3.21.79 \(\int (A+B x) (d+e x)^3 (a+b x+c x^2)^2 \, dx\)

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

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Rubi [A]  time = 0.49, 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)^7 \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 )}{7 e^6}-\frac {(d+e x)^6 \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 )}{6 e^6}+\frac {(d+e x)^5 \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 )}{5 e^6}-\frac {(d+e x)^4 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6}-\frac {c (d+e x)^8 (-A c e-2 b B e+5 B c d)}{8 e^6}+\frac {B c^2 (d+e x)^9}{9 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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Mathematica [A]  time = 0.19, size = 421, normalized size = 1.38 \begin {gather*} a^2 A d^3 x+\frac {1}{7} e x^7 \left (B \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+A c e (2 b e+3 c d)\right )+\frac {1}{6} x^6 \left (A e \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+B \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )\right )+\frac {1}{3} d x^3 \left (A \left (6 a b d e+a \left (3 a e^2+2 c d^2\right )+b^2 d^2\right )+a B d (3 a e+2 b d)\right )+\frac {1}{5} x^5 \left (2 b \left (a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )+A c d \left (6 a e^2+c d^2\right )+a B e \left (a e^2+6 c d^2\right )+3 b^2 d e (A e+B d)\right )+\frac {1}{4} x^4 \left (2 b d \left (3 a A e^2+3 a B d e+A c d^2\right )+a \left (a A e^3+3 a B d e^2+6 A c d^2 e+2 B c d^3\right )+b^2 d^2 (3 A e+B d)\right )+\frac {1}{2} a d^2 x^2 (3 a A e+a B d+2 A b d)+\frac {1}{8} c e^2 x^8 (A c e+2 b B e+3 B c d)+\frac {1}{9} B c^2 e^3 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^3*(a + b*x + c*x^2)^2,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^3 \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)^3*(a + b*x + c*x^2)^2,x]

[Out]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^3*(a + b*x + c*x^2)^2, x]

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fricas [A]  time = 0.36, size = 568, normalized size = 1.87 \begin {gather*} \frac {1}{9} x^{9} e^{3} c^{2} B + \frac {3}{8} x^{8} e^{2} d c^{2} B + \frac {1}{4} x^{8} e^{3} c b B + \frac {1}{8} x^{8} e^{3} c^{2} A + \frac {3}{7} x^{7} e d^{2} c^{2} B + \frac {6}{7} x^{7} e^{2} d c b B + \frac {1}{7} x^{7} e^{3} b^{2} B + \frac {2}{7} x^{7} e^{3} c a B + \frac {3}{7} x^{7} e^{2} d c^{2} A + \frac {2}{7} x^{7} e^{3} c b A + \frac {1}{6} x^{6} d^{3} c^{2} B + x^{6} e d^{2} c b B + \frac {1}{2} x^{6} e^{2} d b^{2} B + x^{6} e^{2} d c a B + \frac {1}{3} x^{6} e^{3} b a B + \frac {1}{2} x^{6} e d^{2} c^{2} A + x^{6} e^{2} d c b A + \frac {1}{6} x^{6} e^{3} b^{2} A + \frac {1}{3} x^{6} e^{3} c a A + \frac {2}{5} x^{5} d^{3} c b B + \frac {3}{5} x^{5} e d^{2} b^{2} B + \frac {6}{5} x^{5} e d^{2} c a B + \frac {6}{5} x^{5} e^{2} d b a B + \frac {1}{5} x^{5} e^{3} a^{2} B + \frac {1}{5} x^{5} d^{3} c^{2} A + \frac {6}{5} x^{5} e d^{2} c b A + \frac {3}{5} x^{5} e^{2} d b^{2} A + \frac {6}{5} x^{5} e^{2} d c a A + \frac {2}{5} x^{5} e^{3} b a A + \frac {1}{4} x^{4} d^{3} b^{2} B + \frac {1}{2} x^{4} d^{3} c a B + \frac {3}{2} x^{4} e d^{2} b a B + \frac {3}{4} x^{4} e^{2} d a^{2} B + \frac {1}{2} x^{4} d^{3} c b A + \frac {3}{4} x^{4} e d^{2} b^{2} A + \frac {3}{2} x^{4} e d^{2} c a A + \frac {3}{2} x^{4} e^{2} d b a A + \frac {1}{4} x^{4} e^{3} a^{2} A + \frac {2}{3} x^{3} d^{3} b a B + x^{3} e d^{2} a^{2} B + \frac {1}{3} x^{3} d^{3} b^{2} A + \frac {2}{3} x^{3} d^{3} c a A + 2 x^{3} e d^{2} b a A + x^{3} e^{2} d a^{2} A + \frac {1}{2} x^{2} d^{3} a^{2} B + x^{2} d^{3} b a A + \frac {3}{2} x^{2} e d^{2} a^{2} A + x d^{3} a^{2} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 556, normalized size = 1.83 \begin {gather*} \frac {1}{9} \, B c^{2} x^{9} e^{3} + \frac {3}{8} \, B c^{2} d x^{8} e^{2} + \frac {3}{7} \, B c^{2} d^{2} x^{7} e + \frac {1}{6} \, B c^{2} d^{3} x^{6} + \frac {1}{4} \, B b c x^{8} e^{3} + \frac {1}{8} \, A c^{2} x^{8} e^{3} + \frac {6}{7} \, B b c d x^{7} e^{2} + \frac {3}{7} \, A c^{2} d x^{7} e^{2} + B b c d^{2} x^{6} e + \frac {1}{2} \, A c^{2} d^{2} x^{6} e + \frac {2}{5} \, B b c d^{3} x^{5} + \frac {1}{5} \, A c^{2} d^{3} x^{5} + \frac {1}{7} \, B b^{2} x^{7} e^{3} + \frac {2}{7} \, B a c x^{7} e^{3} + \frac {2}{7} \, A b c x^{7} e^{3} + \frac {1}{2} \, B b^{2} d x^{6} e^{2} + B a c d x^{6} e^{2} + A b c d x^{6} e^{2} + \frac {3}{5} \, B b^{2} d^{2} x^{5} e + \frac {6}{5} \, B a c d^{2} x^{5} e + \frac {6}{5} \, A b c d^{2} x^{5} e + \frac {1}{4} \, B b^{2} d^{3} x^{4} + \frac {1}{2} \, B a c d^{3} x^{4} + \frac {1}{2} \, A b c d^{3} x^{4} + \frac {1}{3} \, B a b x^{6} e^{3} + \frac {1}{6} \, A b^{2} x^{6} e^{3} + \frac {1}{3} \, A a c x^{6} e^{3} + \frac {6}{5} \, B a b d x^{5} e^{2} + \frac {3}{5} \, A b^{2} d x^{5} e^{2} + \frac {6}{5} \, A a c d x^{5} e^{2} + \frac {3}{2} \, B a b d^{2} x^{4} e + \frac {3}{4} \, A b^{2} d^{2} x^{4} e + \frac {3}{2} \, A a c d^{2} x^{4} e + \frac {2}{3} \, B a b d^{3} x^{3} + \frac {1}{3} \, A b^{2} d^{3} x^{3} + \frac {2}{3} \, A a c d^{3} x^{3} + \frac {1}{5} \, B a^{2} x^{5} e^{3} + \frac {2}{5} \, A a b x^{5} e^{3} + \frac {3}{4} \, B a^{2} d x^{4} e^{2} + \frac {3}{2} \, A a b d x^{4} e^{2} + B a^{2} d^{2} x^{3} e + 2 \, A a b d^{2} x^{3} e + \frac {1}{2} \, B a^{2} d^{3} x^{2} + A a b d^{3} x^{2} + \frac {1}{4} \, A a^{2} x^{4} e^{3} + A a^{2} d x^{3} e^{2} + \frac {3}{2} \, A a^{2} d^{2} x^{2} e + A a^{2} d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 419, normalized size = 1.38 \begin {gather*} \frac {B \,c^{2} e^{3} x^{9}}{9}+\frac {\left (2 B b c \,e^{3}+\left (A \,e^{3}+3 B d \,e^{2}\right ) c^{2}\right ) x^{8}}{8}+A \,a^{2} d^{3} x +\frac {\left (\left (2 a c +b^{2}\right ) B \,e^{3}+2 \left (A \,e^{3}+3 B d \,e^{2}\right ) b c +\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) c^{2}\right ) x^{7}}{7}+\frac {\left (2 B a b \,e^{3}+2 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b c +\left (3 A \,d^{2} e +B \,d^{3}\right ) c^{2}+\left (A \,e^{3}+3 B d \,e^{2}\right ) \left (2 a c +b^{2}\right )\right ) x^{6}}{6}+\frac {\left (A \,c^{2} d^{3}+B \,a^{2} e^{3}+2 \left (A \,e^{3}+3 B d \,e^{2}\right ) a b +2 \left (3 A \,d^{2} e +B \,d^{3}\right ) b c +\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 A b c \,d^{3}+\left (A \,e^{3}+3 B d \,e^{2}\right ) a^{2}+2 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a b +\left (3 A \,d^{2} e +B \,d^{3}\right ) \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (2 a c +b^{2}\right ) A \,d^{3}+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a^{2}+2 \left (3 A \,d^{2} e +B \,d^{3}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 A a b \,d^{3}+\left (3 A \,d^{2} e +B \,d^{3}\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)^3*(c*x^2+b*x+a)^2,x)

[Out]

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

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maxima [A]  time = 0.47, size = 416, normalized size = 1.37 \begin {gather*} \frac {1}{9} \, B c^{2} e^{3} x^{9} + \frac {1}{8} \, {\left (3 \, B c^{2} d e^{2} + {\left (2 \, B b c + A c^{2}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, B c^{2} d^{2} e + 3 \, {\left (2 \, B b c + A c^{2}\right )} d e^{2} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{3}\right )} x^{7} + A a^{2} d^{3} x + \frac {1}{6} \, {\left (B c^{2} d^{3} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left ({\left (2 \, B b c + A c^{2}\right )} d^{3} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e + 3 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{2} e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} + 3 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e + 3 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} d e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{2} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d^{3}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.14, size = 450, normalized size = 1.48 \begin {gather*} x^5\,\left (\frac {B\,a^2\,e^3}{5}+\frac {6\,B\,a\,b\,d\,e^2}{5}+\frac {2\,A\,a\,b\,e^3}{5}+\frac {6\,B\,a\,c\,d^2\,e}{5}+\frac {6\,A\,a\,c\,d\,e^2}{5}+\frac {3\,B\,b^2\,d^2\,e}{5}+\frac {3\,A\,b^2\,d\,e^2}{5}+\frac {2\,B\,b\,c\,d^3}{5}+\frac {6\,A\,b\,c\,d^2\,e}{5}+\frac {A\,c^2\,d^3}{5}\right )+x^3\,\left (B\,a^2\,d^2\,e+A\,a^2\,d\,e^2+\frac {2\,B\,a\,b\,d^3}{3}+2\,A\,a\,b\,d^2\,e+\frac {2\,A\,c\,a\,d^3}{3}+\frac {A\,b^2\,d^3}{3}\right )+x^7\,\left (\frac {B\,b^2\,e^3}{7}+\frac {6\,B\,b\,c\,d\,e^2}{7}+\frac {2\,A\,b\,c\,e^3}{7}+\frac {3\,B\,c^2\,d^2\,e}{7}+\frac {3\,A\,c^2\,d\,e^2}{7}+\frac {2\,B\,a\,c\,e^3}{7}\right )+x^4\,\left (\frac {3\,B\,a^2\,d\,e^2}{4}+\frac {A\,a^2\,e^3}{4}+\frac {3\,B\,a\,b\,d^2\,e}{2}+\frac {3\,A\,a\,b\,d\,e^2}{2}+\frac {B\,c\,a\,d^3}{2}+\frac {3\,A\,c\,a\,d^2\,e}{2}+\frac {B\,b^2\,d^3}{4}+\frac {3\,A\,b^2\,d^2\,e}{4}+\frac {A\,c\,b\,d^3}{2}\right )+x^6\,\left (\frac {B\,b^2\,d\,e^2}{2}+\frac {A\,b^2\,e^3}{6}+B\,b\,c\,d^2\,e+A\,b\,c\,d\,e^2+\frac {B\,a\,b\,e^3}{3}+\frac {B\,c^2\,d^3}{6}+\frac {A\,c^2\,d^2\,e}{2}+B\,a\,c\,d\,e^2+\frac {A\,a\,c\,e^3}{3}\right )+A\,a^2\,d^3\,x+\frac {a\,d^2\,x^2\,\left (3\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {c\,e^2\,x^8\,\left (A\,c\,e+2\,B\,b\,e+3\,B\,c\,d\right )}{8}+\frac {B\,c^2\,e^3\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^3*(a + b*x + c*x^2)^2,x)

[Out]

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

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sympy [A]  time = 0.15, size = 583, normalized size = 1.92 \begin {gather*} A a^{2} d^{3} x + \frac {B c^{2} e^{3} x^{9}}{9} + x^{8} \left (\frac {A c^{2} e^{3}}{8} + \frac {B b c e^{3}}{4} + \frac {3 B c^{2} d e^{2}}{8}\right ) + x^{7} \left (\frac {2 A b c e^{3}}{7} + \frac {3 A c^{2} d e^{2}}{7} + \frac {2 B a c e^{3}}{7} + \frac {B b^{2} e^{3}}{7} + \frac {6 B b c d e^{2}}{7} + \frac {3 B c^{2} d^{2} e}{7}\right ) + x^{6} \left (\frac {A a c e^{3}}{3} + \frac {A b^{2} e^{3}}{6} + A b c d e^{2} + \frac {A c^{2} d^{2} e}{2} + \frac {B a b e^{3}}{3} + B a c d e^{2} + \frac {B b^{2} d e^{2}}{2} + B b c d^{2} e + \frac {B c^{2} d^{3}}{6}\right ) + x^{5} \left (\frac {2 A a b e^{3}}{5} + \frac {6 A a c d e^{2}}{5} + \frac {3 A b^{2} d e^{2}}{5} + \frac {6 A b c d^{2} e}{5} + \frac {A c^{2} d^{3}}{5} + \frac {B a^{2} e^{3}}{5} + \frac {6 B a b d e^{2}}{5} + \frac {6 B a c d^{2} e}{5} + \frac {3 B b^{2} d^{2} e}{5} + \frac {2 B b c d^{3}}{5}\right ) + x^{4} \left (\frac {A a^{2} e^{3}}{4} + \frac {3 A a b d e^{2}}{2} + \frac {3 A a c d^{2} e}{2} + \frac {3 A b^{2} d^{2} e}{4} + \frac {A b c d^{3}}{2} + \frac {3 B a^{2} d e^{2}}{4} + \frac {3 B a b d^{2} e}{2} + \frac {B a c d^{3}}{2} + \frac {B b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{2} d e^{2} + 2 A a b d^{2} e + \frac {2 A a c d^{3}}{3} + \frac {A b^{2} d^{3}}{3} + B a^{2} d^{2} e + \frac {2 B a b d^{3}}{3}\right ) + x^{2} \left (\frac {3 A a^{2} d^{2} e}{2} + A a b d^{3} + \frac {B a^{2} d^{3}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**3*(c*x**2+b*x+a)**2,x)

[Out]

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

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