3.21.78 \(\int (A+B x) (d+e x)^4 (a+b x+c x^2)^2 \, dx\)

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

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Rubi [A]  time = 0.63, 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)^8 \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 )}{8 e^6}-\frac {(d+e x)^7 \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 )}{7 e^6}+\frac {(d+e x)^6 \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 )}{6 e^6}-\frac {(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac {c (d+e x)^9 (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac {B c^2 (d+e x)^{10}}{10 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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Mathematica [A]  time = 0.24, size = 550, normalized size = 1.81 \begin {gather*} \frac {1}{6} x^6 \left (B \left (e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+4 c d^2 e (3 a e+2 b d)+c^2 d^4\right )+2 A e \left (2 c d e (2 a e+3 b d)+b e^2 (a e+2 b d)+2 c^2 d^3\right )\right )+\frac {1}{5} x^5 \left (A \left (a^2 e^4+12 a c d^2 e^2+c^2 d^4\right )+2 b d \left (4 a A e^3+6 a B d e^2+4 A c d^2 e+B c d^3\right )+4 a B d e \left (a e^2+2 c d^2\right )+2 b^2 d^2 e (3 A e+2 B d)\right )+a^2 A d^4 x+\frac {1}{8} e^2 x^8 \left (B \left (2 c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )+2 A c e (b e+2 c d)\right )+\frac {1}{7} e x^7 \left (A e \left (2 c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )+2 B \left (2 c d e (2 a e+3 b d)+b e^2 (a e+2 b d)+2 c^2 d^3\right )\right )+\frac {1}{3} d^2 x^3 \left (A \left (8 a b d e+2 a \left (3 a e^2+c d^2\right )+b^2 d^2\right )+2 a B d (2 a e+b d)\right )+\frac {1}{4} d x^4 \left (2 b d \left (6 a A e^2+4 a B d e+A c d^2\right )+2 a \left (2 a A e^3+3 a B d e^2+4 A c d^2 e+B c d^3\right )+b^2 d^2 (4 A e+B d)\right )+\frac {1}{2} a d^3 x^2 (4 a A e+a B d+2 A b d)+\frac {1}{9} c e^3 x^9 (A c e+2 b B e+4 B c d)+\frac {1}{10} B c^2 e^4 x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^4*(a + b*x + c*x^2)^2,x]

[Out]

a^2*A*d^4*x + (a*d^3*(2*A*b*d + a*B*d + 4*a*A*e)*x^2)/2 + (d^2*(2*a*B*d*(b*d + 2*a*e) + A*(b^2*d^2 + 8*a*b*d*e
 + 2*a*(c*d^2 + 3*a*e^2)))*x^3)/3 + (d*(b^2*d^2*(B*d + 4*A*e) + 2*b*d*(A*c*d^2 + 4*a*B*d*e + 6*a*A*e^2) + 2*a*
(B*c*d^3 + 4*A*c*d^2*e + 3*a*B*d*e^2 + 2*a*A*e^3))*x^4)/4 + ((2*b^2*d^2*e*(2*B*d + 3*A*e) + 4*a*B*d*e*(2*c*d^2
 + a*e^2) + 2*b*d*(B*c*d^3 + 4*A*c*d^2*e + 6*a*B*d*e^2 + 4*a*A*e^3) + A*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4))*
x^5)/5 + ((2*A*e*(2*c^2*d^3 + b*e^2*(2*b*d + a*e) + 2*c*d*e*(3*b*d + 2*a*e)) + B*(c^2*d^4 + 4*c*d^2*e*(2*b*d +
 3*a*e) + e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2)))*x^6)/6 + (e*(A*e*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(4*b*d + a*e))
 + 2*B*(2*c^2*d^3 + b*e^2*(2*b*d + a*e) + 2*c*d*e*(3*b*d + 2*a*e)))*x^7)/7 + (e^2*(2*A*c*e*(2*c*d + b*e) + B*(
6*c^2*d^2 + b^2*e^2 + 2*c*e*(4*b*d + a*e)))*x^8)/8 + (c*e^3*(4*B*c*d + 2*b*B*e + A*c*e)*x^9)/9 + (B*c^2*e^4*x^
10)/10

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^4 \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)^4*(a + b*x + c*x^2)^2,x]

[Out]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^4*(a + b*x + c*x^2)^2, x]

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fricas [B]  time = 0.36, size = 743, normalized size = 2.44 \begin {gather*} \frac {1}{10} x^{10} e^{4} c^{2} B + \frac {4}{9} x^{9} e^{3} d c^{2} B + \frac {2}{9} x^{9} e^{4} c b B + \frac {1}{9} x^{9} e^{4} c^{2} A + \frac {3}{4} x^{8} e^{2} d^{2} c^{2} B + x^{8} e^{3} d c b B + \frac {1}{8} x^{8} e^{4} b^{2} B + \frac {1}{4} x^{8} e^{4} c a B + \frac {1}{2} x^{8} e^{3} d c^{2} A + \frac {1}{4} x^{8} e^{4} c b A + \frac {4}{7} x^{7} e d^{3} c^{2} B + \frac {12}{7} x^{7} e^{2} d^{2} c b B + \frac {4}{7} x^{7} e^{3} d b^{2} B + \frac {8}{7} x^{7} e^{3} d c a B + \frac {2}{7} x^{7} e^{4} b a B + \frac {6}{7} x^{7} e^{2} d^{2} c^{2} A + \frac {8}{7} x^{7} e^{3} d c b A + \frac {1}{7} x^{7} e^{4} b^{2} A + \frac {2}{7} x^{7} e^{4} c a A + \frac {1}{6} x^{6} d^{4} c^{2} B + \frac {4}{3} x^{6} e d^{3} c b B + x^{6} e^{2} d^{2} b^{2} B + 2 x^{6} e^{2} d^{2} c a B + \frac {4}{3} x^{6} e^{3} d b a B + \frac {1}{6} x^{6} e^{4} a^{2} B + \frac {2}{3} x^{6} e d^{3} c^{2} A + 2 x^{6} e^{2} d^{2} c b A + \frac {2}{3} x^{6} e^{3} d b^{2} A + \frac {4}{3} x^{6} e^{3} d c a A + \frac {1}{3} x^{6} e^{4} b a A + \frac {2}{5} x^{5} d^{4} c b B + \frac {4}{5} x^{5} e d^{3} b^{2} B + \frac {8}{5} x^{5} e d^{3} c a B + \frac {12}{5} x^{5} e^{2} d^{2} b a B + \frac {4}{5} x^{5} e^{3} d a^{2} B + \frac {1}{5} x^{5} d^{4} c^{2} A + \frac {8}{5} x^{5} e d^{3} c b A + \frac {6}{5} x^{5} e^{2} d^{2} b^{2} A + \frac {12}{5} x^{5} e^{2} d^{2} c a A + \frac {8}{5} x^{5} e^{3} d b a A + \frac {1}{5} x^{5} e^{4} a^{2} A + \frac {1}{4} x^{4} d^{4} b^{2} B + \frac {1}{2} x^{4} d^{4} c a B + 2 x^{4} e d^{3} b a B + \frac {3}{2} x^{4} e^{2} d^{2} a^{2} B + \frac {1}{2} x^{4} d^{4} c b A + x^{4} e d^{3} b^{2} A + 2 x^{4} e d^{3} c a A + 3 x^{4} e^{2} d^{2} b a A + x^{4} e^{3} d a^{2} A + \frac {2}{3} x^{3} d^{4} b a B + \frac {4}{3} x^{3} e d^{3} a^{2} B + \frac {1}{3} x^{3} d^{4} b^{2} A + \frac {2}{3} x^{3} d^{4} c a A + \frac {8}{3} x^{3} e d^{3} b a A + 2 x^{3} e^{2} d^{2} a^{2} A + \frac {1}{2} x^{2} d^{4} a^{2} B + x^{2} d^{4} b a A + 2 x^{2} e d^{3} a^{2} A + x d^{4} a^{2} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

1/10*x^10*e^4*c^2*B + 4/9*x^9*e^3*d*c^2*B + 2/9*x^9*e^4*c*b*B + 1/9*x^9*e^4*c^2*A + 3/4*x^8*e^2*d^2*c^2*B + x^
8*e^3*d*c*b*B + 1/8*x^8*e^4*b^2*B + 1/4*x^8*e^4*c*a*B + 1/2*x^8*e^3*d*c^2*A + 1/4*x^8*e^4*c*b*A + 4/7*x^7*e*d^
3*c^2*B + 12/7*x^7*e^2*d^2*c*b*B + 4/7*x^7*e^3*d*b^2*B + 8/7*x^7*e^3*d*c*a*B + 2/7*x^7*e^4*b*a*B + 6/7*x^7*e^2
*d^2*c^2*A + 8/7*x^7*e^3*d*c*b*A + 1/7*x^7*e^4*b^2*A + 2/7*x^7*e^4*c*a*A + 1/6*x^6*d^4*c^2*B + 4/3*x^6*e*d^3*c
*b*B + x^6*e^2*d^2*b^2*B + 2*x^6*e^2*d^2*c*a*B + 4/3*x^6*e^3*d*b*a*B + 1/6*x^6*e^4*a^2*B + 2/3*x^6*e*d^3*c^2*A
 + 2*x^6*e^2*d^2*c*b*A + 2/3*x^6*e^3*d*b^2*A + 4/3*x^6*e^3*d*c*a*A + 1/3*x^6*e^4*b*a*A + 2/5*x^5*d^4*c*b*B + 4
/5*x^5*e*d^3*b^2*B + 8/5*x^5*e*d^3*c*a*B + 12/5*x^5*e^2*d^2*b*a*B + 4/5*x^5*e^3*d*a^2*B + 1/5*x^5*d^4*c^2*A +
8/5*x^5*e*d^3*c*b*A + 6/5*x^5*e^2*d^2*b^2*A + 12/5*x^5*e^2*d^2*c*a*A + 8/5*x^5*e^3*d*b*a*A + 1/5*x^5*e^4*a^2*A
 + 1/4*x^4*d^4*b^2*B + 1/2*x^4*d^4*c*a*B + 2*x^4*e*d^3*b*a*B + 3/2*x^4*e^2*d^2*a^2*B + 1/2*x^4*d^4*c*b*A + x^4
*e*d^3*b^2*A + 2*x^4*e*d^3*c*a*A + 3*x^4*e^2*d^2*b*a*A + x^4*e^3*d*a^2*A + 2/3*x^3*d^4*b*a*B + 4/3*x^3*e*d^3*a
^2*B + 1/3*x^3*d^4*b^2*A + 2/3*x^3*d^4*c*a*A + 8/3*x^3*e*d^3*b*a*A + 2*x^3*e^2*d^2*a^2*A + 1/2*x^2*d^4*a^2*B +
 x^2*d^4*b*a*A + 2*x^2*e*d^3*a^2*A + x*d^4*a^2*A

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giac [B]  time = 0.16, size = 719, normalized size = 2.37 \begin {gather*} \frac {1}{10} \, B c^{2} x^{10} e^{4} + \frac {4}{9} \, B c^{2} d x^{9} e^{3} + \frac {3}{4} \, B c^{2} d^{2} x^{8} e^{2} + \frac {4}{7} \, B c^{2} d^{3} x^{7} e + \frac {1}{6} \, B c^{2} d^{4} x^{6} + \frac {2}{9} \, B b c x^{9} e^{4} + \frac {1}{9} \, A c^{2} x^{9} e^{4} + B b c d x^{8} e^{3} + \frac {1}{2} \, A c^{2} d x^{8} e^{3} + \frac {12}{7} \, B b c d^{2} x^{7} e^{2} + \frac {6}{7} \, A c^{2} d^{2} x^{7} e^{2} + \frac {4}{3} \, B b c d^{3} x^{6} e + \frac {2}{3} \, A c^{2} d^{3} x^{6} e + \frac {2}{5} \, B b c d^{4} x^{5} + \frac {1}{5} \, A c^{2} d^{4} x^{5} + \frac {1}{8} \, B b^{2} x^{8} e^{4} + \frac {1}{4} \, B a c x^{8} e^{4} + \frac {1}{4} \, A b c x^{8} e^{4} + \frac {4}{7} \, B b^{2} d x^{7} e^{3} + \frac {8}{7} \, B a c d x^{7} e^{3} + \frac {8}{7} \, A b c d x^{7} e^{3} + B b^{2} d^{2} x^{6} e^{2} + 2 \, B a c d^{2} x^{6} e^{2} + 2 \, A b c d^{2} x^{6} e^{2} + \frac {4}{5} \, B b^{2} d^{3} x^{5} e + \frac {8}{5} \, B a c d^{3} x^{5} e + \frac {8}{5} \, A b c d^{3} x^{5} e + \frac {1}{4} \, B b^{2} d^{4} x^{4} + \frac {1}{2} \, B a c d^{4} x^{4} + \frac {1}{2} \, A b c d^{4} x^{4} + \frac {2}{7} \, B a b x^{7} e^{4} + \frac {1}{7} \, A b^{2} x^{7} e^{4} + \frac {2}{7} \, A a c x^{7} e^{4} + \frac {4}{3} \, B a b d x^{6} e^{3} + \frac {2}{3} \, A b^{2} d x^{6} e^{3} + \frac {4}{3} \, A a c d x^{6} e^{3} + \frac {12}{5} \, B a b d^{2} x^{5} e^{2} + \frac {6}{5} \, A b^{2} d^{2} x^{5} e^{2} + \frac {12}{5} \, A a c d^{2} x^{5} e^{2} + 2 \, B a b d^{3} x^{4} e + A b^{2} d^{3} x^{4} e + 2 \, A a c d^{3} x^{4} e + \frac {2}{3} \, B a b d^{4} x^{3} + \frac {1}{3} \, A b^{2} d^{4} x^{3} + \frac {2}{3} \, A a c d^{4} x^{3} + \frac {1}{6} \, B a^{2} x^{6} e^{4} + \frac {1}{3} \, A a b x^{6} e^{4} + \frac {4}{5} \, B a^{2} d x^{5} e^{3} + \frac {8}{5} \, A a b d x^{5} e^{3} + \frac {3}{2} \, B a^{2} d^{2} x^{4} e^{2} + 3 \, A a b d^{2} x^{4} e^{2} + \frac {4}{3} \, B a^{2} d^{3} x^{3} e + \frac {8}{3} \, A a b d^{3} x^{3} e + \frac {1}{2} \, B a^{2} d^{4} x^{2} + A a b d^{4} x^{2} + \frac {1}{5} \, A a^{2} x^{5} e^{4} + A a^{2} d x^{4} e^{3} + 2 \, A a^{2} d^{2} x^{3} e^{2} + 2 \, A a^{2} d^{3} x^{2} e + A a^{2} d^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

1/10*B*c^2*x^10*e^4 + 4/9*B*c^2*d*x^9*e^3 + 3/4*B*c^2*d^2*x^8*e^2 + 4/7*B*c^2*d^3*x^7*e + 1/6*B*c^2*d^4*x^6 +
2/9*B*b*c*x^9*e^4 + 1/9*A*c^2*x^9*e^4 + B*b*c*d*x^8*e^3 + 1/2*A*c^2*d*x^8*e^3 + 12/7*B*b*c*d^2*x^7*e^2 + 6/7*A
*c^2*d^2*x^7*e^2 + 4/3*B*b*c*d^3*x^6*e + 2/3*A*c^2*d^3*x^6*e + 2/5*B*b*c*d^4*x^5 + 1/5*A*c^2*d^4*x^5 + 1/8*B*b
^2*x^8*e^4 + 1/4*B*a*c*x^8*e^4 + 1/4*A*b*c*x^8*e^4 + 4/7*B*b^2*d*x^7*e^3 + 8/7*B*a*c*d*x^7*e^3 + 8/7*A*b*c*d*x
^7*e^3 + B*b^2*d^2*x^6*e^2 + 2*B*a*c*d^2*x^6*e^2 + 2*A*b*c*d^2*x^6*e^2 + 4/5*B*b^2*d^3*x^5*e + 8/5*B*a*c*d^3*x
^5*e + 8/5*A*b*c*d^3*x^5*e + 1/4*B*b^2*d^4*x^4 + 1/2*B*a*c*d^4*x^4 + 1/2*A*b*c*d^4*x^4 + 2/7*B*a*b*x^7*e^4 + 1
/7*A*b^2*x^7*e^4 + 2/7*A*a*c*x^7*e^4 + 4/3*B*a*b*d*x^6*e^3 + 2/3*A*b^2*d*x^6*e^3 + 4/3*A*a*c*d*x^6*e^3 + 12/5*
B*a*b*d^2*x^5*e^2 + 6/5*A*b^2*d^2*x^5*e^2 + 12/5*A*a*c*d^2*x^5*e^2 + 2*B*a*b*d^3*x^4*e + A*b^2*d^3*x^4*e + 2*A
*a*c*d^3*x^4*e + 2/3*B*a*b*d^4*x^3 + 1/3*A*b^2*d^4*x^3 + 2/3*A*a*c*d^4*x^3 + 1/6*B*a^2*x^6*e^4 + 1/3*A*a*b*x^6
*e^4 + 4/5*B*a^2*d*x^5*e^3 + 8/5*A*a*b*d*x^5*e^3 + 3/2*B*a^2*d^2*x^4*e^2 + 3*A*a*b*d^2*x^4*e^2 + 4/3*B*a^2*d^3
*x^3*e + 8/3*A*a*b*d^3*x^3*e + 1/2*B*a^2*d^4*x^2 + A*a*b*d^4*x^2 + 1/5*A*a^2*x^5*e^4 + A*a^2*d*x^4*e^3 + 2*A*a
^2*d^2*x^3*e^2 + 2*A*a^2*d^3*x^2*e + A*a^2*d^4*x

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maple [A]  time = 0.04, size = 545, normalized size = 1.79 \begin {gather*} \frac {B \,c^{2} e^{4} x^{10}}{10}+\frac {\left (2 B b c \,e^{4}+\left (A \,e^{4}+4 B d \,e^{3}\right ) c^{2}\right ) x^{9}}{9}+A \,a^{2} d^{4} x +\frac {\left (\left (2 a c +b^{2}\right ) B \,e^{4}+2 \left (A \,e^{4}+4 B d \,e^{3}\right ) b c +\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) c^{2}\right ) x^{8}}{8}+\frac {\left (2 B a b \,e^{4}+2 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) b c +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c^{2}+\left (A \,e^{4}+4 B d \,e^{3}\right ) \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (B \,a^{2} e^{4}+2 \left (A \,e^{4}+4 B d \,e^{3}\right ) a b +2 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b c +\left (4 A \,d^{3} e +B \,d^{4}\right ) c^{2}+\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) \left (2 a c +b^{2}\right )\right ) x^{6}}{6}+\frac {\left (A \,c^{2} d^{4}+\left (A \,e^{4}+4 B d \,e^{3}\right ) a^{2}+2 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a b +2 \left (4 A \,d^{3} e +B \,d^{4}\right ) b c +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 A b c \,d^{4}+\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a^{2}+2 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a b +\left (4 A \,d^{3} e +B \,d^{4}\right ) \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (2 a c +b^{2}\right ) A \,d^{4}+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{2}+2 \left (4 A \,d^{3} e +B \,d^{4}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 A a b \,d^{4}+\left (4 A \,d^{3} e +B \,d^{4}\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)^4*(c*x^2+b*x+a)^2,x)

[Out]

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

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maxima [A]  time = 0.52, size = 532, normalized size = 1.75 \begin {gather*} \frac {1}{10} \, B c^{2} e^{4} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{2} d e^{3} + {\left (2 \, B b c + A c^{2}\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (6 \, B c^{2} d^{2} e^{2} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{4}\right )} x^{8} + A a^{2} d^{4} x + \frac {1}{7} \, {\left (4 \, B c^{2} d^{3} e + 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{2} + 4 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{3} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{4} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 6 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{2} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{2} e^{4} + {\left (2 \, B b c + A c^{2}\right )} d^{4} + 4 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{2} d e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{4} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{3} e + 6 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{2} d^{2} e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{4} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{2} d^{3} e + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.48, size = 594, normalized size = 1.95 \begin {gather*} x^3\,\left (\frac {4\,B\,a^2\,d^3\,e}{3}+2\,A\,a^2\,d^2\,e^2+\frac {2\,B\,a\,b\,d^4}{3}+\frac {8\,A\,a\,b\,d^3\,e}{3}+\frac {2\,A\,c\,a\,d^4}{3}+\frac {A\,b^2\,d^4}{3}\right )+x^4\,\left (\frac {3\,B\,a^2\,d^2\,e^2}{2}+A\,a^2\,d\,e^3+2\,B\,a\,b\,d^3\,e+3\,A\,a\,b\,d^2\,e^2+\frac {B\,c\,a\,d^4}{2}+2\,A\,c\,a\,d^3\,e+\frac {B\,b^2\,d^4}{4}+A\,b^2\,d^3\,e+\frac {A\,c\,b\,d^4}{2}\right )+x^8\,\left (\frac {B\,b^2\,e^4}{8}+B\,b\,c\,d\,e^3+\frac {A\,b\,c\,e^4}{4}+\frac {3\,B\,c^2\,d^2\,e^2}{4}+\frac {A\,c^2\,d\,e^3}{2}+\frac {B\,a\,c\,e^4}{4}\right )+x^7\,\left (\frac {4\,B\,b^2\,d\,e^3}{7}+\frac {A\,b^2\,e^4}{7}+\frac {12\,B\,b\,c\,d^2\,e^2}{7}+\frac {8\,A\,b\,c\,d\,e^3}{7}+\frac {2\,B\,a\,b\,e^4}{7}+\frac {4\,B\,c^2\,d^3\,e}{7}+\frac {6\,A\,c^2\,d^2\,e^2}{7}+\frac {8\,B\,a\,c\,d\,e^3}{7}+\frac {2\,A\,a\,c\,e^4}{7}\right )+x^5\,\left (\frac {4\,B\,a^2\,d\,e^3}{5}+\frac {A\,a^2\,e^4}{5}+\frac {12\,B\,a\,b\,d^2\,e^2}{5}+\frac {8\,A\,a\,b\,d\,e^3}{5}+\frac {8\,B\,a\,c\,d^3\,e}{5}+\frac {12\,A\,a\,c\,d^2\,e^2}{5}+\frac {4\,B\,b^2\,d^3\,e}{5}+\frac {6\,A\,b^2\,d^2\,e^2}{5}+\frac {2\,B\,b\,c\,d^4}{5}+\frac {8\,A\,b\,c\,d^3\,e}{5}+\frac {A\,c^2\,d^4}{5}\right )+x^6\,\left (\frac {B\,a^2\,e^4}{6}+\frac {4\,B\,a\,b\,d\,e^3}{3}+\frac {A\,a\,b\,e^4}{3}+2\,B\,a\,c\,d^2\,e^2+\frac {4\,A\,a\,c\,d\,e^3}{3}+B\,b^2\,d^2\,e^2+\frac {2\,A\,b^2\,d\,e^3}{3}+\frac {4\,B\,b\,c\,d^3\,e}{3}+2\,A\,b\,c\,d^2\,e^2+\frac {B\,c^2\,d^4}{6}+\frac {2\,A\,c^2\,d^3\,e}{3}\right )+A\,a^2\,d^4\,x+\frac {a\,d^3\,x^2\,\left (4\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {c\,e^3\,x^9\,\left (A\,c\,e+2\,B\,b\,e+4\,B\,c\,d\right )}{9}+\frac {B\,c^2\,e^4\,x^{10}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^4*(a + b*x + c*x^2)^2,x)

[Out]

x^3*((A*b^2*d^4)/3 + (2*A*a*c*d^4)/3 + (2*B*a*b*d^4)/3 + (4*B*a^2*d^3*e)/3 + 2*A*a^2*d^2*e^2 + (8*A*a*b*d^3*e)
/3) + x^4*((B*b^2*d^4)/4 + (A*b*c*d^4)/2 + (B*a*c*d^4)/2 + A*a^2*d*e^3 + A*b^2*d^3*e + (3*B*a^2*d^2*e^2)/2 + 2
*A*a*c*d^3*e + 2*B*a*b*d^3*e + 3*A*a*b*d^2*e^2) + x^8*((B*b^2*e^4)/8 + (A*b*c*e^4)/4 + (B*a*c*e^4)/4 + (A*c^2*
d*e^3)/2 + (3*B*c^2*d^2*e^2)/4 + B*b*c*d*e^3) + x^7*((A*b^2*e^4)/7 + (2*A*a*c*e^4)/7 + (2*B*a*b*e^4)/7 + (4*B*
b^2*d*e^3)/7 + (4*B*c^2*d^3*e)/7 + (6*A*c^2*d^2*e^2)/7 + (8*A*b*c*d*e^3)/7 + (8*B*a*c*d*e^3)/7 + (12*B*b*c*d^2
*e^2)/7) + x^5*((A*a^2*e^4)/5 + (A*c^2*d^4)/5 + (2*B*b*c*d^4)/5 + (4*B*a^2*d*e^3)/5 + (4*B*b^2*d^3*e)/5 + (6*A
*b^2*d^2*e^2)/5 + (8*A*a*b*d*e^3)/5 + (8*A*b*c*d^3*e)/5 + (8*B*a*c*d^3*e)/5 + (12*A*a*c*d^2*e^2)/5 + (12*B*a*b
*d^2*e^2)/5) + x^6*((B*a^2*e^4)/6 + (B*c^2*d^4)/6 + (A*a*b*e^4)/3 + (2*A*b^2*d*e^3)/3 + (2*A*c^2*d^3*e)/3 + B*
b^2*d^2*e^2 + (4*A*a*c*d*e^3)/3 + (4*B*a*b*d*e^3)/3 + (4*B*b*c*d^3*e)/3 + 2*A*b*c*d^2*e^2 + 2*B*a*c*d^2*e^2) +
 A*a^2*d^4*x + (a*d^3*x^2*(4*A*a*e + 2*A*b*d + B*a*d))/2 + (c*e^3*x^9*(A*c*e + 2*B*b*e + 4*B*c*d))/9 + (B*c^2*
e^4*x^10)/10

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sympy [B]  time = 0.18, size = 765, normalized size = 2.52 \begin {gather*} A a^{2} d^{4} x + \frac {B c^{2} e^{4} x^{10}}{10} + x^{9} \left (\frac {A c^{2} e^{4}}{9} + \frac {2 B b c e^{4}}{9} + \frac {4 B c^{2} d e^{3}}{9}\right ) + x^{8} \left (\frac {A b c e^{4}}{4} + \frac {A c^{2} d e^{3}}{2} + \frac {B a c e^{4}}{4} + \frac {B b^{2} e^{4}}{8} + B b c d e^{3} + \frac {3 B c^{2} d^{2} e^{2}}{4}\right ) + x^{7} \left (\frac {2 A a c e^{4}}{7} + \frac {A b^{2} e^{4}}{7} + \frac {8 A b c d e^{3}}{7} + \frac {6 A c^{2} d^{2} e^{2}}{7} + \frac {2 B a b e^{4}}{7} + \frac {8 B a c d e^{3}}{7} + \frac {4 B b^{2} d e^{3}}{7} + \frac {12 B b c d^{2} e^{2}}{7} + \frac {4 B c^{2} d^{3} e}{7}\right ) + x^{6} \left (\frac {A a b e^{4}}{3} + \frac {4 A a c d e^{3}}{3} + \frac {2 A b^{2} d e^{3}}{3} + 2 A b c d^{2} e^{2} + \frac {2 A c^{2} d^{3} e}{3} + \frac {B a^{2} e^{4}}{6} + \frac {4 B a b d e^{3}}{3} + 2 B a c d^{2} e^{2} + B b^{2} d^{2} e^{2} + \frac {4 B b c d^{3} e}{3} + \frac {B c^{2} d^{4}}{6}\right ) + x^{5} \left (\frac {A a^{2} e^{4}}{5} + \frac {8 A a b d e^{3}}{5} + \frac {12 A a c d^{2} e^{2}}{5} + \frac {6 A b^{2} d^{2} e^{2}}{5} + \frac {8 A b c d^{3} e}{5} + \frac {A c^{2} d^{4}}{5} + \frac {4 B a^{2} d e^{3}}{5} + \frac {12 B a b d^{2} e^{2}}{5} + \frac {8 B a c d^{3} e}{5} + \frac {4 B b^{2} d^{3} e}{5} + \frac {2 B b c d^{4}}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 3 A a b d^{2} e^{2} + 2 A a c d^{3} e + A b^{2} d^{3} e + \frac {A b c d^{4}}{2} + \frac {3 B a^{2} d^{2} e^{2}}{2} + 2 B a b d^{3} e + \frac {B a c d^{4}}{2} + \frac {B b^{2} d^{4}}{4}\right ) + x^{3} \left (2 A a^{2} d^{2} e^{2} + \frac {8 A a b d^{3} e}{3} + \frac {2 A a c d^{4}}{3} + \frac {A b^{2} d^{4}}{3} + \frac {4 B a^{2} d^{3} e}{3} + \frac {2 B a b d^{4}}{3}\right ) + x^{2} \left (2 A a^{2} d^{3} e + A a b d^{4} + \frac {B a^{2} d^{4}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4*(c*x**2+b*x+a)**2,x)

[Out]

A*a**2*d**4*x + B*c**2*e**4*x**10/10 + x**9*(A*c**2*e**4/9 + 2*B*b*c*e**4/9 + 4*B*c**2*d*e**3/9) + x**8*(A*b*c
*e**4/4 + A*c**2*d*e**3/2 + B*a*c*e**4/4 + B*b**2*e**4/8 + B*b*c*d*e**3 + 3*B*c**2*d**2*e**2/4) + x**7*(2*A*a*
c*e**4/7 + A*b**2*e**4/7 + 8*A*b*c*d*e**3/7 + 6*A*c**2*d**2*e**2/7 + 2*B*a*b*e**4/7 + 8*B*a*c*d*e**3/7 + 4*B*b
**2*d*e**3/7 + 12*B*b*c*d**2*e**2/7 + 4*B*c**2*d**3*e/7) + x**6*(A*a*b*e**4/3 + 4*A*a*c*d*e**3/3 + 2*A*b**2*d*
e**3/3 + 2*A*b*c*d**2*e**2 + 2*A*c**2*d**3*e/3 + B*a**2*e**4/6 + 4*B*a*b*d*e**3/3 + 2*B*a*c*d**2*e**2 + B*b**2
*d**2*e**2 + 4*B*b*c*d**3*e/3 + B*c**2*d**4/6) + x**5*(A*a**2*e**4/5 + 8*A*a*b*d*e**3/5 + 12*A*a*c*d**2*e**2/5
 + 6*A*b**2*d**2*e**2/5 + 8*A*b*c*d**3*e/5 + A*c**2*d**4/5 + 4*B*a**2*d*e**3/5 + 12*B*a*b*d**2*e**2/5 + 8*B*a*
c*d**3*e/5 + 4*B*b**2*d**3*e/5 + 2*B*b*c*d**4/5) + x**4*(A*a**2*d*e**3 + 3*A*a*b*d**2*e**2 + 2*A*a*c*d**3*e +
A*b**2*d**3*e + A*b*c*d**4/2 + 3*B*a**2*d**2*e**2/2 + 2*B*a*b*d**3*e + B*a*c*d**4/2 + B*b**2*d**4/4) + x**3*(2
*A*a**2*d**2*e**2 + 8*A*a*b*d**3*e/3 + 2*A*a*c*d**4/3 + A*b**2*d**4/3 + 4*B*a**2*d**3*e/3 + 2*B*a*b*d**4/3) +
x**2*(2*A*a**2*d**3*e + A*a*b*d**4 + B*a**2*d**4/2)

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