+7 a e)+A \left (6 b^2 d^2+28 a b e d+21 a^2 e^2\right )\right ) x^3+\frac {1}{2} a^3 d^6 (4 A b d+a B d+7 a A e) x^2+a^4 A d^7 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*A*d^7*x + (a^3*d^6*(4*A*b*d + a*B*d + 7*a*A*e)*x^2)/2 + (a^2*d^5*(a*B*d*(4*b*d + 7*a*e) + A*(6*b^2*d^2 + 2
8*a*b*d*e + 21*a^2*e^2))*x^3)/3 + (a*d^4*(a*B*d*(6*b^2*d^2 + 28*a*b*d*e + 21*a^2*e^2) + A*(4*b^3*d^3 + 42*a*b^
2*d^2*e + 84*a^2*b*d*e^2 + 35*a^3*e^3))*x^4)/4 + (d^3*(a*B*d*(4*b^3*d^3 + 42*a*b^2*d^2*e + 84*a^2*b*d*e^2 + 35
*a^3*e^3) + A*(b^4*d^4 + 28*a*b^3*d^3*e + 126*a^2*b^2*d^2*e^2 + 140*a^3*b*d*e^3 + 35*a^4*e^4))*x^5)/5 + (d^2*(
140*a^3*b*d*e^3*(B*d + A*e) + 28*a*b^3*d^3*e*(B*d + 3*A*e) + 7*a^4*e^4*(5*B*d + 3*A*e) + 42*a^2*b^2*d^2*e^2*(3
*B*d + 5*A*e) + b^4*d^4*(B*d + 7*A*e))*x^6)/6 + d*e*(30*a^2*b^2*d^2*e^2*(B*d + A*e) + a^4*e^4*(3*B*d + A*e) +
b^4*d^4*(B*d + 3*A*e) + 4*a^3*b*d*e^3*(5*B*d + 3*A*e) + 4*a*b^3*d^3*e*(3*B*d + 5*A*e))*x^7 + (e^2*(140*a*b^3*d
^3*e*(B*d + A*e) + 28*a^3*b*d*e^3*(3*B*d + A*e) + a^4*e^4*(7*B*d + A*e) + 42*a^2*b^2*d^2*e^2*(5*B*d + 3*A*e) +
7*b^4*d^4*(3*B*d + 5*A*e))*x^8)/8 + (e^3*(a^4*B*e^4 + 35*b^4*d^3*(B*d + A*e) + 42*a^2*b^2*d*e^2*(3*B*d + A*e)
+ 4*a^3*b*e^3*(7*B*d + A*e) + 28*a*b^3*d^2*e*(5*B*d + 3*A*e))*x^9)/9 + (b*e^4*(4*a^3*B*e^3 + 28*a*b^2*d*e*(3*
B*d + A*e) + 6*a^2*b*e^2*(7*B*d + A*e) + 7*b^3*d^2*(5*B*d + 3*A*e))*x^10)/10 + (b^2*e^5*(6*a^2*B*e^2 + 7*b^2*d
*(3*B*d + A*e) + 4*a*b*e*(7*B*d + A*e))*x^11)/11 + (b^3*e^6*(7*b*B*d + A*b*e + 4*a*B*e)*x^12)/12 + (b^4*B*e^7*
x^13)/13

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^2, x]

________________________________________________________________________________________

fricas [B]  time = 0.36, size = 1175, normalized size = 5.70 \begin {gather*} \frac {1}{13} x^{13} e^{7} b^{4} B + \frac {7}{12} x^{12} e^{6} d b^{4} B + \frac {1}{3} x^{12} e^{7} b^{3} a B + \frac {1}{12} x^{12} e^{7} b^{4} A + \frac {21}{11} x^{11} e^{5} d^{2} b^{4} B + \frac {28}{11} x^{11} e^{6} d b^{3} a B + \frac {6}{11} x^{11} e^{7} b^{2} a^{2} B + \frac {7}{11} x^{11} e^{6} d b^{4} A + \frac {4}{11} x^{11} e^{7} b^{3} a A + \frac {7}{2} x^{10} e^{4} d^{3} b^{4} B + \frac {42}{5} x^{10} e^{5} d^{2} b^{3} a B + \frac {21}{5} x^{10} e^{6} d b^{2} a^{2} B + \frac {2}{5} x^{10} e^{7} b a^{3} B + \frac {21}{10} x^{10} e^{5} d^{2} b^{4} A + \frac {14}{5} x^{10} e^{6} d b^{3} a A + \frac {3}{5} x^{10} e^{7} b^{2} a^{2} A + \frac {35}{9} x^{9} e^{3} d^{4} b^{4} B + \frac {140}{9} x^{9} e^{4} d^{3} b^{3} a B + 14 x^{9} e^{5} d^{2} b^{2} a^{2} B + \frac {28}{9} x^{9} e^{6} d b a^{3} B + \frac {1}{9} x^{9} e^{7} a^{4} B + \frac {35}{9} x^{9} e^{4} d^{3} b^{4} A + \frac {28}{3} x^{9} e^{5} d^{2} b^{3} a A + \frac {14}{3} x^{9} e^{6} d b^{2} a^{2} A + \frac {4}{9} x^{9} e^{7} b a^{3} A + \frac {21}{8} x^{8} e^{2} d^{5} b^{4} B + \frac {35}{2} x^{8} e^{3} d^{4} b^{3} a B + \frac {105}{4} x^{8} e^{4} d^{3} b^{2} a^{2} B + \frac {21}{2} x^{8} e^{5} d^{2} b a^{3} B + \frac {7}{8} x^{8} e^{6} d a^{4} B + \frac {35}{8} x^{8} e^{3} d^{4} b^{4} A + \frac {35}{2} x^{8} e^{4} d^{3} b^{3} a A + \frac {63}{4} x^{8} e^{5} d^{2} b^{2} a^{2} A + \frac {7}{2} x^{8} e^{6} d b a^{3} A + \frac {1}{8} x^{8} e^{7} a^{4} A + x^{7} e d^{6} b^{4} B + 12 x^{7} e^{2} d^{5} b^{3} a B + 30 x^{7} e^{3} d^{4} b^{2} a^{2} B + 20 x^{7} e^{4} d^{3} b a^{3} B + 3 x^{7} e^{5} d^{2} a^{4} B + 3 x^{7} e^{2} d^{5} b^{4} A + 20 x^{7} e^{3} d^{4} b^{3} a A + 30 x^{7} e^{4} d^{3} b^{2} a^{2} A + 12 x^{7} e^{5} d^{2} b a^{3} A + x^{7} e^{6} d a^{4} A + \frac {1}{6} x^{6} d^{7} b^{4} B + \frac {14}{3} x^{6} e d^{6} b^{3} a B + 21 x^{6} e^{2} d^{5} b^{2} a^{2} B + \frac {70}{3} x^{6} e^{3} d^{4} b a^{3} B + \frac {35}{6} x^{6} e^{4} d^{3} a^{4} B + \frac {7}{6} x^{6} e d^{6} b^{4} A + 14 x^{6} e^{2} d^{5} b^{3} a A + 35 x^{6} e^{3} d^{4} b^{2} a^{2} A + \frac {70}{3} x^{6} e^{4} d^{3} b a^{3} A + \frac {7}{2} x^{6} e^{5} d^{2} a^{4} A + \frac {4}{5} x^{5} d^{7} b^{3} a B + \frac {42}{5} x^{5} e d^{6} b^{2} a^{2} B + \frac {84}{5} x^{5} e^{2} d^{5} b a^{3} B + 7 x^{5} e^{3} d^{4} a^{4} B + \frac {1}{5} x^{5} d^{7} b^{4} A + \frac {28}{5} x^{5} e d^{6} b^{3} a A + \frac {126}{5} x^{5} e^{2} d^{5} b^{2} a^{2} A + 28 x^{5} e^{3} d^{4} b a^{3} A + 7 x^{5} e^{4} d^{3} a^{4} A + \frac {3}{2} x^{4} d^{7} b^{2} a^{2} B + 7 x^{4} e d^{6} b a^{3} B + \frac {21}{4} x^{4} e^{2} d^{5} a^{4} B + x^{4} d^{7} b^{3} a A + \frac {21}{2} x^{4} e d^{6} b^{2} a^{2} A + 21 x^{4} e^{2} d^{5} b a^{3} A + \frac {35}{4} x^{4} e^{3} d^{4} a^{4} A + \frac {4}{3} x^{3} d^{7} b a^{3} B + \frac {7}{3} x^{3} e d^{6} a^{4} B + 2 x^{3} d^{7} b^{2} a^{2} A + \frac {28}{3} x^{3} e d^{6} b a^{3} A + 7 x^{3} e^{2} d^{5} a^{4} A + \frac {1}{2} x^{2} d^{7} a^{4} B + 2 x^{2} d^{7} b a^{3} A + \frac {7}{2} x^{2} e d^{6} a^{4} A + x d^{7} a^{4} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/13*x^13*e^7*b^4*B + 7/12*x^12*e^6*d*b^4*B + 1/3*x^12*e^7*b^3*a*B + 1/12*x^12*e^7*b^4*A + 21/11*x^11*e^5*d^2*
b^4*B + 28/11*x^11*e^6*d*b^3*a*B + 6/11*x^11*e^7*b^2*a^2*B + 7/11*x^11*e^6*d*b^4*A + 4/11*x^11*e^7*b^3*a*A + 7
/2*x^10*e^4*d^3*b^4*B + 42/5*x^10*e^5*d^2*b^3*a*B + 21/5*x^10*e^6*d*b^2*a^2*B + 2/5*x^10*e^7*b*a^3*B + 21/10*x
^10*e^5*d^2*b^4*A + 14/5*x^10*e^6*d*b^3*a*A + 3/5*x^10*e^7*b^2*a^2*A + 35/9*x^9*e^3*d^4*b^4*B + 140/9*x^9*e^4*
d^3*b^3*a*B + 14*x^9*e^5*d^2*b^2*a^2*B + 28/9*x^9*e^6*d*b*a^3*B + 1/9*x^9*e^7*a^4*B + 35/9*x^9*e^4*d^3*b^4*A +
28/3*x^9*e^5*d^2*b^3*a*A + 14/3*x^9*e^6*d*b^2*a^2*A + 4/9*x^9*e^7*b*a^3*A + 21/8*x^8*e^2*d^5*b^4*B + 35/2*x^8
*e^3*d^4*b^3*a*B + 105/4*x^8*e^4*d^3*b^2*a^2*B + 21/2*x^8*e^5*d^2*b*a^3*B + 7/8*x^8*e^6*d*a^4*B + 35/8*x^8*e^3
*d^4*b^4*A + 35/2*x^8*e^4*d^3*b^3*a*A + 63/4*x^8*e^5*d^2*b^2*a^2*A + 7/2*x^8*e^6*d*b*a^3*A + 1/8*x^8*e^7*a^4*A
+ x^7*e*d^6*b^4*B + 12*x^7*e^2*d^5*b^3*a*B + 30*x^7*e^3*d^4*b^2*a^2*B + 20*x^7*e^4*d^3*b*a^3*B + 3*x^7*e^5*d^
2*a^4*B + 3*x^7*e^2*d^5*b^4*A + 20*x^7*e^3*d^4*b^3*a*A + 30*x^7*e^4*d^3*b^2*a^2*A + 12*x^7*e^5*d^2*b*a^3*A + x
^7*e^6*d*a^4*A + 1/6*x^6*d^7*b^4*B + 14/3*x^6*e*d^6*b^3*a*B + 21*x^6*e^2*d^5*b^2*a^2*B + 70/3*x^6*e^3*d^4*b*a^
3*B + 35/6*x^6*e^4*d^3*a^4*B + 7/6*x^6*e*d^6*b^4*A + 14*x^6*e^2*d^5*b^3*a*A + 35*x^6*e^3*d^4*b^2*a^2*A + 70/3*
x^6*e^4*d^3*b*a^3*A + 7/2*x^6*e^5*d^2*a^4*A + 4/5*x^5*d^7*b^3*a*B + 42/5*x^5*e*d^6*b^2*a^2*B + 84/5*x^5*e^2*d^
5*b*a^3*B + 7*x^5*e^3*d^4*a^4*B + 1/5*x^5*d^7*b^4*A + 28/5*x^5*e*d^6*b^3*a*A + 126/5*x^5*e^2*d^5*b^2*a^2*A + 2
8*x^5*e^3*d^4*b*a^3*A + 7*x^5*e^4*d^3*a^4*A + 3/2*x^4*d^7*b^2*a^2*B + 7*x^4*e*d^6*b*a^3*B + 21/4*x^4*e^2*d^5*a
^4*B + x^4*d^7*b^3*a*A + 21/2*x^4*e*d^6*b^2*a^2*A + 21*x^4*e^2*d^5*b*a^3*A + 35/4*x^4*e^3*d^4*a^4*A + 4/3*x^3*
d^7*b*a^3*B + 7/3*x^3*e*d^6*a^4*B + 2*x^3*d^7*b^2*a^2*A + 28/3*x^3*e*d^6*b*a^3*A + 7*x^3*e^2*d^5*a^4*A + 1/2*x
^2*d^7*a^4*B + 2*x^2*d^7*b*a^3*A + 7/2*x^2*e*d^6*a^4*A + x*d^7*a^4*A

________________________________________________________________________________________

giac [B]  time = 0.17, size = 1125, normalized size = 5.46

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/13*B*b^4*x^13*e^7 + 7/12*B*b^4*d*x^12*e^6 + 21/11*B*b^4*d^2*x^11*e^5 + 7/2*B*b^4*d^3*x^10*e^4 + 35/9*B*b^4*d
^4*x^9*e^3 + 21/8*B*b^4*d^5*x^8*e^2 + B*b^4*d^6*x^7*e + 1/6*B*b^4*d^7*x^6 + 1/3*B*a*b^3*x^12*e^7 + 1/12*A*b^4*
x^12*e^7 + 28/11*B*a*b^3*d*x^11*e^6 + 7/11*A*b^4*d*x^11*e^6 + 42/5*B*a*b^3*d^2*x^10*e^5 + 21/10*A*b^4*d^2*x^10
*e^5 + 140/9*B*a*b^3*d^3*x^9*e^4 + 35/9*A*b^4*d^3*x^9*e^4 + 35/2*B*a*b^3*d^4*x^8*e^3 + 35/8*A*b^4*d^4*x^8*e^3
+ 12*B*a*b^3*d^5*x^7*e^2 + 3*A*b^4*d^5*x^7*e^2 + 14/3*B*a*b^3*d^6*x^6*e + 7/6*A*b^4*d^6*x^6*e + 4/5*B*a*b^3*d^
7*x^5 + 1/5*A*b^4*d^7*x^5 + 6/11*B*a^2*b^2*x^11*e^7 + 4/11*A*a*b^3*x^11*e^7 + 21/5*B*a^2*b^2*d*x^10*e^6 + 14/5
*A*a*b^3*d*x^10*e^6 + 14*B*a^2*b^2*d^2*x^9*e^5 + 28/3*A*a*b^3*d^2*x^9*e^5 + 105/4*B*a^2*b^2*d^3*x^8*e^4 + 35/2
*A*a*b^3*d^3*x^8*e^4 + 30*B*a^2*b^2*d^4*x^7*e^3 + 20*A*a*b^3*d^4*x^7*e^3 + 21*B*a^2*b^2*d^5*x^6*e^2 + 14*A*a*b
^3*d^5*x^6*e^2 + 42/5*B*a^2*b^2*d^6*x^5*e + 28/5*A*a*b^3*d^6*x^5*e + 3/2*B*a^2*b^2*d^7*x^4 + A*a*b^3*d^7*x^4 +
2/5*B*a^3*b*x^10*e^7 + 3/5*A*a^2*b^2*x^10*e^7 + 28/9*B*a^3*b*d*x^9*e^6 + 14/3*A*a^2*b^2*d*x^9*e^6 + 21/2*B*a^
3*b*d^2*x^8*e^5 + 63/4*A*a^2*b^2*d^2*x^8*e^5 + 20*B*a^3*b*d^3*x^7*e^4 + 30*A*a^2*b^2*d^3*x^7*e^4 + 70/3*B*a^3*
b*d^4*x^6*e^3 + 35*A*a^2*b^2*d^4*x^6*e^3 + 84/5*B*a^3*b*d^5*x^5*e^2 + 126/5*A*a^2*b^2*d^5*x^5*e^2 + 7*B*a^3*b*
d^6*x^4*e + 21/2*A*a^2*b^2*d^6*x^4*e + 4/3*B*a^3*b*d^7*x^3 + 2*A*a^2*b^2*d^7*x^3 + 1/9*B*a^4*x^9*e^7 + 4/9*A*a
^3*b*x^9*e^7 + 7/8*B*a^4*d*x^8*e^6 + 7/2*A*a^3*b*d*x^8*e^6 + 3*B*a^4*d^2*x^7*e^5 + 12*A*a^3*b*d^2*x^7*e^5 + 35
/6*B*a^4*d^3*x^6*e^4 + 70/3*A*a^3*b*d^3*x^6*e^4 + 7*B*a^4*d^4*x^5*e^3 + 28*A*a^3*b*d^4*x^5*e^3 + 21/4*B*a^4*d^
5*x^4*e^2 + 21*A*a^3*b*d^5*x^4*e^2 + 7/3*B*a^4*d^6*x^3*e + 28/3*A*a^3*b*d^6*x^3*e + 1/2*B*a^4*d^7*x^2 + 2*A*a^
3*b*d^7*x^2 + 1/8*A*a^4*x^8*e^7 + A*a^4*d*x^7*e^6 + 7/2*A*a^4*d^2*x^6*e^5 + 7*A*a^4*d^3*x^5*e^4 + 35/4*A*a^4*d
^4*x^4*e^3 + 7*A*a^4*d^5*x^3*e^2 + 7/2*A*a^4*d^6*x^2*e + A*a^4*d^7*x

________________________________________________________________________________________

maple [B]  time = 0.04, size = 950, normalized size = 4.61 \begin {gather*} \frac {B \,b^{4} e^{7} x^{13}}{13}+A \,a^{4} d^{7} x +\frac {\left (4 B a \,b^{3} e^{7}+\left (A \,e^{7}+7 B d \,e^{6}\right ) b^{4}\right ) x^{12}}{12}+\frac {\left (6 B \,a^{2} b^{2} e^{7}+4 \left (A \,e^{7}+7 B d \,e^{6}\right ) a \,b^{3}+\left (7 A d \,e^{6}+21 B \,d^{2} e^{5}\right ) b^{4}\right ) x^{11}}{11}+\frac {\left (4 B \,a^{3} b \,e^{7}+6 \left (A \,e^{7}+7 B d \,e^{6}\right ) a^{2} b^{2}+4 \left (7 A d \,e^{6}+21 B \,d^{2} e^{5}\right ) a \,b^{3}+\left (21 A \,d^{2} e^{5}+35 B \,d^{3} e^{4}\right ) b^{4}\right ) x^{10}}{10}+\frac {\left (B \,a^{4} e^{7}+4 \left (A \,e^{7}+7 B d \,e^{6}\right ) a^{3} b +6 \left (7 A d \,e^{6}+21 B \,d^{2} e^{5}\right ) a^{2} b^{2}+4 \left (21 A \,d^{2} e^{5}+35 B \,d^{3} e^{4}\right ) a \,b^{3}+\left (35 A \,d^{3} e^{4}+35 B \,d^{4} e^{3}\right ) b^{4}\right ) x^{9}}{9}+\frac {\left (\left (A \,e^{7}+7 B d \,e^{6}\right ) a^{4}+4 \left (7 A d \,e^{6}+21 B \,d^{2} e^{5}\right ) a^{3} b +6 \left (21 A \,d^{2} e^{5}+35 B \,d^{3} e^{4}\right ) a^{2} b^{2}+4 \left (35 A \,d^{3} e^{4}+35 B \,d^{4} e^{3}\right ) a \,b^{3}+\left (35 A \,d^{4} e^{3}+21 B \,d^{5} e^{2}\right ) b^{4}\right ) x^{8}}{8}+\frac {\left (\left (7 A d \,e^{6}+21 B \,d^{2} e^{5}\right ) a^{4}+4 \left (21 A \,d^{2} e^{5}+35 B \,d^{3} e^{4}\right ) a^{3} b +6 \left (35 A \,d^{3} e^{4}+35 B \,d^{4} e^{3}\right ) a^{2} b^{2}+4 \left (35 A \,d^{4} e^{3}+21 B \,d^{5} e^{2}\right ) a \,b^{3}+\left (21 A \,d^{5} e^{2}+7 B \,d^{6} e \right ) b^{4}\right ) x^{7}}{7}+\frac {\left (\left (21 A \,d^{2} e^{5}+35 B \,d^{3} e^{4}\right ) a^{4}+4 \left (35 A \,d^{3} e^{4}+35 B \,d^{4} e^{3}\right ) a^{3} b +6 \left (35 A \,d^{4} e^{3}+21 B \,d^{5} e^{2}\right ) a^{2} b^{2}+4 \left (21 A \,d^{5} e^{2}+7 B \,d^{6} e \right ) a \,b^{3}+\left (7 A \,d^{6} e +B \,d^{7}\right ) b^{4}\right ) x^{6}}{6}+\frac {\left (A \,b^{4} d^{7}+\left (35 A \,d^{3} e^{4}+35 B \,d^{4} e^{3}\right ) a^{4}+4 \left (35 A \,d^{4} e^{3}+21 B \,d^{5} e^{2}\right ) a^{3} b +6 \left (21 A \,d^{5} e^{2}+7 B \,d^{6} e \right ) a^{2} b^{2}+4 \left (7 A \,d^{6} e +B \,d^{7}\right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 A a \,b^{3} d^{7}+\left (35 A \,d^{4} e^{3}+21 B \,d^{5} e^{2}\right ) a^{4}+4 \left (21 A \,d^{5} e^{2}+7 B \,d^{6} e \right ) a^{3} b +6 \left (7 A \,d^{6} e +B \,d^{7}\right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (6 A \,a^{2} b^{2} d^{7}+\left (21 A \,d^{5} e^{2}+7 B \,d^{6} e \right ) a^{4}+4 \left (7 A \,d^{6} e +B \,d^{7}\right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 A \,a^{3} b \,d^{7}+\left (7 A \,d^{6} e +B \,d^{7}\right ) a^{4}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/13*B*e^7*b^4*x^13+1/12*((A*e^7+7*B*d*e^6)*b^4+4*B*e^7*a*b^3)*x^12+1/11*((7*A*d*e^6+21*B*d^2*e^5)*b^4+4*(A*e^
7+7*B*d*e^6)*a*b^3+6*B*e^7*a^2*b^2)*x^11+1/10*((21*A*d^2*e^5+35*B*d^3*e^4)*b^4+4*(7*A*d*e^6+21*B*d^2*e^5)*a*b^
3+6*(A*e^7+7*B*d*e^6)*a^2*b^2+4*B*e^7*a^3*b)*x^10+1/9*((35*A*d^3*e^4+35*B*d^4*e^3)*b^4+4*(21*A*d^2*e^5+35*B*d^
3*e^4)*a*b^3+6*(7*A*d*e^6+21*B*d^2*e^5)*a^2*b^2+4*(A*e^7+7*B*d*e^6)*a^3*b+B*e^7*a^4)*x^9+1/8*((35*A*d^4*e^3+21
*B*d^5*e^2)*b^4+4*(35*A*d^3*e^4+35*B*d^4*e^3)*a*b^3+6*(21*A*d^2*e^5+35*B*d^3*e^4)*a^2*b^2+4*(7*A*d*e^6+21*B*d^
2*e^5)*a^3*b+(A*e^7+7*B*d*e^6)*a^4)*x^8+1/7*((21*A*d^5*e^2+7*B*d^6*e)*b^4+4*(35*A*d^4*e^3+21*B*d^5*e^2)*a*b^3+
6*(35*A*d^3*e^4+35*B*d^4*e^3)*a^2*b^2+4*(21*A*d^2*e^5+35*B*d^3*e^4)*a^3*b+(7*A*d*e^6+21*B*d^2*e^5)*a^4)*x^7+1/
6*((7*A*d^6*e+B*d^7)*b^4+4*(21*A*d^5*e^2+7*B*d^6*e)*a*b^3+6*(35*A*d^4*e^3+21*B*d^5*e^2)*a^2*b^2+4*(35*A*d^3*e^
4+35*B*d^4*e^3)*a^3*b+(21*A*d^2*e^5+35*B*d^3*e^4)*a^4)*x^6+1/5*(A*d^7*b^4+4*(7*A*d^6*e+B*d^7)*a*b^3+6*(21*A*d^
5*e^2+7*B*d^6*e)*a^2*b^2+4*(35*A*d^4*e^3+21*B*d^5*e^2)*a^3*b+(35*A*d^3*e^4+35*B*d^4*e^3)*a^4)*x^5+1/4*(4*A*d^7
*a*b^3+6*(7*A*d^6*e+B*d^7)*a^2*b^2+4*(21*A*d^5*e^2+7*B*d^6*e)*a^3*b+(35*A*d^4*e^3+21*B*d^5*e^2)*a^4)*x^4+1/3*(
6*A*d^7*a^2*b^2+4*(7*A*d^6*e+B*d^7)*a^3*b+(21*A*d^5*e^2+7*B*d^6*e)*a^4)*x^3+1/2*(4*A*d^7*a^3*b+(7*A*d^6*e+B*d^
7)*a^4)*x^2+A*d^7*a^4*x

________________________________________________________________________________________

maxima [B]  time = 0.65, size = 929, normalized size = 4.51 \begin {gather*} \frac {1}{13} \, B b^{4} e^{7} x^{13} + A a^{4} d^{7} x + \frac {1}{12} \, {\left (7 \, B b^{4} d e^{6} + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{7}\right )} x^{12} + \frac {1}{11} \, {\left (21 \, B b^{4} d^{2} e^{5} + 7 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{6} + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{7}\right )} x^{11} + \frac {1}{10} \, {\left (35 \, B b^{4} d^{3} e^{4} + 21 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{5} + 14 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{6} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{7}\right )} x^{10} + \frac {1}{9} \, {\left (35 \, B b^{4} d^{4} e^{3} + 35 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{4} + 42 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{5} + 14 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{6} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{7}\right )} x^{9} + \frac {1}{8} \, {\left (21 \, B b^{4} d^{5} e^{2} + A a^{4} e^{7} + 35 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e^{3} + 70 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{4} + 42 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{5} + 7 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{6}\right )} x^{8} + {\left (B b^{4} d^{6} e + A a^{4} d e^{6} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{5} e^{2} + 10 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} e^{3} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e^{4} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{4} d^{7} + 21 \, A a^{4} d^{2} e^{5} + 7 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{6} e + 42 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{5} e^{2} + 70 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{4} e^{3} + 35 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{3} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (35 \, A a^{4} d^{3} e^{4} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{7} + 14 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{6} e + 42 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{5} e^{2} + 35 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{4} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (35 \, A a^{4} d^{4} e^{3} + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{7} + 14 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{6} e + 21 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{5} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, A a^{4} d^{5} e^{2} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{7} + 7 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{6} e\right )} x^{3} + \frac {1}{2} \, {\left (7 \, A a^{4} d^{6} e + {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{7}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/13*B*b^4*e^7*x^13 + A*a^4*d^7*x + 1/12*(7*B*b^4*d*e^6 + (4*B*a*b^3 + A*b^4)*e^7)*x^12 + 1/11*(21*B*b^4*d^2*e
^5 + 7*(4*B*a*b^3 + A*b^4)*d*e^6 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*e^7)*x^11 + 1/10*(35*B*b^4*d^3*e^4 + 21*(4*B*a*
b^3 + A*b^4)*d^2*e^5 + 14*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^6 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*e^7)*x^10 + 1/9*(35*B*
b^4*d^4*e^3 + 35*(4*B*a*b^3 + A*b^4)*d^3*e^4 + 42*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^5 + 14*(2*B*a^3*b + 3*A*a^2*
b^2)*d*e^6 + (B*a^4 + 4*A*a^3*b)*e^7)*x^9 + 1/8*(21*B*b^4*d^5*e^2 + A*a^4*e^7 + 35*(4*B*a*b^3 + A*b^4)*d^4*e^3
+ 70*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^4 + 42*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^5 + 7*(B*a^4 + 4*A*a^3*b)*d*e^6)*
x^8 + (B*b^4*d^6*e + A*a^4*d*e^6 + 3*(4*B*a*b^3 + A*b^4)*d^5*e^2 + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e^3 + 10*(
2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^4 + 3*(B*a^4 + 4*A*a^3*b)*d^2*e^5)*x^7 + 1/6*(B*b^4*d^7 + 21*A*a^4*d^2*e^5 + 7*
(4*B*a*b^3 + A*b^4)*d^6*e + 42*(3*B*a^2*b^2 + 2*A*a*b^3)*d^5*e^2 + 70*(2*B*a^3*b + 3*A*a^2*b^2)*d^4*e^3 + 35*(
B*a^4 + 4*A*a^3*b)*d^3*e^4)*x^6 + 1/5*(35*A*a^4*d^3*e^4 + (4*B*a*b^3 + A*b^4)*d^7 + 14*(3*B*a^2*b^2 + 2*A*a*b^
3)*d^6*e + 42*(2*B*a^3*b + 3*A*a^2*b^2)*d^5*e^2 + 35*(B*a^4 + 4*A*a^3*b)*d^4*e^3)*x^5 + 1/4*(35*A*a^4*d^4*e^3
+ 2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^7 + 14*(2*B*a^3*b + 3*A*a^2*b^2)*d^6*e + 21*(B*a^4 + 4*A*a^3*b)*d^5*e^2)*x^4 +
1/3*(21*A*a^4*d^5*e^2 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^7 + 7*(B*a^4 + 4*A*a^3*b)*d^6*e)*x^3 + 1/2*(7*A*a^4*d^6
*e + (B*a^4 + 4*A*a^3*b)*d^7)*x^2

________________________________________________________________________________________

mupad [B]  time = 2.30, size = 980, normalized size = 4.76 \begin {gather*} x^7\,\left (3\,B\,a^4\,d^2\,e^5+A\,a^4\,d\,e^6+20\,B\,a^3\,b\,d^3\,e^4+12\,A\,a^3\,b\,d^2\,e^5+30\,B\,a^2\,b^2\,d^4\,e^3+30\,A\,a^2\,b^2\,d^3\,e^4+12\,B\,a\,b^3\,d^5\,e^2+20\,A\,a\,b^3\,d^4\,e^3+B\,b^4\,d^6\,e+3\,A\,b^4\,d^5\,e^2\right )+x^6\,\left (\frac {35\,B\,a^4\,d^3\,e^4}{6}+\frac {7\,A\,a^4\,d^2\,e^5}{2}+\frac {70\,B\,a^3\,b\,d^4\,e^3}{3}+\frac {70\,A\,a^3\,b\,d^3\,e^4}{3}+21\,B\,a^2\,b^2\,d^5\,e^2+35\,A\,a^2\,b^2\,d^4\,e^3+\frac {14\,B\,a\,b^3\,d^6\,e}{3}+14\,A\,a\,b^3\,d^5\,e^2+\frac {B\,b^4\,d^7}{6}+\frac {7\,A\,b^4\,d^6\,e}{6}\right )+x^8\,\left (\frac {7\,B\,a^4\,d\,e^6}{8}+\frac {A\,a^4\,e^7}{8}+\frac {21\,B\,a^3\,b\,d^2\,e^5}{2}+\frac {7\,A\,a^3\,b\,d\,e^6}{2}+\frac {105\,B\,a^2\,b^2\,d^3\,e^4}{4}+\frac {63\,A\,a^2\,b^2\,d^2\,e^5}{4}+\frac {35\,B\,a\,b^3\,d^4\,e^3}{2}+\frac {35\,A\,a\,b^3\,d^3\,e^4}{2}+\frac {21\,B\,b^4\,d^5\,e^2}{8}+\frac {35\,A\,b^4\,d^4\,e^3}{8}\right )+x^4\,\left (\frac {21\,B\,a^4\,d^5\,e^2}{4}+\frac {35\,A\,a^4\,d^4\,e^3}{4}+7\,B\,a^3\,b\,d^6\,e+21\,A\,a^3\,b\,d^5\,e^2+\frac {3\,B\,a^2\,b^2\,d^7}{2}+\frac {21\,A\,a^2\,b^2\,d^6\,e}{2}+A\,a\,b^3\,d^7\right )+x^{10}\,\left (\frac {2\,B\,a^3\,b\,e^7}{5}+\frac {21\,B\,a^2\,b^2\,d\,e^6}{5}+\frac {3\,A\,a^2\,b^2\,e^7}{5}+\frac {42\,B\,a\,b^3\,d^2\,e^5}{5}+\frac {14\,A\,a\,b^3\,d\,e^6}{5}+\frac {7\,B\,b^4\,d^3\,e^4}{2}+\frac {21\,A\,b^4\,d^2\,e^5}{10}\right )+x^3\,\left (\frac {7\,B\,a^4\,d^6\,e}{3}+7\,A\,a^4\,d^5\,e^2+\frac {4\,B\,a^3\,b\,d^7}{3}+\frac {28\,A\,a^3\,b\,d^6\,e}{3}+2\,A\,a^2\,b^2\,d^7\right )+x^{11}\,\left (\frac {6\,B\,a^2\,b^2\,e^7}{11}+\frac {28\,B\,a\,b^3\,d\,e^6}{11}+\frac {4\,A\,a\,b^3\,e^7}{11}+\frac {21\,B\,b^4\,d^2\,e^5}{11}+\frac {7\,A\,b^4\,d\,e^6}{11}\right )+x^5\,\left (7\,B\,a^4\,d^4\,e^3+7\,A\,a^4\,d^3\,e^4+\frac {84\,B\,a^3\,b\,d^5\,e^2}{5}+28\,A\,a^3\,b\,d^4\,e^3+\frac {42\,B\,a^2\,b^2\,d^6\,e}{5}+\frac {126\,A\,a^2\,b^2\,d^5\,e^2}{5}+\frac {4\,B\,a\,b^3\,d^7}{5}+\frac {28\,A\,a\,b^3\,d^6\,e}{5}+\frac {A\,b^4\,d^7}{5}\right )+x^9\,\left (\frac {B\,a^4\,e^7}{9}+\frac {28\,B\,a^3\,b\,d\,e^6}{9}+\frac {4\,A\,a^3\,b\,e^7}{9}+14\,B\,a^2\,b^2\,d^2\,e^5+\frac {14\,A\,a^2\,b^2\,d\,e^6}{3}+\frac {140\,B\,a\,b^3\,d^3\,e^4}{9}+\frac {28\,A\,a\,b^3\,d^2\,e^5}{3}+\frac {35\,B\,b^4\,d^4\,e^3}{9}+\frac {35\,A\,b^4\,d^3\,e^4}{9}\right )+\frac {a^3\,d^6\,x^2\,\left (7\,A\,a\,e+4\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^3\,e^6\,x^{12}\,\left (A\,b\,e+4\,B\,a\,e+7\,B\,b\,d\right )}{12}+A\,a^4\,d^7\,x+\frac {B\,b^4\,e^7\,x^{13}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^7*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

x^7*(A*a^4*d*e^6 + B*b^4*d^6*e + 3*A*b^4*d^5*e^2 + 3*B*a^4*d^2*e^5 + 20*A*a*b^3*d^4*e^3 + 12*A*a^3*b*d^2*e^5 +
12*B*a*b^3*d^5*e^2 + 20*B*a^3*b*d^3*e^4 + 30*A*a^2*b^2*d^3*e^4 + 30*B*a^2*b^2*d^4*e^3) + x^6*((B*b^4*d^7)/6 +
(7*A*b^4*d^6*e)/6 + (7*A*a^4*d^2*e^5)/2 + (35*B*a^4*d^3*e^4)/6 + 14*A*a*b^3*d^5*e^2 + (70*A*a^3*b*d^3*e^4)/3
+ (70*B*a^3*b*d^4*e^3)/3 + 35*A*a^2*b^2*d^4*e^3 + 21*B*a^2*b^2*d^5*e^2 + (14*B*a*b^3*d^6*e)/3) + x^8*((A*a^4*e
^7)/8 + (7*B*a^4*d*e^6)/8 + (35*A*b^4*d^4*e^3)/8 + (21*B*b^4*d^5*e^2)/8 + (35*A*a*b^3*d^3*e^4)/2 + (35*B*a*b^3
*d^4*e^3)/2 + (21*B*a^3*b*d^2*e^5)/2 + (63*A*a^2*b^2*d^2*e^5)/4 + (105*B*a^2*b^2*d^3*e^4)/4 + (7*A*a^3*b*d*e^6
)/2) + x^4*(A*a*b^3*d^7 + (3*B*a^2*b^2*d^7)/2 + (35*A*a^4*d^4*e^3)/4 + (21*B*a^4*d^5*e^2)/4 + (21*A*a^2*b^2*d^
6*e)/2 + 21*A*a^3*b*d^5*e^2 + 7*B*a^3*b*d^6*e) + x^10*((2*B*a^3*b*e^7)/5 + (3*A*a^2*b^2*e^7)/5 + (21*A*b^4*d^2
*e^5)/10 + (7*B*b^4*d^3*e^4)/2 + (42*B*a*b^3*d^2*e^5)/5 + (21*B*a^2*b^2*d*e^6)/5 + (14*A*a*b^3*d*e^6)/5) + x^3
*((4*B*a^3*b*d^7)/3 + (7*B*a^4*d^6*e)/3 + 2*A*a^2*b^2*d^7 + 7*A*a^4*d^5*e^2 + (28*A*a^3*b*d^6*e)/3) + x^11*((4
*A*a*b^3*e^7)/11 + (7*A*b^4*d*e^6)/11 + (6*B*a^2*b^2*e^7)/11 + (21*B*b^4*d^2*e^5)/11 + (28*B*a*b^3*d*e^6)/11)
+ x^5*((A*b^4*d^7)/5 + (4*B*a*b^3*d^7)/5 + 7*A*a^4*d^3*e^4 + 7*B*a^4*d^4*e^3 + 28*A*a^3*b*d^4*e^3 + (42*B*a^2*
b^2*d^6*e)/5 + (84*B*a^3*b*d^5*e^2)/5 + (126*A*a^2*b^2*d^5*e^2)/5 + (28*A*a*b^3*d^6*e)/5) + x^9*((B*a^4*e^7)/9
+ (4*A*a^3*b*e^7)/9 + (35*A*b^4*d^3*e^4)/9 + (35*B*b^4*d^4*e^3)/9 + (28*A*a*b^3*d^2*e^5)/3 + (14*A*a^2*b^2*d*
e^6)/3 + (140*B*a*b^3*d^3*e^4)/9 + 14*B*a^2*b^2*d^2*e^5 + (28*B*a^3*b*d*e^6)/9) + (a^3*d^6*x^2*(7*A*a*e + 4*A*
b*d + B*a*d))/2 + (b^3*e^6*x^12*(A*b*e + 4*B*a*e + 7*B*b*d))/12 + A*a^4*d^7*x + (B*b^4*e^7*x^13)/13

________________________________________________________________________________________

sympy [B]  time = 0.22, size = 1210, normalized size = 5.87 \begin {gather*} A a^{4} d^{7} x + \frac {B b^{4} e^{7} x^{13}}{13} + x^{12} \left (\frac {A b^{4} e^{7}}{12} + \frac {B a b^{3} e^{7}}{3} + \frac {7 B b^{4} d e^{6}}{12}\right ) + x^{11} \left (\frac {4 A a b^{3} e^{7}}{11} + \frac {7 A b^{4} d e^{6}}{11} + \frac {6 B a^{2} b^{2} e^{7}}{11} + \frac {28 B a b^{3} d e^{6}}{11} + \frac {21 B b^{4} d^{2} e^{5}}{11}\right ) + x^{10} \left (\frac {3 A a^{2} b^{2} e^{7}}{5} + \frac {14 A a b^{3} d e^{6}}{5} + \frac {21 A b^{4} d^{2} e^{5}}{10} + \frac {2 B a^{3} b e^{7}}{5} + \frac {21 B a^{2} b^{2} d e^{6}}{5} + \frac {42 B a b^{3} d^{2} e^{5}}{5} + \frac {7 B b^{4} d^{3} e^{4}}{2}\right ) + x^{9} \left (\frac {4 A a^{3} b e^{7}}{9} + \frac {14 A a^{2} b^{2} d e^{6}}{3} + \frac {28 A a b^{3} d^{2} e^{5}}{3} + \frac {35 A b^{4} d^{3} e^{4}}{9} + \frac {B a^{4} e^{7}}{9} + \frac {28 B a^{3} b d e^{6}}{9} + 14 B a^{2} b^{2} d^{2} e^{5} + \frac {140 B a b^{3} d^{3} e^{4}}{9} + \frac {35 B b^{4} d^{4} e^{3}}{9}\right ) + x^{8} \left (\frac {A a^{4} e^{7}}{8} + \frac {7 A a^{3} b d e^{6}}{2} + \frac {63 A a^{2} b^{2} d^{2} e^{5}}{4} + \frac {35 A a b^{3} d^{3} e^{4}}{2} + \frac {35 A b^{4} d^{4} e^{3}}{8} + \frac {7 B a^{4} d e^{6}}{8} + \frac {21 B a^{3} b d^{2} e^{5}}{2} + \frac {105 B a^{2} b^{2} d^{3} e^{4}}{4} + \frac {35 B a b^{3} d^{4} e^{3}}{2} + \frac {21 B b^{4} d^{5} e^{2}}{8}\right ) + x^{7} \left (A a^{4} d e^{6} + 12 A a^{3} b d^{2} e^{5} + 30 A a^{2} b^{2} d^{3} e^{4} + 20 A a b^{3} d^{4} e^{3} + 3 A b^{4} d^{5} e^{2} + 3 B a^{4} d^{2} e^{5} + 20 B a^{3} b d^{3} e^{4} + 30 B a^{2} b^{2} d^{4} e^{3} + 12 B a b^{3} d^{5} e^{2} + B b^{4} d^{6} e\right ) + x^{6} \left (\frac {7 A a^{4} d^{2} e^{5}}{2} + \frac {70 A a^{3} b d^{3} e^{4}}{3} + 35 A a^{2} b^{2} d^{4} e^{3} + 14 A a b^{3} d^{5} e^{2} + \frac {7 A b^{4} d^{6} e}{6} + \frac {35 B a^{4} d^{3} e^{4}}{6} + \frac {70 B a^{3} b d^{4} e^{3}}{3} + 21 B a^{2} b^{2} d^{5} e^{2} + \frac {14 B a b^{3} d^{6} e}{3} + \frac {B b^{4} d^{7}}{6}\right ) + x^{5} \left (7 A a^{4} d^{3} e^{4} + 28 A a^{3} b d^{4} e^{3} + \frac {126 A a^{2} b^{2} d^{5} e^{2}}{5} + \frac {28 A a b^{3} d^{6} e}{5} + \frac {A b^{4} d^{7}}{5} + 7 B a^{4} d^{4} e^{3} + \frac {84 B a^{3} b d^{5} e^{2}}{5} + \frac {42 B a^{2} b^{2} d^{6} e}{5} + \frac {4 B a b^{3} d^{7}}{5}\right ) + x^{4} \left (\frac {35 A a^{4} d^{4} e^{3}}{4} + 21 A a^{3} b d^{5} e^{2} + \frac {21 A a^{2} b^{2} d^{6} e}{2} + A a b^{3} d^{7} + \frac {21 B a^{4} d^{5} e^{2}}{4} + 7 B a^{3} b d^{6} e + \frac {3 B a^{2} b^{2} d^{7}}{2}\right ) + x^{3} \left (7 A a^{4} d^{5} e^{2} + \frac {28 A a^{3} b d^{6} e}{3} + 2 A a^{2} b^{2} d^{7} + \frac {7 B a^{4} d^{6} e}{3} + \frac {4 B a^{3} b d^{7}}{3}\right ) + x^{2} \left (\frac {7 A a^{4} d^{6} e}{2} + 2 A a^{3} b d^{7} + \frac {B a^{4} d^{7}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**7*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*d**7*x + B*b**4*e**7*x**13/13 + x**12*(A*b**4*e**7/12 + B*a*b**3*e**7/3 + 7*B*b**4*d*e**6/12) + x**11*(
4*A*a*b**3*e**7/11 + 7*A*b**4*d*e**6/11 + 6*B*a**2*b**2*e**7/11 + 28*B*a*b**3*d*e**6/11 + 21*B*b**4*d**2*e**5/
11) + x**10*(3*A*a**2*b**2*e**7/5 + 14*A*a*b**3*d*e**6/5 + 21*A*b**4*d**2*e**5/10 + 2*B*a**3*b*e**7/5 + 21*B*a
**2*b**2*d*e**6/5 + 42*B*a*b**3*d**2*e**5/5 + 7*B*b**4*d**3*e**4/2) + x**9*(4*A*a**3*b*e**7/9 + 14*A*a**2*b**2
*d*e**6/3 + 28*A*a*b**3*d**2*e**5/3 + 35*A*b**4*d**3*e**4/9 + B*a**4*e**7/9 + 28*B*a**3*b*d*e**6/9 + 14*B*a**2
*b**2*d**2*e**5 + 140*B*a*b**3*d**3*e**4/9 + 35*B*b**4*d**4*e**3/9) + x**8*(A*a**4*e**7/8 + 7*A*a**3*b*d*e**6/
2 + 63*A*a**2*b**2*d**2*e**5/4 + 35*A*a*b**3*d**3*e**4/2 + 35*A*b**4*d**4*e**3/8 + 7*B*a**4*d*e**6/8 + 21*B*a*
*3*b*d**2*e**5/2 + 105*B*a**2*b**2*d**3*e**4/4 + 35*B*a*b**3*d**4*e**3/2 + 21*B*b**4*d**5*e**2/8) + x**7*(A*a*
*4*d*e**6 + 12*A*a**3*b*d**2*e**5 + 30*A*a**2*b**2*d**3*e**4 + 20*A*a*b**3*d**4*e**3 + 3*A*b**4*d**5*e**2 + 3*
B*a**4*d**2*e**5 + 20*B*a**3*b*d**3*e**4 + 30*B*a**2*b**2*d**4*e**3 + 12*B*a*b**3*d**5*e**2 + B*b**4*d**6*e) +
x**6*(7*A*a**4*d**2*e**5/2 + 70*A*a**3*b*d**3*e**4/3 + 35*A*a**2*b**2*d**4*e**3 + 14*A*a*b**3*d**5*e**2 + 7*A
*b**4*d**6*e/6 + 35*B*a**4*d**3*e**4/6 + 70*B*a**3*b*d**4*e**3/3 + 21*B*a**2*b**2*d**5*e**2 + 14*B*a*b**3*d**6
*e/3 + B*b**4*d**7/6) + x**5*(7*A*a**4*d**3*e**4 + 28*A*a**3*b*d**4*e**3 + 126*A*a**2*b**2*d**5*e**2/5 + 28*A*
a*b**3*d**6*e/5 + A*b**4*d**7/5 + 7*B*a**4*d**4*e**3 + 84*B*a**3*b*d**5*e**2/5 + 42*B*a**2*b**2*d**6*e/5 + 4*B
*a*b**3*d**7/5) + x**4*(35*A*a**4*d**4*e**3/4 + 21*A*a**3*b*d**5*e**2 + 21*A*a**2*b**2*d**6*e/2 + A*a*b**3*d**
7 + 21*B*a**4*d**5*e**2/4 + 7*B*a**3*b*d**6*e + 3*B*a**2*b**2*d**7/2) + x**3*(7*A*a**4*d**5*e**2 + 28*A*a**3*b
*d**6*e/3 + 2*A*a**2*b**2*d**7 + 7*B*a**4*d**6*e/3 + 4*B*a**3*b*d**7/3) + x**2*(7*A*a**4*d**6*e/2 + 2*A*a**3*b
*d**7 + B*a**4*d**7/2)

________________________________________________________________________________________