3.21.77 \(\int (A+B x) (d+e x)^5 (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=304 \[ -\frac {(d+e x)^9 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{9 e^6}-\frac {(d+e x)^8 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{8 e^6}-\frac {(d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{7 e^6}-\frac {(d+e x)^6 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{6 e^6}-\frac {c (d+e x)^{10} (-A c e-2 b B e+5 B c d)}{10 e^6}+\frac {B c^2 (d+e x)^{11}}{11 e^6} \]

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Rubi [A]  time = 0.82, antiderivative size = 302, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \begin {gather*} -\frac {(d+e x)^9 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{9 e^6}-\frac {(d+e x)^8 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{8 e^6}+\frac {(d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{7 e^6}-\frac {(d+e x)^6 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{6 e^6}-\frac {c (d+e x)^{10} (-A c e-2 b B e+5 B c d)}{10 e^6}+\frac {B c^2 (d+e x)^{11}}{11 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^5*(a + b*x + c*x^2)^2,x]

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [B]  time = 0.30, size = 665, normalized size = 2.19 \begin {gather*} \frac {1}{7} e x^7 \left (B \left (e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )+20 c d^2 e (a e+b d)+5 c^2 d^4\right )+A e \left (10 c d e (a e+2 b d)+b e^2 (2 a e+5 b d)+10 c^2 d^3\right )\right )+\frac {1}{5} d x^5 \left (A \left (5 a^2 e^4+20 a c d^2 e^2+c^2 d^4\right )+2 b d \left (10 a A e^3+10 a B d e^2+5 A c d^2 e+B c d^3\right )+10 a B d e \left (a e^2+c d^2\right )+5 b^2 d^2 e (2 A e+B d)\right )+\frac {1}{6} x^6 \left (A e \left (e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )+20 c d^2 e (a e+b d)+5 c^2 d^4\right )+B \left (5 d e^2 \left (a^2 e^2+4 a b d e+2 b^2 d^2\right )+10 c d^3 e (2 a e+b d)+c^2 d^5\right )\right )+a^2 A d^5 x+\frac {1}{9} e^3 x^9 \left (B \left (2 c e (a e+5 b d)+b^2 e^2+10 c^2 d^2\right )+A c e (2 b e+5 c d)\right )+\frac {1}{8} e^2 x^8 \left (A e \left (2 c e (a e+5 b d)+b^2 e^2+10 c^2 d^2\right )+B \left (10 c d e (a e+2 b d)+b e^2 (2 a e+5 b d)+10 c^2 d^3\right )\right )+\frac {1}{3} d^3 x^3 \left (A \left (10 a b d e+2 a \left (5 a e^2+c d^2\right )+b^2 d^2\right )+a B d (5 a e+2 b d)\right )+\frac {1}{4} d^2 x^4 \left (2 b d \left (10 a A e^2+5 a B d e+A c d^2\right )+2 a \left (5 a A e^3+5 a B d e^2+5 A c d^2 e+B c d^3\right )+b^2 d^2 (5 A e+B d)\right )+\frac {1}{2} a d^4 x^2 (5 a A e+a B d+2 A b d)+\frac {1}{10} c e^4 x^{10} (A c e+2 b B e+5 B c d)+\frac {1}{11} B c^2 e^5 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^5*(a + b*x + c*x^2)^2,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^5 \left (a+b x+c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^5*(a + b*x + c*x^2)^2,x]

[Out]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^5*(a + b*x + c*x^2)^2, x]

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fricas [B]  time = 0.34, size = 919, normalized size = 3.02 \begin {gather*} \frac {1}{11} x^{11} e^{5} c^{2} B + \frac {1}{2} x^{10} e^{4} d c^{2} B + \frac {1}{5} x^{10} e^{5} c b B + \frac {1}{10} x^{10} e^{5} c^{2} A + \frac {10}{9} x^{9} e^{3} d^{2} c^{2} B + \frac {10}{9} x^{9} e^{4} d c b B + \frac {1}{9} x^{9} e^{5} b^{2} B + \frac {2}{9} x^{9} e^{5} c a B + \frac {5}{9} x^{9} e^{4} d c^{2} A + \frac {2}{9} x^{9} e^{5} c b A + \frac {5}{4} x^{8} e^{2} d^{3} c^{2} B + \frac {5}{2} x^{8} e^{3} d^{2} c b B + \frac {5}{8} x^{8} e^{4} d b^{2} B + \frac {5}{4} x^{8} e^{4} d c a B + \frac {1}{4} x^{8} e^{5} b a B + \frac {5}{4} x^{8} e^{3} d^{2} c^{2} A + \frac {5}{4} x^{8} e^{4} d c b A + \frac {1}{8} x^{8} e^{5} b^{2} A + \frac {1}{4} x^{8} e^{5} c a A + \frac {5}{7} x^{7} e d^{4} c^{2} B + \frac {20}{7} x^{7} e^{2} d^{3} c b B + \frac {10}{7} x^{7} e^{3} d^{2} b^{2} B + \frac {20}{7} x^{7} e^{3} d^{2} c a B + \frac {10}{7} x^{7} e^{4} d b a B + \frac {1}{7} x^{7} e^{5} a^{2} B + \frac {10}{7} x^{7} e^{2} d^{3} c^{2} A + \frac {20}{7} x^{7} e^{3} d^{2} c b A + \frac {5}{7} x^{7} e^{4} d b^{2} A + \frac {10}{7} x^{7} e^{4} d c a A + \frac {2}{7} x^{7} e^{5} b a A + \frac {1}{6} x^{6} d^{5} c^{2} B + \frac {5}{3} x^{6} e d^{4} c b B + \frac {5}{3} x^{6} e^{2} d^{3} b^{2} B + \frac {10}{3} x^{6} e^{2} d^{3} c a B + \frac {10}{3} x^{6} e^{3} d^{2} b a B + \frac {5}{6} x^{6} e^{4} d a^{2} B + \frac {5}{6} x^{6} e d^{4} c^{2} A + \frac {10}{3} x^{6} e^{2} d^{3} c b A + \frac {5}{3} x^{6} e^{3} d^{2} b^{2} A + \frac {10}{3} x^{6} e^{3} d^{2} c a A + \frac {5}{3} x^{6} e^{4} d b a A + \frac {1}{6} x^{6} e^{5} a^{2} A + \frac {2}{5} x^{5} d^{5} c b B + x^{5} e d^{4} b^{2} B + 2 x^{5} e d^{4} c a B + 4 x^{5} e^{2} d^{3} b a B + 2 x^{5} e^{3} d^{2} a^{2} B + \frac {1}{5} x^{5} d^{5} c^{2} A + 2 x^{5} e d^{4} c b A + 2 x^{5} e^{2} d^{3} b^{2} A + 4 x^{5} e^{2} d^{3} c a A + 4 x^{5} e^{3} d^{2} b a A + x^{5} e^{4} d a^{2} A + \frac {1}{4} x^{4} d^{5} b^{2} B + \frac {1}{2} x^{4} d^{5} c a B + \frac {5}{2} x^{4} e d^{4} b a B + \frac {5}{2} x^{4} e^{2} d^{3} a^{2} B + \frac {1}{2} x^{4} d^{5} c b A + \frac {5}{4} x^{4} e d^{4} b^{2} A + \frac {5}{2} x^{4} e d^{4} c a A + 5 x^{4} e^{2} d^{3} b a A + \frac {5}{2} x^{4} e^{3} d^{2} a^{2} A + \frac {2}{3} x^{3} d^{5} b a B + \frac {5}{3} x^{3} e d^{4} a^{2} B + \frac {1}{3} x^{3} d^{5} b^{2} A + \frac {2}{3} x^{3} d^{5} c a A + \frac {10}{3} x^{3} e d^{4} b a A + \frac {10}{3} x^{3} e^{2} d^{3} a^{2} A + \frac {1}{2} x^{2} d^{5} a^{2} B + x^{2} d^{5} b a A + \frac {5}{2} x^{2} e d^{4} a^{2} A + x d^{5} a^{2} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

1/11*x^11*e^5*c^2*B + 1/2*x^10*e^4*d*c^2*B + 1/5*x^10*e^5*c*b*B + 1/10*x^10*e^5*c^2*A + 10/9*x^9*e^3*d^2*c^2*B
+ 10/9*x^9*e^4*d*c*b*B + 1/9*x^9*e^5*b^2*B + 2/9*x^9*e^5*c*a*B + 5/9*x^9*e^4*d*c^2*A + 2/9*x^9*e^5*c*b*A + 5/
4*x^8*e^2*d^3*c^2*B + 5/2*x^8*e^3*d^2*c*b*B + 5/8*x^8*e^4*d*b^2*B + 5/4*x^8*e^4*d*c*a*B + 1/4*x^8*e^5*b*a*B +
5/4*x^8*e^3*d^2*c^2*A + 5/4*x^8*e^4*d*c*b*A + 1/8*x^8*e^5*b^2*A + 1/4*x^8*e^5*c*a*A + 5/7*x^7*e*d^4*c^2*B + 20
/7*x^7*e^2*d^3*c*b*B + 10/7*x^7*e^3*d^2*b^2*B + 20/7*x^7*e^3*d^2*c*a*B + 10/7*x^7*e^4*d*b*a*B + 1/7*x^7*e^5*a^
2*B + 10/7*x^7*e^2*d^3*c^2*A + 20/7*x^7*e^3*d^2*c*b*A + 5/7*x^7*e^4*d*b^2*A + 10/7*x^7*e^4*d*c*a*A + 2/7*x^7*e
^5*b*a*A + 1/6*x^6*d^5*c^2*B + 5/3*x^6*e*d^4*c*b*B + 5/3*x^6*e^2*d^3*b^2*B + 10/3*x^6*e^2*d^3*c*a*B + 10/3*x^6
*e^3*d^2*b*a*B + 5/6*x^6*e^4*d*a^2*B + 5/6*x^6*e*d^4*c^2*A + 10/3*x^6*e^2*d^3*c*b*A + 5/3*x^6*e^3*d^2*b^2*A +
10/3*x^6*e^3*d^2*c*a*A + 5/3*x^6*e^4*d*b*a*A + 1/6*x^6*e^5*a^2*A + 2/5*x^5*d^5*c*b*B + x^5*e*d^4*b^2*B + 2*x^5
*e*d^4*c*a*B + 4*x^5*e^2*d^3*b*a*B + 2*x^5*e^3*d^2*a^2*B + 1/5*x^5*d^5*c^2*A + 2*x^5*e*d^4*c*b*A + 2*x^5*e^2*d
^3*b^2*A + 4*x^5*e^2*d^3*c*a*A + 4*x^5*e^3*d^2*b*a*A + x^5*e^4*d*a^2*A + 1/4*x^4*d^5*b^2*B + 1/2*x^4*d^5*c*a*B
+ 5/2*x^4*e*d^4*b*a*B + 5/2*x^4*e^2*d^3*a^2*B + 1/2*x^4*d^5*c*b*A + 5/4*x^4*e*d^4*b^2*A + 5/2*x^4*e*d^4*c*a*A
+ 5*x^4*e^2*d^3*b*a*A + 5/2*x^4*e^3*d^2*a^2*A + 2/3*x^3*d^5*b*a*B + 5/3*x^3*e*d^4*a^2*B + 1/3*x^3*d^5*b^2*A +
2/3*x^3*d^5*c*a*A + 10/3*x^3*e*d^4*b*a*A + 10/3*x^3*e^2*d^3*a^2*A + 1/2*x^2*d^5*a^2*B + x^2*d^5*b*a*A + 5/2*x
^2*e*d^4*a^2*A + x*d^5*a^2*A

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giac [B]  time = 0.17, size = 883, normalized size = 2.90 \begin {gather*} \frac {1}{11} \, B c^{2} x^{11} e^{5} + \frac {1}{2} \, B c^{2} d x^{10} e^{4} + \frac {10}{9} \, B c^{2} d^{2} x^{9} e^{3} + \frac {5}{4} \, B c^{2} d^{3} x^{8} e^{2} + \frac {5}{7} \, B c^{2} d^{4} x^{7} e + \frac {1}{6} \, B c^{2} d^{5} x^{6} + \frac {1}{5} \, B b c x^{10} e^{5} + \frac {1}{10} \, A c^{2} x^{10} e^{5} + \frac {10}{9} \, B b c d x^{9} e^{4} + \frac {5}{9} \, A c^{2} d x^{9} e^{4} + \frac {5}{2} \, B b c d^{2} x^{8} e^{3} + \frac {5}{4} \, A c^{2} d^{2} x^{8} e^{3} + \frac {20}{7} \, B b c d^{3} x^{7} e^{2} + \frac {10}{7} \, A c^{2} d^{3} x^{7} e^{2} + \frac {5}{3} \, B b c d^{4} x^{6} e + \frac {5}{6} \, A c^{2} d^{4} x^{6} e + \frac {2}{5} \, B b c d^{5} x^{5} + \frac {1}{5} \, A c^{2} d^{5} x^{5} + \frac {1}{9} \, B b^{2} x^{9} e^{5} + \frac {2}{9} \, B a c x^{9} e^{5} + \frac {2}{9} \, A b c x^{9} e^{5} + \frac {5}{8} \, B b^{2} d x^{8} e^{4} + \frac {5}{4} \, B a c d x^{8} e^{4} + \frac {5}{4} \, A b c d x^{8} e^{4} + \frac {10}{7} \, B b^{2} d^{2} x^{7} e^{3} + \frac {20}{7} \, B a c d^{2} x^{7} e^{3} + \frac {20}{7} \, A b c d^{2} x^{7} e^{3} + \frac {5}{3} \, B b^{2} d^{3} x^{6} e^{2} + \frac {10}{3} \, B a c d^{3} x^{6} e^{2} + \frac {10}{3} \, A b c d^{3} x^{6} e^{2} + B b^{2} d^{4} x^{5} e + 2 \, B a c d^{4} x^{5} e + 2 \, A b c d^{4} x^{5} e + \frac {1}{4} \, B b^{2} d^{5} x^{4} + \frac {1}{2} \, B a c d^{5} x^{4} + \frac {1}{2} \, A b c d^{5} x^{4} + \frac {1}{4} \, B a b x^{8} e^{5} + \frac {1}{8} \, A b^{2} x^{8} e^{5} + \frac {1}{4} \, A a c x^{8} e^{5} + \frac {10}{7} \, B a b d x^{7} e^{4} + \frac {5}{7} \, A b^{2} d x^{7} e^{4} + \frac {10}{7} \, A a c d x^{7} e^{4} + \frac {10}{3} \, B a b d^{2} x^{6} e^{3} + \frac {5}{3} \, A b^{2} d^{2} x^{6} e^{3} + \frac {10}{3} \, A a c d^{2} x^{6} e^{3} + 4 \, B a b d^{3} x^{5} e^{2} + 2 \, A b^{2} d^{3} x^{5} e^{2} + 4 \, A a c d^{3} x^{5} e^{2} + \frac {5}{2} \, B a b d^{4} x^{4} e + \frac {5}{4} \, A b^{2} d^{4} x^{4} e + \frac {5}{2} \, A a c d^{4} x^{4} e + \frac {2}{3} \, B a b d^{5} x^{3} + \frac {1}{3} \, A b^{2} d^{5} x^{3} + \frac {2}{3} \, A a c d^{5} x^{3} + \frac {1}{7} \, B a^{2} x^{7} e^{5} + \frac {2}{7} \, A a b x^{7} e^{5} + \frac {5}{6} \, B a^{2} d x^{6} e^{4} + \frac {5}{3} \, A a b d x^{6} e^{4} + 2 \, B a^{2} d^{2} x^{5} e^{3} + 4 \, A a b d^{2} x^{5} e^{3} + \frac {5}{2} \, B a^{2} d^{3} x^{4} e^{2} + 5 \, A a b d^{3} x^{4} e^{2} + \frac {5}{3} \, B a^{2} d^{4} x^{3} e + \frac {10}{3} \, A a b d^{4} x^{3} e + \frac {1}{2} \, B a^{2} d^{5} x^{2} + A a b d^{5} x^{2} + \frac {1}{6} \, A a^{2} x^{6} e^{5} + A a^{2} d x^{5} e^{4} + \frac {5}{2} \, A a^{2} d^{2} x^{4} e^{3} + \frac {10}{3} \, A a^{2} d^{3} x^{3} e^{2} + \frac {5}{2} \, A a^{2} d^{4} x^{2} e + A a^{2} d^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

1/11*B*c^2*x^11*e^5 + 1/2*B*c^2*d*x^10*e^4 + 10/9*B*c^2*d^2*x^9*e^3 + 5/4*B*c^2*d^3*x^8*e^2 + 5/7*B*c^2*d^4*x^
7*e + 1/6*B*c^2*d^5*x^6 + 1/5*B*b*c*x^10*e^5 + 1/10*A*c^2*x^10*e^5 + 10/9*B*b*c*d*x^9*e^4 + 5/9*A*c^2*d*x^9*e^
4 + 5/2*B*b*c*d^2*x^8*e^3 + 5/4*A*c^2*d^2*x^8*e^3 + 20/7*B*b*c*d^3*x^7*e^2 + 10/7*A*c^2*d^3*x^7*e^2 + 5/3*B*b*
c*d^4*x^6*e + 5/6*A*c^2*d^4*x^6*e + 2/5*B*b*c*d^5*x^5 + 1/5*A*c^2*d^5*x^5 + 1/9*B*b^2*x^9*e^5 + 2/9*B*a*c*x^9*
e^5 + 2/9*A*b*c*x^9*e^5 + 5/8*B*b^2*d*x^8*e^4 + 5/4*B*a*c*d*x^8*e^4 + 5/4*A*b*c*d*x^8*e^4 + 10/7*B*b^2*d^2*x^7
*e^3 + 20/7*B*a*c*d^2*x^7*e^3 + 20/7*A*b*c*d^2*x^7*e^3 + 5/3*B*b^2*d^3*x^6*e^2 + 10/3*B*a*c*d^3*x^6*e^2 + 10/3
*A*b*c*d^3*x^6*e^2 + B*b^2*d^4*x^5*e + 2*B*a*c*d^4*x^5*e + 2*A*b*c*d^4*x^5*e + 1/4*B*b^2*d^5*x^4 + 1/2*B*a*c*d
^5*x^4 + 1/2*A*b*c*d^5*x^4 + 1/4*B*a*b*x^8*e^5 + 1/8*A*b^2*x^8*e^5 + 1/4*A*a*c*x^8*e^5 + 10/7*B*a*b*d*x^7*e^4
+ 5/7*A*b^2*d*x^7*e^4 + 10/7*A*a*c*d*x^7*e^4 + 10/3*B*a*b*d^2*x^6*e^3 + 5/3*A*b^2*d^2*x^6*e^3 + 10/3*A*a*c*d^2
*x^6*e^3 + 4*B*a*b*d^3*x^5*e^2 + 2*A*b^2*d^3*x^5*e^2 + 4*A*a*c*d^3*x^5*e^2 + 5/2*B*a*b*d^4*x^4*e + 5/4*A*b^2*d
^4*x^4*e + 5/2*A*a*c*d^4*x^4*e + 2/3*B*a*b*d^5*x^3 + 1/3*A*b^2*d^5*x^3 + 2/3*A*a*c*d^5*x^3 + 1/7*B*a^2*x^7*e^5
+ 2/7*A*a*b*x^7*e^5 + 5/6*B*a^2*d*x^6*e^4 + 5/3*A*a*b*d*x^6*e^4 + 2*B*a^2*d^2*x^5*e^3 + 4*A*a*b*d^2*x^5*e^3 +
5/2*B*a^2*d^3*x^4*e^2 + 5*A*a*b*d^3*x^4*e^2 + 5/3*B*a^2*d^4*x^3*e + 10/3*A*a*b*d^4*x^3*e + 1/2*B*a^2*d^5*x^2
+ A*a*b*d^5*x^2 + 1/6*A*a^2*x^6*e^5 + A*a^2*d*x^5*e^4 + 5/2*A*a^2*d^2*x^4*e^3 + 10/3*A*a^2*d^3*x^3*e^2 + 5/2*A
*a^2*d^4*x^2*e + A*a^2*d^5*x

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maple [B]  time = 0.04, size = 671, normalized size = 2.21 \begin {gather*} \frac {B \,c^{2} e^{5} x^{11}}{11}+\frac {\left (2 B b c \,e^{5}+\left (A \,e^{5}+5 B d \,e^{4}\right ) c^{2}\right ) x^{10}}{10}+A \,a^{2} d^{5} x +\frac {\left (\left (2 a c +b^{2}\right ) B \,e^{5}+2 \left (A \,e^{5}+5 B d \,e^{4}\right ) b c +\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) c^{2}\right ) x^{9}}{9}+\frac {\left (2 B a b \,e^{5}+2 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) b c +\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) c^{2}+\left (A \,e^{5}+5 B d \,e^{4}\right ) \left (2 a c +b^{2}\right )\right ) x^{8}}{8}+\frac {\left (B \,a^{2} e^{5}+2 \left (A \,e^{5}+5 B d \,e^{4}\right ) a b +2 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) b c +\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) c^{2}+\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (\left (A \,e^{5}+5 B d \,e^{4}\right ) a^{2}+2 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a b +2 \left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) b c +\left (5 A \,d^{4} e +B \,d^{5}\right ) c^{2}+\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) \left (2 a c +b^{2}\right )\right ) x^{6}}{6}+\frac {\left (A \,c^{2} d^{5}+\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a^{2}+2 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a b +2 \left (5 A \,d^{4} e +B \,d^{5}\right ) b c +\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 A b c \,d^{5}+\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a^{2}+2 \left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a b +\left (5 A \,d^{4} e +B \,d^{5}\right ) \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (2 a c +b^{2}\right ) A \,d^{5}+\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a^{2}+2 \left (5 A \,d^{4} e +B \,d^{5}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 A a b \,d^{5}+\left (5 A \,d^{4} e +B \,d^{5}\right ) a^{2}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^5*(c*x^2+b*x+a)^2,x)

[Out]

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

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maxima [B]  time = 0.49, size = 648, normalized size = 2.13 \begin {gather*} \frac {1}{11} \, B c^{2} e^{5} x^{11} + \frac {1}{10} \, {\left (5 \, B c^{2} d e^{4} + {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{10} + \frac {1}{9} \, {\left (10 \, B c^{2} d^{2} e^{3} + 5 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{9} + A a^{2} d^{5} x + \frac {1}{8} \, {\left (10 \, B c^{2} d^{3} e^{2} + 10 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 5 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (5 \, B c^{2} d^{4} e + 10 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 10 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{5} + A a^{2} e^{5} + 5 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 10 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + 10 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, A a^{2} d e^{4} + {\left (2 \, B b c + A c^{2}\right )} d^{5} + 5 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{4} e + 10 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{3} e^{2} + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, A a^{2} d^{2} e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{5} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{4} e + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, A a^{2} d^{3} e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{5} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} d^{4} e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, A a^{2} d^{4} e + {\left (B a^{2} + 2 \, A a b\right )} d^{5}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.27, size = 739, normalized size = 2.43 \begin {gather*} x^3\,\left (\frac {5\,B\,a^2\,d^4\,e}{3}+\frac {10\,A\,a^2\,d^3\,e^2}{3}+\frac {2\,B\,a\,b\,d^5}{3}+\frac {10\,A\,a\,b\,d^4\,e}{3}+\frac {2\,A\,c\,a\,d^5}{3}+\frac {A\,b^2\,d^5}{3}\right )+x^9\,\left (\frac {B\,b^2\,e^5}{9}+\frac {10\,B\,b\,c\,d\,e^4}{9}+\frac {2\,A\,b\,c\,e^5}{9}+\frac {10\,B\,c^2\,d^2\,e^3}{9}+\frac {5\,A\,c^2\,d\,e^4}{9}+\frac {2\,B\,a\,c\,e^5}{9}\right )+x^4\,\left (\frac {5\,B\,a^2\,d^3\,e^2}{2}+\frac {5\,A\,a^2\,d^2\,e^3}{2}+\frac {5\,B\,a\,b\,d^4\,e}{2}+5\,A\,a\,b\,d^3\,e^2+\frac {B\,c\,a\,d^5}{2}+\frac {5\,A\,c\,a\,d^4\,e}{2}+\frac {B\,b^2\,d^5}{4}+\frac {5\,A\,b^2\,d^4\,e}{4}+\frac {A\,c\,b\,d^5}{2}\right )+x^8\,\left (\frac {5\,B\,b^2\,d\,e^4}{8}+\frac {A\,b^2\,e^5}{8}+\frac {5\,B\,b\,c\,d^2\,e^3}{2}+\frac {5\,A\,b\,c\,d\,e^4}{4}+\frac {B\,a\,b\,e^5}{4}+\frac {5\,B\,c^2\,d^3\,e^2}{4}+\frac {5\,A\,c^2\,d^2\,e^3}{4}+\frac {5\,B\,a\,c\,d\,e^4}{4}+\frac {A\,a\,c\,e^5}{4}\right )+x^6\,\left (\frac {5\,B\,a^2\,d\,e^4}{6}+\frac {A\,a^2\,e^5}{6}+\frac {10\,B\,a\,b\,d^2\,e^3}{3}+\frac {5\,A\,a\,b\,d\,e^4}{3}+\frac {10\,B\,a\,c\,d^3\,e^2}{3}+\frac {10\,A\,a\,c\,d^2\,e^3}{3}+\frac {5\,B\,b^2\,d^3\,e^2}{3}+\frac {5\,A\,b^2\,d^2\,e^3}{3}+\frac {5\,B\,b\,c\,d^4\,e}{3}+\frac {10\,A\,b\,c\,d^3\,e^2}{3}+\frac {B\,c^2\,d^5}{6}+\frac {5\,A\,c^2\,d^4\,e}{6}\right )+x^5\,\left (2\,B\,a^2\,d^2\,e^3+A\,a^2\,d\,e^4+4\,B\,a\,b\,d^3\,e^2+4\,A\,a\,b\,d^2\,e^3+2\,B\,a\,c\,d^4\,e+4\,A\,a\,c\,d^3\,e^2+B\,b^2\,d^4\,e+2\,A\,b^2\,d^3\,e^2+\frac {2\,B\,b\,c\,d^5}{5}+2\,A\,b\,c\,d^4\,e+\frac {A\,c^2\,d^5}{5}\right )+x^7\,\left (\frac {B\,a^2\,e^5}{7}+\frac {10\,B\,a\,b\,d\,e^4}{7}+\frac {2\,A\,a\,b\,e^5}{7}+\frac {20\,B\,a\,c\,d^2\,e^3}{7}+\frac {10\,A\,a\,c\,d\,e^4}{7}+\frac {10\,B\,b^2\,d^2\,e^3}{7}+\frac {5\,A\,b^2\,d\,e^4}{7}+\frac {20\,B\,b\,c\,d^3\,e^2}{7}+\frac {20\,A\,b\,c\,d^2\,e^3}{7}+\frac {5\,B\,c^2\,d^4\,e}{7}+\frac {10\,A\,c^2\,d^3\,e^2}{7}\right )+A\,a^2\,d^5\,x+\frac {a\,d^4\,x^2\,\left (5\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {c\,e^4\,x^{10}\,\left (A\,c\,e+2\,B\,b\,e+5\,B\,c\,d\right )}{10}+\frac {B\,c^2\,e^5\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^5*(a + b*x + c*x^2)^2,x)

[Out]

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

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sympy [B]  time = 0.24, size = 957, normalized size = 3.15 \begin {gather*} A a^{2} d^{5} x + \frac {B c^{2} e^{5} x^{11}}{11} + x^{10} \left (\frac {A c^{2} e^{5}}{10} + \frac {B b c e^{5}}{5} + \frac {B c^{2} d e^{4}}{2}\right ) + x^{9} \left (\frac {2 A b c e^{5}}{9} + \frac {5 A c^{2} d e^{4}}{9} + \frac {2 B a c e^{5}}{9} + \frac {B b^{2} e^{5}}{9} + \frac {10 B b c d e^{4}}{9} + \frac {10 B c^{2} d^{2} e^{3}}{9}\right ) + x^{8} \left (\frac {A a c e^{5}}{4} + \frac {A b^{2} e^{5}}{8} + \frac {5 A b c d e^{4}}{4} + \frac {5 A c^{2} d^{2} e^{3}}{4} + \frac {B a b e^{5}}{4} + \frac {5 B a c d e^{4}}{4} + \frac {5 B b^{2} d e^{4}}{8} + \frac {5 B b c d^{2} e^{3}}{2} + \frac {5 B c^{2} d^{3} e^{2}}{4}\right ) + x^{7} \left (\frac {2 A a b e^{5}}{7} + \frac {10 A a c d e^{4}}{7} + \frac {5 A b^{2} d e^{4}}{7} + \frac {20 A b c d^{2} e^{3}}{7} + \frac {10 A c^{2} d^{3} e^{2}}{7} + \frac {B a^{2} e^{5}}{7} + \frac {10 B a b d e^{4}}{7} + \frac {20 B a c d^{2} e^{3}}{7} + \frac {10 B b^{2} d^{2} e^{3}}{7} + \frac {20 B b c d^{3} e^{2}}{7} + \frac {5 B c^{2} d^{4} e}{7}\right ) + x^{6} \left (\frac {A a^{2} e^{5}}{6} + \frac {5 A a b d e^{4}}{3} + \frac {10 A a c d^{2} e^{3}}{3} + \frac {5 A b^{2} d^{2} e^{3}}{3} + \frac {10 A b c d^{3} e^{2}}{3} + \frac {5 A c^{2} d^{4} e}{6} + \frac {5 B a^{2} d e^{4}}{6} + \frac {10 B a b d^{2} e^{3}}{3} + \frac {10 B a c d^{3} e^{2}}{3} + \frac {5 B b^{2} d^{3} e^{2}}{3} + \frac {5 B b c d^{4} e}{3} + \frac {B c^{2} d^{5}}{6}\right ) + x^{5} \left (A a^{2} d e^{4} + 4 A a b d^{2} e^{3} + 4 A a c d^{3} e^{2} + 2 A b^{2} d^{3} e^{2} + 2 A b c d^{4} e + \frac {A c^{2} d^{5}}{5} + 2 B a^{2} d^{2} e^{3} + 4 B a b d^{3} e^{2} + 2 B a c d^{4} e + B b^{2} d^{4} e + \frac {2 B b c d^{5}}{5}\right ) + x^{4} \left (\frac {5 A a^{2} d^{2} e^{3}}{2} + 5 A a b d^{3} e^{2} + \frac {5 A a c d^{4} e}{2} + \frac {5 A b^{2} d^{4} e}{4} + \frac {A b c d^{5}}{2} + \frac {5 B a^{2} d^{3} e^{2}}{2} + \frac {5 B a b d^{4} e}{2} + \frac {B a c d^{5}}{2} + \frac {B b^{2} d^{5}}{4}\right ) + x^{3} \left (\frac {10 A a^{2} d^{3} e^{2}}{3} + \frac {10 A a b d^{4} e}{3} + \frac {2 A a c d^{5}}{3} + \frac {A b^{2} d^{5}}{3} + \frac {5 B a^{2} d^{4} e}{3} + \frac {2 B a b d^{5}}{3}\right ) + x^{2} \left (\frac {5 A a^{2} d^{4} e}{2} + A a b d^{5} + \frac {B a^{2} d^{5}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**5*(c*x**2+b*x+a)**2,x)

[Out]

A*a**2*d**5*x + B*c**2*e**5*x**11/11 + x**10*(A*c**2*e**5/10 + B*b*c*e**5/5 + B*c**2*d*e**4/2) + x**9*(2*A*b*c
*e**5/9 + 5*A*c**2*d*e**4/9 + 2*B*a*c*e**5/9 + B*b**2*e**5/9 + 10*B*b*c*d*e**4/9 + 10*B*c**2*d**2*e**3/9) + x*
*8*(A*a*c*e**5/4 + A*b**2*e**5/8 + 5*A*b*c*d*e**4/4 + 5*A*c**2*d**2*e**3/4 + B*a*b*e**5/4 + 5*B*a*c*d*e**4/4 +
5*B*b**2*d*e**4/8 + 5*B*b*c*d**2*e**3/2 + 5*B*c**2*d**3*e**2/4) + x**7*(2*A*a*b*e**5/7 + 10*A*a*c*d*e**4/7 +
5*A*b**2*d*e**4/7 + 20*A*b*c*d**2*e**3/7 + 10*A*c**2*d**3*e**2/7 + B*a**2*e**5/7 + 10*B*a*b*d*e**4/7 + 20*B*a*
c*d**2*e**3/7 + 10*B*b**2*d**2*e**3/7 + 20*B*b*c*d**3*e**2/7 + 5*B*c**2*d**4*e/7) + x**6*(A*a**2*e**5/6 + 5*A*
a*b*d*e**4/3 + 10*A*a*c*d**2*e**3/3 + 5*A*b**2*d**2*e**3/3 + 10*A*b*c*d**3*e**2/3 + 5*A*c**2*d**4*e/6 + 5*B*a*
*2*d*e**4/6 + 10*B*a*b*d**2*e**3/3 + 10*B*a*c*d**3*e**2/3 + 5*B*b**2*d**3*e**2/3 + 5*B*b*c*d**4*e/3 + B*c**2*d
**5/6) + x**5*(A*a**2*d*e**4 + 4*A*a*b*d**2*e**3 + 4*A*a*c*d**3*e**2 + 2*A*b**2*d**3*e**2 + 2*A*b*c*d**4*e + A
*c**2*d**5/5 + 2*B*a**2*d**2*e**3 + 4*B*a*b*d**3*e**2 + 2*B*a*c*d**4*e + B*b**2*d**4*e + 2*B*b*c*d**5/5) + x**
4*(5*A*a**2*d**2*e**3/2 + 5*A*a*b*d**3*e**2 + 5*A*a*c*d**4*e/2 + 5*A*b**2*d**4*e/4 + A*b*c*d**5/2 + 5*B*a**2*d
**3*e**2/2 + 5*B*a*b*d**4*e/2 + B*a*c*d**5/2 + B*b**2*d**5/4) + x**3*(10*A*a**2*d**3*e**2/3 + 10*A*a*b*d**4*e/
3 + 2*A*a*c*d**5/3 + A*b**2*d**5/3 + 5*B*a**2*d**4*e/3 + 2*B*a*b*d**5/3) + x**2*(5*A*a**2*d**4*e/2 + A*a*b*d**
5 + B*a**2*d**5/2)

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