3.15.54 \(\int (A+B x) (d+e x)^6 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=206 \[ -\frac {b^3 (d+e x)^{11} (-4 a B e-A b e+5 b B d)}{11 e^6}+\frac {b^2 (d+e x)^{10} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{5 e^6}-\frac {2 b (d+e x)^9 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6}+\frac {(d+e x)^8 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{8 e^6}-\frac {(d+e x)^7 (b d-a e)^4 (B d-A e)}{7 e^6}+\frac {b^4 B (d+e x)^{12}}{12 e^6} \]

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Rubi [A]  time = 0.66, 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)^{11} (-4 a B e-A b e+5 b B d)}{11 e^6}+\frac {b^2 (d+e x)^{10} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{5 e^6}-\frac {2 b (d+e x)^9 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6}+\frac {(d+e x)^8 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{8 e^6}-\frac {(d+e x)^7 (b d-a e)^4 (B d-A e)}{7 e^6}+\frac {b^4 B (d+e x)^{12}}{12 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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Mathematica [B]  time = 0.25, size = 737, normalized size = 3.58 \begin {gather*} a^4 A d^6 x+\frac {1}{2} a^3 d^5 x^2 (6 a A e+a B d+4 A b d)+\frac {1}{3} a^2 d^4 x^3 \left (3 A \left (5 a^2 e^2+8 a b d e+2 b^2 d^2\right )+2 a B d (3 a e+2 b d)\right )+\frac {1}{10} b^2 e^4 x^{10} \left (6 a^2 B e^2+4 a b e (A e+6 B d)+3 b^2 d (2 A e+5 B d)\right )+\frac {1}{9} b e^3 x^9 \left (4 a^3 B e^3+6 a^2 b e^2 (A e+6 B d)+12 a b^2 d e (2 A e+5 B d)+5 b^3 d^2 (3 A e+4 B d)\right )+\frac {1}{4} a d^3 x^4 \left (3 a B d \left (5 a^2 e^2+8 a b d e+2 b^2 d^2\right )+4 A \left (5 a^3 e^3+15 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right )\right )+\frac {1}{8} e^2 x^8 \left (a^4 B e^4+4 a^3 b e^3 (A e+6 B d)+18 a^2 b^2 d e^2 (2 A e+5 B d)+20 a b^3 d^2 e (3 A e+4 B d)+5 b^4 d^3 (4 A e+3 B d)\right )+\frac {1}{7} e x^7 \left (a^4 e^4 (A e+6 B d)+12 a^3 b d e^3 (2 A e+5 B d)+30 a^2 b^2 d^2 e^2 (3 A e+4 B d)+20 a b^3 d^3 e (4 A e+3 B d)+3 b^4 d^4 (5 A e+2 B d)\right )+\frac {1}{6} d x^6 \left (3 a^4 e^4 (2 A e+5 B d)+20 a^3 b d e^3 (3 A e+4 B d)+30 a^2 b^2 d^2 e^2 (4 A e+3 B d)+12 a b^3 d^3 e (5 A e+2 B d)+b^4 d^4 (6 A e+B d)\right )+\frac {1}{5} d^2 x^5 \left (4 a B d \left (5 a^3 e^3+15 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right )+A \left (15 a^4 e^4+80 a^3 b d e^3+90 a^2 b^2 d^2 e^2+24 a b^3 d^3 e+b^4 d^4\right )\right )+\frac {1}{11} b^3 e^5 x^{11} (4 a B e+A b e+6 b B d)+\frac {1}{12} b^4 B e^6 x^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*A*d^6*x + (a^3*d^5*(4*A*b*d + a*B*d + 6*a*A*e)*x^2)/2 + (a^2*d^4*(2*a*B*d*(2*b*d + 3*a*e) + 3*A*(2*b^2*d^2
+ 8*a*b*d*e + 5*a^2*e^2))*x^3)/3 + (a*d^3*(3*a*B*d*(2*b^2*d^2 + 8*a*b*d*e + 5*a^2*e^2) + 4*A*(b^3*d^3 + 9*a*b
^2*d^2*e + 15*a^2*b*d*e^2 + 5*a^3*e^3))*x^4)/4 + (d^2*(4*a*B*d*(b^3*d^3 + 9*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 5*a
^3*e^3) + A*(b^4*d^4 + 24*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 80*a^3*b*d*e^3 + 15*a^4*e^4))*x^5)/5 + (d*(3*a^4*
e^4*(5*B*d + 2*A*e) + 20*a^3*b*d*e^3*(4*B*d + 3*A*e) + 30*a^2*b^2*d^2*e^2*(3*B*d + 4*A*e) + 12*a*b^3*d^3*e*(2*
B*d + 5*A*e) + b^4*d^4*(B*d + 6*A*e))*x^6)/6 + (e*(a^4*e^4*(6*B*d + A*e) + 12*a^3*b*d*e^3*(5*B*d + 2*A*e) + 30
*a^2*b^2*d^2*e^2*(4*B*d + 3*A*e) + 20*a*b^3*d^3*e*(3*B*d + 4*A*e) + 3*b^4*d^4*(2*B*d + 5*A*e))*x^7)/7 + (e^2*(
a^4*B*e^4 + 4*a^3*b*e^3*(6*B*d + A*e) + 18*a^2*b^2*d*e^2*(5*B*d + 2*A*e) + 20*a*b^3*d^2*e*(4*B*d + 3*A*e) + 5*
b^4*d^3*(3*B*d + 4*A*e))*x^8)/8 + (b*e^3*(4*a^3*B*e^3 + 6*a^2*b*e^2*(6*B*d + A*e) + 12*a*b^2*d*e*(5*B*d + 2*A*
e) + 5*b^3*d^2*(4*B*d + 3*A*e))*x^9)/9 + (b^2*e^4*(6*a^2*B*e^2 + 4*a*b*e*(6*B*d + A*e) + 3*b^2*d*(5*B*d + 2*A*
e))*x^10)/10 + (b^3*e^5*(6*b*B*d + A*b*e + 4*a*B*e)*x^11)/11 + (b^4*B*e^6*x^12)/12

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^6 \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)^6*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^2, x]

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fricas [B]  time = 0.38, size = 1015, normalized size = 4.93 \begin {gather*} \frac {1}{12} x^{12} e^{6} b^{4} B + \frac {6}{11} x^{11} e^{5} d b^{4} B + \frac {4}{11} x^{11} e^{6} b^{3} a B + \frac {1}{11} x^{11} e^{6} b^{4} A + \frac {3}{2} x^{10} e^{4} d^{2} b^{4} B + \frac {12}{5} x^{10} e^{5} d b^{3} a B + \frac {3}{5} x^{10} e^{6} b^{2} a^{2} B + \frac {3}{5} x^{10} e^{5} d b^{4} A + \frac {2}{5} x^{10} e^{6} b^{3} a A + \frac {20}{9} x^{9} e^{3} d^{3} b^{4} B + \frac {20}{3} x^{9} e^{4} d^{2} b^{3} a B + 4 x^{9} e^{5} d b^{2} a^{2} B + \frac {4}{9} x^{9} e^{6} b a^{3} B + \frac {5}{3} x^{9} e^{4} d^{2} b^{4} A + \frac {8}{3} x^{9} e^{5} d b^{3} a A + \frac {2}{3} x^{9} e^{6} b^{2} a^{2} A + \frac {15}{8} x^{8} e^{2} d^{4} b^{4} B + 10 x^{8} e^{3} d^{3} b^{3} a B + \frac {45}{4} x^{8} e^{4} d^{2} b^{2} a^{2} B + 3 x^{8} e^{5} d b a^{3} B + \frac {1}{8} x^{8} e^{6} a^{4} B + \frac {5}{2} x^{8} e^{3} d^{3} b^{4} A + \frac {15}{2} x^{8} e^{4} d^{2} b^{3} a A + \frac {9}{2} x^{8} e^{5} d b^{2} a^{2} A + \frac {1}{2} x^{8} e^{6} b a^{3} A + \frac {6}{7} x^{7} e d^{5} b^{4} B + \frac {60}{7} x^{7} e^{2} d^{4} b^{3} a B + \frac {120}{7} x^{7} e^{3} d^{3} b^{2} a^{2} B + \frac {60}{7} x^{7} e^{4} d^{2} b a^{3} B + \frac {6}{7} x^{7} e^{5} d a^{4} B + \frac {15}{7} x^{7} e^{2} d^{4} b^{4} A + \frac {80}{7} x^{7} e^{3} d^{3} b^{3} a A + \frac {90}{7} x^{7} e^{4} d^{2} b^{2} a^{2} A + \frac {24}{7} x^{7} e^{5} d b a^{3} A + \frac {1}{7} x^{7} e^{6} a^{4} A + \frac {1}{6} x^{6} d^{6} b^{4} B + 4 x^{6} e d^{5} b^{3} a B + 15 x^{6} e^{2} d^{4} b^{2} a^{2} B + \frac {40}{3} x^{6} e^{3} d^{3} b a^{3} B + \frac {5}{2} x^{6} e^{4} d^{2} a^{4} B + x^{6} e d^{5} b^{4} A + 10 x^{6} e^{2} d^{4} b^{3} a A + 20 x^{6} e^{3} d^{3} b^{2} a^{2} A + 10 x^{6} e^{4} d^{2} b a^{3} A + x^{6} e^{5} d a^{4} A + \frac {4}{5} x^{5} d^{6} b^{3} a B + \frac {36}{5} x^{5} e d^{5} b^{2} a^{2} B + 12 x^{5} e^{2} d^{4} b a^{3} B + 4 x^{5} e^{3} d^{3} a^{4} B + \frac {1}{5} x^{5} d^{6} b^{4} A + \frac {24}{5} x^{5} e d^{5} b^{3} a A + 18 x^{5} e^{2} d^{4} b^{2} a^{2} A + 16 x^{5} e^{3} d^{3} b a^{3} A + 3 x^{5} e^{4} d^{2} a^{4} A + \frac {3}{2} x^{4} d^{6} b^{2} a^{2} B + 6 x^{4} e d^{5} b a^{3} B + \frac {15}{4} x^{4} e^{2} d^{4} a^{4} B + x^{4} d^{6} b^{3} a A + 9 x^{4} e d^{5} b^{2} a^{2} A + 15 x^{4} e^{2} d^{4} b a^{3} A + 5 x^{4} e^{3} d^{3} a^{4} A + \frac {4}{3} x^{3} d^{6} b a^{3} B + 2 x^{3} e d^{5} a^{4} B + 2 x^{3} d^{6} b^{2} a^{2} A + 8 x^{3} e d^{5} b a^{3} A + 5 x^{3} e^{2} d^{4} a^{4} A + \frac {1}{2} x^{2} d^{6} a^{4} B + 2 x^{2} d^{6} b a^{3} A + 3 x^{2} e d^{5} a^{4} A + x d^{6} a^{4} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/12*x^12*e^6*b^4*B + 6/11*x^11*e^5*d*b^4*B + 4/11*x^11*e^6*b^3*a*B + 1/11*x^11*e^6*b^4*A + 3/2*x^10*e^4*d^2*b
^4*B + 12/5*x^10*e^5*d*b^3*a*B + 3/5*x^10*e^6*b^2*a^2*B + 3/5*x^10*e^5*d*b^4*A + 2/5*x^10*e^6*b^3*a*A + 20/9*x
^9*e^3*d^3*b^4*B + 20/3*x^9*e^4*d^2*b^3*a*B + 4*x^9*e^5*d*b^2*a^2*B + 4/9*x^9*e^6*b*a^3*B + 5/3*x^9*e^4*d^2*b^
4*A + 8/3*x^9*e^5*d*b^3*a*A + 2/3*x^9*e^6*b^2*a^2*A + 15/8*x^8*e^2*d^4*b^4*B + 10*x^8*e^3*d^3*b^3*a*B + 45/4*x
^8*e^4*d^2*b^2*a^2*B + 3*x^8*e^5*d*b*a^3*B + 1/8*x^8*e^6*a^4*B + 5/2*x^8*e^3*d^3*b^4*A + 15/2*x^8*e^4*d^2*b^3*
a*A + 9/2*x^8*e^5*d*b^2*a^2*A + 1/2*x^8*e^6*b*a^3*A + 6/7*x^7*e*d^5*b^4*B + 60/7*x^7*e^2*d^4*b^3*a*B + 120/7*x
^7*e^3*d^3*b^2*a^2*B + 60/7*x^7*e^4*d^2*b*a^3*B + 6/7*x^7*e^5*d*a^4*B + 15/7*x^7*e^2*d^4*b^4*A + 80/7*x^7*e^3*
d^3*b^3*a*A + 90/7*x^7*e^4*d^2*b^2*a^2*A + 24/7*x^7*e^5*d*b*a^3*A + 1/7*x^7*e^6*a^4*A + 1/6*x^6*d^6*b^4*B + 4*
x^6*e*d^5*b^3*a*B + 15*x^6*e^2*d^4*b^2*a^2*B + 40/3*x^6*e^3*d^3*b*a^3*B + 5/2*x^6*e^4*d^2*a^4*B + x^6*e*d^5*b^
4*A + 10*x^6*e^2*d^4*b^3*a*A + 20*x^6*e^3*d^3*b^2*a^2*A + 10*x^6*e^4*d^2*b*a^3*A + x^6*e^5*d*a^4*A + 4/5*x^5*d
^6*b^3*a*B + 36/5*x^5*e*d^5*b^2*a^2*B + 12*x^5*e^2*d^4*b*a^3*B + 4*x^5*e^3*d^3*a^4*B + 1/5*x^5*d^6*b^4*A + 24/
5*x^5*e*d^5*b^3*a*A + 18*x^5*e^2*d^4*b^2*a^2*A + 16*x^5*e^3*d^3*b*a^3*A + 3*x^5*e^4*d^2*a^4*A + 3/2*x^4*d^6*b^
2*a^2*B + 6*x^4*e*d^5*b*a^3*B + 15/4*x^4*e^2*d^4*a^4*B + x^4*d^6*b^3*a*A + 9*x^4*e*d^5*b^2*a^2*A + 15*x^4*e^2*
d^4*b*a^3*A + 5*x^4*e^3*d^3*a^4*A + 4/3*x^3*d^6*b*a^3*B + 2*x^3*e*d^5*a^4*B + 2*x^3*d^6*b^2*a^2*A + 8*x^3*e*d^
5*b*a^3*A + 5*x^3*e^2*d^4*a^4*A + 1/2*x^2*d^6*a^4*B + 2*x^2*d^6*b*a^3*A + 3*x^2*e*d^5*a^4*A + x*d^6*a^4*A

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giac [B]  time = 0.16, size = 975, normalized size = 4.73 \begin {gather*} \frac {1}{12} \, B b^{4} x^{12} e^{6} + \frac {6}{11} \, B b^{4} d x^{11} e^{5} + \frac {3}{2} \, B b^{4} d^{2} x^{10} e^{4} + \frac {20}{9} \, B b^{4} d^{3} x^{9} e^{3} + \frac {15}{8} \, B b^{4} d^{4} x^{8} e^{2} + \frac {6}{7} \, B b^{4} d^{5} x^{7} e + \frac {1}{6} \, B b^{4} d^{6} x^{6} + \frac {4}{11} \, B a b^{3} x^{11} e^{6} + \frac {1}{11} \, A b^{4} x^{11} e^{6} + \frac {12}{5} \, B a b^{3} d x^{10} e^{5} + \frac {3}{5} \, A b^{4} d x^{10} e^{5} + \frac {20}{3} \, B a b^{3} d^{2} x^{9} e^{4} + \frac {5}{3} \, A b^{4} d^{2} x^{9} e^{4} + 10 \, B a b^{3} d^{3} x^{8} e^{3} + \frac {5}{2} \, A b^{4} d^{3} x^{8} e^{3} + \frac {60}{7} \, B a b^{3} d^{4} x^{7} e^{2} + \frac {15}{7} \, A b^{4} d^{4} x^{7} e^{2} + 4 \, B a b^{3} d^{5} x^{6} e + A b^{4} d^{5} x^{6} e + \frac {4}{5} \, B a b^{3} d^{6} x^{5} + \frac {1}{5} \, A b^{4} d^{6} x^{5} + \frac {3}{5} \, B a^{2} b^{2} x^{10} e^{6} + \frac {2}{5} \, A a b^{3} x^{10} e^{6} + 4 \, B a^{2} b^{2} d x^{9} e^{5} + \frac {8}{3} \, A a b^{3} d x^{9} e^{5} + \frac {45}{4} \, B a^{2} b^{2} d^{2} x^{8} e^{4} + \frac {15}{2} \, A a b^{3} d^{2} x^{8} e^{4} + \frac {120}{7} \, B a^{2} b^{2} d^{3} x^{7} e^{3} + \frac {80}{7} \, A a b^{3} d^{3} x^{7} e^{3} + 15 \, B a^{2} b^{2} d^{4} x^{6} e^{2} + 10 \, A a b^{3} d^{4} x^{6} e^{2} + \frac {36}{5} \, B a^{2} b^{2} d^{5} x^{5} e + \frac {24}{5} \, A a b^{3} d^{5} x^{5} e + \frac {3}{2} \, B a^{2} b^{2} d^{6} x^{4} + A a b^{3} d^{6} x^{4} + \frac {4}{9} \, B a^{3} b x^{9} e^{6} + \frac {2}{3} \, A a^{2} b^{2} x^{9} e^{6} + 3 \, B a^{3} b d x^{8} e^{5} + \frac {9}{2} \, A a^{2} b^{2} d x^{8} e^{5} + \frac {60}{7} \, B a^{3} b d^{2} x^{7} e^{4} + \frac {90}{7} \, A a^{2} b^{2} d^{2} x^{7} e^{4} + \frac {40}{3} \, B a^{3} b d^{3} x^{6} e^{3} + 20 \, A a^{2} b^{2} d^{3} x^{6} e^{3} + 12 \, B a^{3} b d^{4} x^{5} e^{2} + 18 \, A a^{2} b^{2} d^{4} x^{5} e^{2} + 6 \, B a^{3} b d^{5} x^{4} e + 9 \, A a^{2} b^{2} d^{5} x^{4} e + \frac {4}{3} \, B a^{3} b d^{6} x^{3} + 2 \, A a^{2} b^{2} d^{6} x^{3} + \frac {1}{8} \, B a^{4} x^{8} e^{6} + \frac {1}{2} \, A a^{3} b x^{8} e^{6} + \frac {6}{7} \, B a^{4} d x^{7} e^{5} + \frac {24}{7} \, A a^{3} b d x^{7} e^{5} + \frac {5}{2} \, B a^{4} d^{2} x^{6} e^{4} + 10 \, A a^{3} b d^{2} x^{6} e^{4} + 4 \, B a^{4} d^{3} x^{5} e^{3} + 16 \, A a^{3} b d^{3} x^{5} e^{3} + \frac {15}{4} \, B a^{4} d^{4} x^{4} e^{2} + 15 \, A a^{3} b d^{4} x^{4} e^{2} + 2 \, B a^{4} d^{5} x^{3} e + 8 \, A a^{3} b d^{5} x^{3} e + \frac {1}{2} \, B a^{4} d^{6} x^{2} + 2 \, A a^{3} b d^{6} x^{2} + \frac {1}{7} \, A a^{4} x^{7} e^{6} + A a^{4} d x^{6} e^{5} + 3 \, A a^{4} d^{2} x^{5} e^{4} + 5 \, A a^{4} d^{3} x^{4} e^{3} + 5 \, A a^{4} d^{4} x^{3} e^{2} + 3 \, A a^{4} d^{5} x^{2} e + A a^{4} d^{6} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/12*B*b^4*x^12*e^6 + 6/11*B*b^4*d*x^11*e^5 + 3/2*B*b^4*d^2*x^10*e^4 + 20/9*B*b^4*d^3*x^9*e^3 + 15/8*B*b^4*d^4
*x^8*e^2 + 6/7*B*b^4*d^5*x^7*e + 1/6*B*b^4*d^6*x^6 + 4/11*B*a*b^3*x^11*e^6 + 1/11*A*b^4*x^11*e^6 + 12/5*B*a*b^
3*d*x^10*e^5 + 3/5*A*b^4*d*x^10*e^5 + 20/3*B*a*b^3*d^2*x^9*e^4 + 5/3*A*b^4*d^2*x^9*e^4 + 10*B*a*b^3*d^3*x^8*e^
3 + 5/2*A*b^4*d^3*x^8*e^3 + 60/7*B*a*b^3*d^4*x^7*e^2 + 15/7*A*b^4*d^4*x^7*e^2 + 4*B*a*b^3*d^5*x^6*e + A*b^4*d^
5*x^6*e + 4/5*B*a*b^3*d^6*x^5 + 1/5*A*b^4*d^6*x^5 + 3/5*B*a^2*b^2*x^10*e^6 + 2/5*A*a*b^3*x^10*e^6 + 4*B*a^2*b^
2*d*x^9*e^5 + 8/3*A*a*b^3*d*x^9*e^5 + 45/4*B*a^2*b^2*d^2*x^8*e^4 + 15/2*A*a*b^3*d^2*x^8*e^4 + 120/7*B*a^2*b^2*
d^3*x^7*e^3 + 80/7*A*a*b^3*d^3*x^7*e^3 + 15*B*a^2*b^2*d^4*x^6*e^2 + 10*A*a*b^3*d^4*x^6*e^2 + 36/5*B*a^2*b^2*d^
5*x^5*e + 24/5*A*a*b^3*d^5*x^5*e + 3/2*B*a^2*b^2*d^6*x^4 + A*a*b^3*d^6*x^4 + 4/9*B*a^3*b*x^9*e^6 + 2/3*A*a^2*b
^2*x^9*e^6 + 3*B*a^3*b*d*x^8*e^5 + 9/2*A*a^2*b^2*d*x^8*e^5 + 60/7*B*a^3*b*d^2*x^7*e^4 + 90/7*A*a^2*b^2*d^2*x^7
*e^4 + 40/3*B*a^3*b*d^3*x^6*e^3 + 20*A*a^2*b^2*d^3*x^6*e^3 + 12*B*a^3*b*d^4*x^5*e^2 + 18*A*a^2*b^2*d^4*x^5*e^2
+ 6*B*a^3*b*d^5*x^4*e + 9*A*a^2*b^2*d^5*x^4*e + 4/3*B*a^3*b*d^6*x^3 + 2*A*a^2*b^2*d^6*x^3 + 1/8*B*a^4*x^8*e^6
+ 1/2*A*a^3*b*x^8*e^6 + 6/7*B*a^4*d*x^7*e^5 + 24/7*A*a^3*b*d*x^7*e^5 + 5/2*B*a^4*d^2*x^6*e^4 + 10*A*a^3*b*d^2
*x^6*e^4 + 4*B*a^4*d^3*x^5*e^3 + 16*A*a^3*b*d^3*x^5*e^3 + 15/4*B*a^4*d^4*x^4*e^2 + 15*A*a^3*b*d^4*x^4*e^2 + 2*
B*a^4*d^5*x^3*e + 8*A*a^3*b*d^5*x^3*e + 1/2*B*a^4*d^6*x^2 + 2*A*a^3*b*d^6*x^2 + 1/7*A*a^4*x^7*e^6 + A*a^4*d*x^
6*e^5 + 3*A*a^4*d^2*x^5*e^4 + 5*A*a^4*d^3*x^4*e^3 + 5*A*a^4*d^4*x^3*e^2 + 3*A*a^4*d^5*x^2*e + A*a^4*d^6*x

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maple [B]  time = 0.04, size = 821, normalized size = 3.99 \begin {gather*} \frac {B \,b^{4} e^{6} x^{12}}{12}+A \,a^{4} d^{6} x +\frac {\left (4 B a \,b^{3} e^{6}+\left (A \,e^{6}+6 B d \,e^{5}\right ) b^{4}\right ) x^{11}}{11}+\frac {\left (6 B \,a^{2} b^{2} e^{6}+4 \left (A \,e^{6}+6 B d \,e^{5}\right ) a \,b^{3}+\left (6 A d \,e^{5}+15 B \,d^{2} e^{4}\right ) b^{4}\right ) x^{10}}{10}+\frac {\left (4 B \,a^{3} b \,e^{6}+6 \left (A \,e^{6}+6 B d \,e^{5}\right ) a^{2} b^{2}+4 \left (6 A d \,e^{5}+15 B \,d^{2} e^{4}\right ) a \,b^{3}+\left (15 A \,d^{2} e^{4}+20 B \,d^{3} e^{3}\right ) b^{4}\right ) x^{9}}{9}+\frac {\left (B \,a^{4} e^{6}+4 \left (A \,e^{6}+6 B d \,e^{5}\right ) a^{3} b +6 \left (6 A d \,e^{5}+15 B \,d^{2} e^{4}\right ) a^{2} b^{2}+4 \left (15 A \,d^{2} e^{4}+20 B \,d^{3} e^{3}\right ) a \,b^{3}+\left (20 A \,d^{3} e^{3}+15 B \,d^{4} e^{2}\right ) b^{4}\right ) x^{8}}{8}+\frac {\left (\left (A \,e^{6}+6 B d \,e^{5}\right ) a^{4}+4 \left (6 A d \,e^{5}+15 B \,d^{2} e^{4}\right ) a^{3} b +6 \left (15 A \,d^{2} e^{4}+20 B \,d^{3} e^{3}\right ) a^{2} b^{2}+4 \left (20 A \,d^{3} e^{3}+15 B \,d^{4} e^{2}\right ) a \,b^{3}+\left (15 A \,d^{4} e^{2}+6 B \,d^{5} e \right ) b^{4}\right ) x^{7}}{7}+\frac {\left (\left (6 A d \,e^{5}+15 B \,d^{2} e^{4}\right ) a^{4}+4 \left (15 A \,d^{2} e^{4}+20 B \,d^{3} e^{3}\right ) a^{3} b +6 \left (20 A \,d^{3} e^{3}+15 B \,d^{4} e^{2}\right ) a^{2} b^{2}+4 \left (15 A \,d^{4} e^{2}+6 B \,d^{5} e \right ) a \,b^{3}+\left (6 A \,d^{5} e +B \,d^{6}\right ) b^{4}\right ) x^{6}}{6}+\frac {\left (A \,b^{4} d^{6}+\left (15 A \,d^{2} e^{4}+20 B \,d^{3} e^{3}\right ) a^{4}+4 \left (20 A \,d^{3} e^{3}+15 B \,d^{4} e^{2}\right ) a^{3} b +6 \left (15 A \,d^{4} e^{2}+6 B \,d^{5} e \right ) a^{2} b^{2}+4 \left (6 A \,d^{5} e +B \,d^{6}\right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 A a \,b^{3} d^{6}+\left (20 A \,d^{3} e^{3}+15 B \,d^{4} e^{2}\right ) a^{4}+4 \left (15 A \,d^{4} e^{2}+6 B \,d^{5} e \right ) a^{3} b +6 \left (6 A \,d^{5} e +B \,d^{6}\right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (6 A \,a^{2} b^{2} d^{6}+\left (15 A \,d^{4} e^{2}+6 B \,d^{5} e \right ) a^{4}+4 \left (6 A \,d^{5} e +B \,d^{6}\right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 A \,a^{3} b \,d^{6}+\left (6 A \,d^{5} e +B \,d^{6}\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)^6*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/12*B*e^6*b^4*x^12+1/11*((A*e^6+6*B*d*e^5)*b^4+4*B*e^6*a*b^3)*x^11+1/10*((6*A*d*e^5+15*B*d^2*e^4)*b^4+4*(A*e^
6+6*B*d*e^5)*a*b^3+6*B*e^6*a^2*b^2)*x^10+1/9*((15*A*d^2*e^4+20*B*d^3*e^3)*b^4+4*(6*A*d*e^5+15*B*d^2*e^4)*a*b^3
+6*(A*e^6+6*B*d*e^5)*a^2*b^2+4*B*e^6*a^3*b)*x^9+1/8*((20*A*d^3*e^3+15*B*d^4*e^2)*b^4+4*(15*A*d^2*e^4+20*B*d^3*
e^3)*a*b^3+6*(6*A*d*e^5+15*B*d^2*e^4)*a^2*b^2+4*(A*e^6+6*B*d*e^5)*a^3*b+B*e^6*a^4)*x^8+1/7*((15*A*d^4*e^2+6*B*
d^5*e)*b^4+4*(20*A*d^3*e^3+15*B*d^4*e^2)*a*b^3+6*(15*A*d^2*e^4+20*B*d^3*e^3)*a^2*b^2+4*(6*A*d*e^5+15*B*d^2*e^4
)*a^3*b+(A*e^6+6*B*d*e^5)*a^4)*x^7+1/6*((6*A*d^5*e+B*d^6)*b^4+4*(15*A*d^4*e^2+6*B*d^5*e)*a*b^3+6*(20*A*d^3*e^3
+15*B*d^4*e^2)*a^2*b^2+4*(15*A*d^2*e^4+20*B*d^3*e^3)*a^3*b+(6*A*d*e^5+15*B*d^2*e^4)*a^4)*x^6+1/5*(A*d^6*b^4+4*
(6*A*d^5*e+B*d^6)*a*b^3+6*(15*A*d^4*e^2+6*B*d^5*e)*a^2*b^2+4*(20*A*d^3*e^3+15*B*d^4*e^2)*a^3*b+(15*A*d^2*e^4+2
0*B*d^3*e^3)*a^4)*x^5+1/4*(4*A*d^6*a*b^3+6*(6*A*d^5*e+B*d^6)*a^2*b^2+4*(15*A*d^4*e^2+6*B*d^5*e)*a^3*b+(20*A*d^
3*e^3+15*B*d^4*e^2)*a^4)*x^4+1/3*(6*A*d^6*a^2*b^2+4*(6*A*d^5*e+B*d^6)*a^3*b+(15*A*d^4*e^2+6*B*d^5*e)*a^4)*x^3+
1/2*(4*A*d^6*a^3*b+(6*A*d^5*e+B*d^6)*a^4)*x^2+A*d^6*a^4*x

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maxima [B]  time = 0.78, size = 810, normalized size = 3.93 \begin {gather*} \frac {1}{12} \, B b^{4} e^{6} x^{12} + A a^{4} d^{6} x + \frac {1}{11} \, {\left (6 \, B b^{4} d e^{5} + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{6}\right )} x^{11} + \frac {1}{10} \, {\left (15 \, B b^{4} d^{2} e^{4} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{5} + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{6}\right )} x^{10} + \frac {1}{9} \, {\left (20 \, B b^{4} d^{3} e^{3} + 15 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{4} + 12 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{5} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{6}\right )} x^{9} + \frac {1}{8} \, {\left (15 \, B b^{4} d^{4} e^{2} + 20 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{3} + 30 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{4} + 12 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{5} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, B b^{4} d^{5} e + A a^{4} e^{6} + 15 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e^{2} + 40 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{3} + 30 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{4} + 6 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{4} d^{6} + 6 \, A a^{4} d e^{5} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{5} e + 30 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} e^{2} + 40 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e^{3} + 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (15 \, A a^{4} d^{2} e^{4} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{6} + 12 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{5} e + 30 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{4} e^{2} + 20 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{3} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, A a^{4} d^{3} e^{3} + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{6} + 12 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{5} e + 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{4} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (15 \, A a^{4} d^{4} e^{2} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{6} + 6 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{5} e\right )} x^{3} + \frac {1}{2} \, {\left (6 \, A a^{4} d^{5} e + {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{6}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/12*B*b^4*e^6*x^12 + A*a^4*d^6*x + 1/11*(6*B*b^4*d*e^5 + (4*B*a*b^3 + A*b^4)*e^6)*x^11 + 1/10*(15*B*b^4*d^2*e
^4 + 6*(4*B*a*b^3 + A*b^4)*d*e^5 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*x^10 + 1/9*(20*B*b^4*d^3*e^3 + 15*(4*B*a*b
^3 + A*b^4)*d^2*e^4 + 12*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^5 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*e^6)*x^9 + 1/8*(15*B*b^
4*d^4*e^2 + 20*(4*B*a*b^3 + A*b^4)*d^3*e^3 + 30*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^4 + 12*(2*B*a^3*b + 3*A*a^2*b^
2)*d*e^5 + (B*a^4 + 4*A*a^3*b)*e^6)*x^8 + 1/7*(6*B*b^4*d^5*e + A*a^4*e^6 + 15*(4*B*a*b^3 + A*b^4)*d^4*e^2 + 40
*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^3 + 30*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^4 + 6*(B*a^4 + 4*A*a^3*b)*d*e^5)*x^7 +
1/6*(B*b^4*d^6 + 6*A*a^4*d*e^5 + 6*(4*B*a*b^3 + A*b^4)*d^5*e + 30*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e^2 + 40*(2*B
*a^3*b + 3*A*a^2*b^2)*d^3*e^3 + 15*(B*a^4 + 4*A*a^3*b)*d^2*e^4)*x^6 + 1/5*(15*A*a^4*d^2*e^4 + (4*B*a*b^3 + A*b
^4)*d^6 + 12*(3*B*a^2*b^2 + 2*A*a*b^3)*d^5*e + 30*(2*B*a^3*b + 3*A*a^2*b^2)*d^4*e^2 + 20*(B*a^4 + 4*A*a^3*b)*d
^3*e^3)*x^5 + 1/4*(20*A*a^4*d^3*e^3 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^6 + 12*(2*B*a^3*b + 3*A*a^2*b^2)*d^5*e + 1
5*(B*a^4 + 4*A*a^3*b)*d^4*e^2)*x^4 + 1/3*(15*A*a^4*d^4*e^2 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^6 + 6*(B*a^4 + 4*A*
a^3*b)*d^5*e)*x^3 + 1/2*(6*A*a^4*d^5*e + (B*a^4 + 4*A*a^3*b)*d^6)*x^2

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mupad [B]  time = 2.22, size = 845, normalized size = 4.10 \begin {gather*} x^4\,\left (\frac {15\,B\,a^4\,d^4\,e^2}{4}+5\,A\,a^4\,d^3\,e^3+6\,B\,a^3\,b\,d^5\,e+15\,A\,a^3\,b\,d^4\,e^2+\frac {3\,B\,a^2\,b^2\,d^6}{2}+9\,A\,a^2\,b^2\,d^5\,e+A\,a\,b^3\,d^6\right )+x^9\,\left (\frac {4\,B\,a^3\,b\,e^6}{9}+4\,B\,a^2\,b^2\,d\,e^5+\frac {2\,A\,a^2\,b^2\,e^6}{3}+\frac {20\,B\,a\,b^3\,d^2\,e^4}{3}+\frac {8\,A\,a\,b^3\,d\,e^5}{3}+\frac {20\,B\,b^4\,d^3\,e^3}{9}+\frac {5\,A\,b^4\,d^2\,e^4}{3}\right )+x^3\,\left (2\,B\,a^4\,d^5\,e+5\,A\,a^4\,d^4\,e^2+\frac {4\,B\,a^3\,b\,d^6}{3}+8\,A\,a^3\,b\,d^5\,e+2\,A\,a^2\,b^2\,d^6\right )+x^{10}\,\left (\frac {3\,B\,a^2\,b^2\,e^6}{5}+\frac {12\,B\,a\,b^3\,d\,e^5}{5}+\frac {2\,A\,a\,b^3\,e^6}{5}+\frac {3\,B\,b^4\,d^2\,e^4}{2}+\frac {3\,A\,b^4\,d\,e^5}{5}\right )+x^5\,\left (4\,B\,a^4\,d^3\,e^3+3\,A\,a^4\,d^2\,e^4+12\,B\,a^3\,b\,d^4\,e^2+16\,A\,a^3\,b\,d^3\,e^3+\frac {36\,B\,a^2\,b^2\,d^5\,e}{5}+18\,A\,a^2\,b^2\,d^4\,e^2+\frac {4\,B\,a\,b^3\,d^6}{5}+\frac {24\,A\,a\,b^3\,d^5\,e}{5}+\frac {A\,b^4\,d^6}{5}\right )+x^8\,\left (\frac {B\,a^4\,e^6}{8}+3\,B\,a^3\,b\,d\,e^5+\frac {A\,a^3\,b\,e^6}{2}+\frac {45\,B\,a^2\,b^2\,d^2\,e^4}{4}+\frac {9\,A\,a^2\,b^2\,d\,e^5}{2}+10\,B\,a\,b^3\,d^3\,e^3+\frac {15\,A\,a\,b^3\,d^2\,e^4}{2}+\frac {15\,B\,b^4\,d^4\,e^2}{8}+\frac {5\,A\,b^4\,d^3\,e^3}{2}\right )+x^6\,\left (\frac {5\,B\,a^4\,d^2\,e^4}{2}+A\,a^4\,d\,e^5+\frac {40\,B\,a^3\,b\,d^3\,e^3}{3}+10\,A\,a^3\,b\,d^2\,e^4+15\,B\,a^2\,b^2\,d^4\,e^2+20\,A\,a^2\,b^2\,d^3\,e^3+4\,B\,a\,b^3\,d^5\,e+10\,A\,a\,b^3\,d^4\,e^2+\frac {B\,b^4\,d^6}{6}+A\,b^4\,d^5\,e\right )+x^7\,\left (\frac {6\,B\,a^4\,d\,e^5}{7}+\frac {A\,a^4\,e^6}{7}+\frac {60\,B\,a^3\,b\,d^2\,e^4}{7}+\frac {24\,A\,a^3\,b\,d\,e^5}{7}+\frac {120\,B\,a^2\,b^2\,d^3\,e^3}{7}+\frac {90\,A\,a^2\,b^2\,d^2\,e^4}{7}+\frac {60\,B\,a\,b^3\,d^4\,e^2}{7}+\frac {80\,A\,a\,b^3\,d^3\,e^3}{7}+\frac {6\,B\,b^4\,d^5\,e}{7}+\frac {15\,A\,b^4\,d^4\,e^2}{7}\right )+\frac {a^3\,d^5\,x^2\,\left (6\,A\,a\,e+4\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^3\,e^5\,x^{11}\,\left (A\,b\,e+4\,B\,a\,e+6\,B\,b\,d\right )}{11}+A\,a^4\,d^6\,x+\frac {B\,b^4\,e^6\,x^{12}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^6*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

x^4*(A*a*b^3*d^6 + (3*B*a^2*b^2*d^6)/2 + 5*A*a^4*d^3*e^3 + (15*B*a^4*d^4*e^2)/4 + 9*A*a^2*b^2*d^5*e + 15*A*a^3
*b*d^4*e^2 + 6*B*a^3*b*d^5*e) + x^9*((4*B*a^3*b*e^6)/9 + (2*A*a^2*b^2*e^6)/3 + (5*A*b^4*d^2*e^4)/3 + (20*B*b^4
*d^3*e^3)/9 + (20*B*a*b^3*d^2*e^4)/3 + 4*B*a^2*b^2*d*e^5 + (8*A*a*b^3*d*e^5)/3) + x^3*((4*B*a^3*b*d^6)/3 + 2*B
*a^4*d^5*e + 2*A*a^2*b^2*d^6 + 5*A*a^4*d^4*e^2 + 8*A*a^3*b*d^5*e) + x^10*((2*A*a*b^3*e^6)/5 + (3*A*b^4*d*e^5)/
5 + (3*B*a^2*b^2*e^6)/5 + (3*B*b^4*d^2*e^4)/2 + (12*B*a*b^3*d*e^5)/5) + x^5*((A*b^4*d^6)/5 + (4*B*a*b^3*d^6)/5
+ 3*A*a^4*d^2*e^4 + 4*B*a^4*d^3*e^3 + 16*A*a^3*b*d^3*e^3 + (36*B*a^2*b^2*d^5*e)/5 + 12*B*a^3*b*d^4*e^2 + 18*A
*a^2*b^2*d^4*e^2 + (24*A*a*b^3*d^5*e)/5) + x^8*((B*a^4*e^6)/8 + (A*a^3*b*e^6)/2 + (5*A*b^4*d^3*e^3)/2 + (15*B*
b^4*d^4*e^2)/8 + (15*A*a*b^3*d^2*e^4)/2 + (9*A*a^2*b^2*d*e^5)/2 + 10*B*a*b^3*d^3*e^3 + (45*B*a^2*b^2*d^2*e^4)/
4 + 3*B*a^3*b*d*e^5) + x^6*((B*b^4*d^6)/6 + A*a^4*d*e^5 + A*b^4*d^5*e + (5*B*a^4*d^2*e^4)/2 + 10*A*a*b^3*d^4*e
^2 + 10*A*a^3*b*d^2*e^4 + (40*B*a^3*b*d^3*e^3)/3 + 20*A*a^2*b^2*d^3*e^3 + 15*B*a^2*b^2*d^4*e^2 + 4*B*a*b^3*d^5
*e) + x^7*((A*a^4*e^6)/7 + (6*B*a^4*d*e^5)/7 + (6*B*b^4*d^5*e)/7 + (15*A*b^4*d^4*e^2)/7 + (80*A*a*b^3*d^3*e^3)
/7 + (60*B*a*b^3*d^4*e^2)/7 + (60*B*a^3*b*d^2*e^4)/7 + (90*A*a^2*b^2*d^2*e^4)/7 + (120*B*a^2*b^2*d^3*e^3)/7 +
(24*A*a^3*b*d*e^5)/7) + (a^3*d^5*x^2*(6*A*a*e + 4*A*b*d + B*a*d))/2 + (b^3*e^5*x^11*(A*b*e + 4*B*a*e + 6*B*b*d
))/11 + A*a^4*d^6*x + (B*b^4*e^6*x^12)/12

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sympy [B]  time = 0.20, size = 1035, normalized size = 5.02 \begin {gather*} A a^{4} d^{6} x + \frac {B b^{4} e^{6} x^{12}}{12} + x^{11} \left (\frac {A b^{4} e^{6}}{11} + \frac {4 B a b^{3} e^{6}}{11} + \frac {6 B b^{4} d e^{5}}{11}\right ) + x^{10} \left (\frac {2 A a b^{3} e^{6}}{5} + \frac {3 A b^{4} d e^{5}}{5} + \frac {3 B a^{2} b^{2} e^{6}}{5} + \frac {12 B a b^{3} d e^{5}}{5} + \frac {3 B b^{4} d^{2} e^{4}}{2}\right ) + x^{9} \left (\frac {2 A a^{2} b^{2} e^{6}}{3} + \frac {8 A a b^{3} d e^{5}}{3} + \frac {5 A b^{4} d^{2} e^{4}}{3} + \frac {4 B a^{3} b e^{6}}{9} + 4 B a^{2} b^{2} d e^{5} + \frac {20 B a b^{3} d^{2} e^{4}}{3} + \frac {20 B b^{4} d^{3} e^{3}}{9}\right ) + x^{8} \left (\frac {A a^{3} b e^{6}}{2} + \frac {9 A a^{2} b^{2} d e^{5}}{2} + \frac {15 A a b^{3} d^{2} e^{4}}{2} + \frac {5 A b^{4} d^{3} e^{3}}{2} + \frac {B a^{4} e^{6}}{8} + 3 B a^{3} b d e^{5} + \frac {45 B a^{2} b^{2} d^{2} e^{4}}{4} + 10 B a b^{3} d^{3} e^{3} + \frac {15 B b^{4} d^{4} e^{2}}{8}\right ) + x^{7} \left (\frac {A a^{4} e^{6}}{7} + \frac {24 A a^{3} b d e^{5}}{7} + \frac {90 A a^{2} b^{2} d^{2} e^{4}}{7} + \frac {80 A a b^{3} d^{3} e^{3}}{7} + \frac {15 A b^{4} d^{4} e^{2}}{7} + \frac {6 B a^{4} d e^{5}}{7} + \frac {60 B a^{3} b d^{2} e^{4}}{7} + \frac {120 B a^{2} b^{2} d^{3} e^{3}}{7} + \frac {60 B a b^{3} d^{4} e^{2}}{7} + \frac {6 B b^{4} d^{5} e}{7}\right ) + x^{6} \left (A a^{4} d e^{5} + 10 A a^{3} b d^{2} e^{4} + 20 A a^{2} b^{2} d^{3} e^{3} + 10 A a b^{3} d^{4} e^{2} + A b^{4} d^{5} e + \frac {5 B a^{4} d^{2} e^{4}}{2} + \frac {40 B a^{3} b d^{3} e^{3}}{3} + 15 B a^{2} b^{2} d^{4} e^{2} + 4 B a b^{3} d^{5} e + \frac {B b^{4} d^{6}}{6}\right ) + x^{5} \left (3 A a^{4} d^{2} e^{4} + 16 A a^{3} b d^{3} e^{3} + 18 A a^{2} b^{2} d^{4} e^{2} + \frac {24 A a b^{3} d^{5} e}{5} + \frac {A b^{4} d^{6}}{5} + 4 B a^{4} d^{3} e^{3} + 12 B a^{3} b d^{4} e^{2} + \frac {36 B a^{2} b^{2} d^{5} e}{5} + \frac {4 B a b^{3} d^{6}}{5}\right ) + x^{4} \left (5 A a^{4} d^{3} e^{3} + 15 A a^{3} b d^{4} e^{2} + 9 A a^{2} b^{2} d^{5} e + A a b^{3} d^{6} + \frac {15 B a^{4} d^{4} e^{2}}{4} + 6 B a^{3} b d^{5} e + \frac {3 B a^{2} b^{2} d^{6}}{2}\right ) + x^{3} \left (5 A a^{4} d^{4} e^{2} + 8 A a^{3} b d^{5} e + 2 A a^{2} b^{2} d^{6} + 2 B a^{4} d^{5} e + \frac {4 B a^{3} b d^{6}}{3}\right ) + x^{2} \left (3 A a^{4} d^{5} e + 2 A a^{3} b d^{6} + \frac {B a^{4} d^{6}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*d**6*x + B*b**4*e**6*x**12/12 + x**11*(A*b**4*e**6/11 + 4*B*a*b**3*e**6/11 + 6*B*b**4*d*e**5/11) + x**1
0*(2*A*a*b**3*e**6/5 + 3*A*b**4*d*e**5/5 + 3*B*a**2*b**2*e**6/5 + 12*B*a*b**3*d*e**5/5 + 3*B*b**4*d**2*e**4/2)
+ x**9*(2*A*a**2*b**2*e**6/3 + 8*A*a*b**3*d*e**5/3 + 5*A*b**4*d**2*e**4/3 + 4*B*a**3*b*e**6/9 + 4*B*a**2*b**2
*d*e**5 + 20*B*a*b**3*d**2*e**4/3 + 20*B*b**4*d**3*e**3/9) + x**8*(A*a**3*b*e**6/2 + 9*A*a**2*b**2*d*e**5/2 +
15*A*a*b**3*d**2*e**4/2 + 5*A*b**4*d**3*e**3/2 + B*a**4*e**6/8 + 3*B*a**3*b*d*e**5 + 45*B*a**2*b**2*d**2*e**4/
4 + 10*B*a*b**3*d**3*e**3 + 15*B*b**4*d**4*e**2/8) + x**7*(A*a**4*e**6/7 + 24*A*a**3*b*d*e**5/7 + 90*A*a**2*b*
*2*d**2*e**4/7 + 80*A*a*b**3*d**3*e**3/7 + 15*A*b**4*d**4*e**2/7 + 6*B*a**4*d*e**5/7 + 60*B*a**3*b*d**2*e**4/7
+ 120*B*a**2*b**2*d**3*e**3/7 + 60*B*a*b**3*d**4*e**2/7 + 6*B*b**4*d**5*e/7) + x**6*(A*a**4*d*e**5 + 10*A*a**
3*b*d**2*e**4 + 20*A*a**2*b**2*d**3*e**3 + 10*A*a*b**3*d**4*e**2 + A*b**4*d**5*e + 5*B*a**4*d**2*e**4/2 + 40*B
*a**3*b*d**3*e**3/3 + 15*B*a**2*b**2*d**4*e**2 + 4*B*a*b**3*d**5*e + B*b**4*d**6/6) + x**5*(3*A*a**4*d**2*e**4
+ 16*A*a**3*b*d**3*e**3 + 18*A*a**2*b**2*d**4*e**2 + 24*A*a*b**3*d**5*e/5 + A*b**4*d**6/5 + 4*B*a**4*d**3*e**
3 + 12*B*a**3*b*d**4*e**2 + 36*B*a**2*b**2*d**5*e/5 + 4*B*a*b**3*d**6/5) + x**4*(5*A*a**4*d**3*e**3 + 15*A*a**
3*b*d**4*e**2 + 9*A*a**2*b**2*d**5*e + A*a*b**3*d**6 + 15*B*a**4*d**4*e**2/4 + 6*B*a**3*b*d**5*e + 3*B*a**2*b*
*2*d**6/2) + x**3*(5*A*a**4*d**4*e**2 + 8*A*a**3*b*d**5*e + 2*A*a**2*b**2*d**6 + 2*B*a**4*d**5*e + 4*B*a**3*b*
d**6/3) + x**2*(3*A*a**4*d**5*e + 2*A*a**3*b*d**6 + B*a**4*d**6/2)

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