3.2337 \(\int (A+B x) (d+e x) (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=310 \[ a^3 A d x+\frac {1}{4} x^4 \left (A \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+3 a B \left (a b e+a c d+b^2 d\right )\right )+\frac {1}{2} a^2 x^2 (a A e+a B d+3 A b d)+\frac {1}{7} c x^7 \left (c (3 a B e+A c d)+3 b c (A e+B d)+3 b^2 B e\right )+\frac {1}{3} a x^3 \left (3 A \left (a b e+a c d+b^2 d\right )+a B (a e+3 b d)\right )+\frac {1}{6} x^6 \left (3 b c (2 a B e+A c d)+3 a c^2 (A e+B d)+3 b^2 c (A e+B d)+b^3 B e\right )+\frac {1}{5} x^5 \left (3 b^2 (a B e+A c d)+6 a b c (A e+B d)+3 a c (a B e+A c d)+b^3 (A e+B d)\right )+\frac {1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac {1}{9} B c^3 e x^9 \]

[Out]

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

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Rubi [A]  time = 0.64, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {771} \[ \frac {1}{4} x^4 \left (A \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+3 a B \left (a b e+a c d+b^2 d\right )\right )+\frac {1}{2} a^2 x^2 (a A e+a B d+3 A b d)+a^3 A d x+\frac {1}{6} x^6 \left (3 b c (2 a B e+A c d)+3 a c^2 (A e+B d)+3 b^2 c (A e+B d)+b^3 B e\right )+\frac {1}{7} c x^7 \left (c (3 a B e+A c d)+3 b c (A e+B d)+3 b^2 B e\right )+\frac {1}{5} x^5 \left (3 b^2 (a B e+A c d)+6 a b c (A e+B d)+3 a c (a B e+A c d)+b^3 (A e+B d)\right )+\frac {1}{3} a x^3 \left (3 A \left (a b e+a c d+b^2 d\right )+a B (a e+3 b d)\right )+\frac {1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac {1}{9} B c^3 e x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 A d+a^2 (3 A b d+a B d+a A e) x+a \left (a B (3 b d+a e)+3 A \left (b^2 d+a c d+a b e\right )\right ) x^2+\left (3 a B \left (b^2 d+a c d+a b e\right )+A \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right )\right ) x^3+\left (b^3 (B d+A e)+6 a b c (B d+A e)+3 b^2 (A c d+a B e)+3 a c (A c d+a B e)\right ) x^4+\left (b^3 B e+3 b^2 c (B d+A e)+3 a c^2 (B d+A e)+3 b c (A c d+2 a B e)\right ) x^5+c \left (3 b^2 B e+3 b c (B d+A e)+c (A c d+3 a B e)\right ) x^6+c^2 (B c d+3 b B e+A c e) x^7+B c^3 e x^8\right ) \, dx\\ &=a^3 A d x+\frac {1}{2} a^2 (3 A b d+a B d+a A e) x^2+\frac {1}{3} a \left (a B (3 b d+a e)+3 A \left (b^2 d+a c d+a b e\right )\right ) x^3+\frac {1}{4} \left (3 a B \left (b^2 d+a c d+a b e\right )+A \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right )\right ) x^4+\frac {1}{5} \left (b^3 (B d+A e)+6 a b c (B d+A e)+3 b^2 (A c d+a B e)+3 a c (A c d+a B e)\right ) x^5+\frac {1}{6} \left (b^3 B e+3 b^2 c (B d+A e)+3 a c^2 (B d+A e)+3 b c (A c d+2 a B e)\right ) x^6+\frac {1}{7} c \left (3 b^2 B e+3 b c (B d+A e)+c (A c d+3 a B e)\right ) x^7+\frac {1}{8} c^2 (B c d+3 b B e+A c e) x^8+\frac {1}{9} B c^3 e x^9\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 310, normalized size = 1.00 \[ a^3 A d x+\frac {1}{4} x^4 \left (A \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+3 a B \left (a b e+a c d+b^2 d\right )\right )+\frac {1}{2} a^2 x^2 (a A e+a B d+3 A b d)+\frac {1}{7} c x^7 \left (c (3 a B e+A c d)+3 b c (A e+B d)+3 b^2 B e\right )+\frac {1}{3} a x^3 \left (3 A \left (a b e+a c d+b^2 d\right )+a B (a e+3 b d)\right )+\frac {1}{6} x^6 \left (3 b c (2 a B e+A c d)+3 a c^2 (A e+B d)+3 b^2 c (A e+B d)+b^3 B e\right )+\frac {1}{5} x^5 \left (3 b^2 (a B e+A c d)+6 a b c (A e+B d)+3 a c (a B e+A c d)+b^3 (A e+B d)\right )+\frac {1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac {1}{9} B c^3 e x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.65, size = 417, normalized size = 1.35 \[ \frac {1}{9} x^{9} e c^{3} B + \frac {1}{8} x^{8} d c^{3} B + \frac {3}{8} x^{8} e c^{2} b B + \frac {1}{8} x^{8} e c^{3} A + \frac {3}{7} x^{7} d c^{2} b B + \frac {3}{7} x^{7} e c b^{2} B + \frac {3}{7} x^{7} e c^{2} a B + \frac {1}{7} x^{7} d c^{3} A + \frac {3}{7} x^{7} e c^{2} b A + \frac {1}{2} x^{6} d c b^{2} B + \frac {1}{6} x^{6} e b^{3} B + \frac {1}{2} x^{6} d c^{2} a B + x^{6} e c b a B + \frac {1}{2} x^{6} d c^{2} b A + \frac {1}{2} x^{6} e c b^{2} A + \frac {1}{2} x^{6} e c^{2} a A + \frac {1}{5} x^{5} d b^{3} B + \frac {6}{5} x^{5} d c b a B + \frac {3}{5} x^{5} e b^{2} a B + \frac {3}{5} x^{5} e c a^{2} B + \frac {3}{5} x^{5} d c b^{2} A + \frac {1}{5} x^{5} e b^{3} A + \frac {3}{5} x^{5} d c^{2} a A + \frac {6}{5} x^{5} e c b a A + \frac {3}{4} x^{4} d b^{2} a B + \frac {3}{4} x^{4} d c a^{2} B + \frac {3}{4} x^{4} e b a^{2} B + \frac {1}{4} x^{4} d b^{3} A + \frac {3}{2} x^{4} d c b a A + \frac {3}{4} x^{4} e b^{2} a A + \frac {3}{4} x^{4} e c a^{2} A + x^{3} d b a^{2} B + \frac {1}{3} x^{3} e a^{3} B + x^{3} d b^{2} a A + x^{3} d c a^{2} A + x^{3} e b a^{2} A + \frac {1}{2} x^{2} d a^{3} B + \frac {3}{2} x^{2} d b a^{2} A + \frac {1}{2} x^{2} e a^{3} A + x d a^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 437, normalized size = 1.41 \[ \frac {1}{9} \, B c^{3} x^{9} e + \frac {1}{8} \, B c^{3} d x^{8} + \frac {3}{8} \, B b c^{2} x^{8} e + \frac {1}{8} \, A c^{3} x^{8} e + \frac {3}{7} \, B b c^{2} d x^{7} + \frac {1}{7} \, A c^{3} d x^{7} + \frac {3}{7} \, B b^{2} c x^{7} e + \frac {3}{7} \, B a c^{2} x^{7} e + \frac {3}{7} \, A b c^{2} x^{7} e + \frac {1}{2} \, B b^{2} c d x^{6} + \frac {1}{2} \, B a c^{2} d x^{6} + \frac {1}{2} \, A b c^{2} d x^{6} + \frac {1}{6} \, B b^{3} x^{6} e + B a b c x^{6} e + \frac {1}{2} \, A b^{2} c x^{6} e + \frac {1}{2} \, A a c^{2} x^{6} e + \frac {1}{5} \, B b^{3} d x^{5} + \frac {6}{5} \, B a b c d x^{5} + \frac {3}{5} \, A b^{2} c d x^{5} + \frac {3}{5} \, A a c^{2} d x^{5} + \frac {3}{5} \, B a b^{2} x^{5} e + \frac {1}{5} \, A b^{3} x^{5} e + \frac {3}{5} \, B a^{2} c x^{5} e + \frac {6}{5} \, A a b c x^{5} e + \frac {3}{4} \, B a b^{2} d x^{4} + \frac {1}{4} \, A b^{3} d x^{4} + \frac {3}{4} \, B a^{2} c d x^{4} + \frac {3}{2} \, A a b c d x^{4} + \frac {3}{4} \, B a^{2} b x^{4} e + \frac {3}{4} \, A a b^{2} x^{4} e + \frac {3}{4} \, A a^{2} c x^{4} e + B a^{2} b d x^{3} + A a b^{2} d x^{3} + A a^{2} c d x^{3} + \frac {1}{3} \, B a^{3} x^{3} e + A a^{2} b x^{3} e + \frac {1}{2} \, B a^{3} d x^{2} + \frac {3}{2} \, A a^{2} b d x^{2} + \frac {1}{2} \, A a^{3} x^{2} e + A a^{3} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 375, normalized size = 1.21 \[ \frac {B \,c^{3} e \,x^{9}}{9}+\frac {\left (3 B b \,c^{2} e +\left (A e +B d \right ) c^{3}\right ) x^{8}}{8}+\frac {\left (A \,c^{3} d +3 \left (A e +B d \right ) b \,c^{2}+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) B e \right ) x^{7}}{7}+A \,a^{3} d x +\frac {\left (3 A b \,c^{2} d +\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) B e +\left (A e +B d \right ) \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right )\right ) x^{6}}{6}+\frac {\left (\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) A d +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) B e +\left (A e +B d \right ) \left (4 a b c +\left (2 a c +b^{2}\right ) b \right )\right ) x^{5}}{5}+\frac {\left (3 B \,a^{2} b e +\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) A d +\left (A e +B d \right ) \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right )\right ) x^{4}}{4}+\frac {\left (B \,a^{3} e +3 \left (A e +B d \right ) a^{2} b +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) A d \right ) x^{3}}{3}+\frac {\left (3 A \,a^{2} b d +\left (A e +B d \right ) a^{3}\right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.60, size = 334, normalized size = 1.08 \[ \frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, {\left (B c^{3} d + {\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e\right )} x^{6} + A a^{3} d x + \frac {1}{5} \, {\left ({\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.13, size = 338, normalized size = 1.09 \[ x^7\,\left (\frac {A\,c^3\,d}{7}+\frac {3\,A\,b\,c^2\,e}{7}+\frac {3\,B\,a\,c^2\,e}{7}+\frac {3\,B\,b\,c^2\,d}{7}+\frac {3\,B\,b^2\,c\,e}{7}\right )+x^5\,\left (\frac {A\,b^3\,e}{5}+\frac {B\,b^3\,d}{5}+\frac {3\,A\,a\,c^2\,d}{5}+\frac {3\,A\,b^2\,c\,d}{5}+\frac {3\,B\,a\,b^2\,e}{5}+\frac {3\,B\,a^2\,c\,e}{5}+\frac {6\,A\,a\,b\,c\,e}{5}+\frac {6\,B\,a\,b\,c\,d}{5}\right )+x^2\,\left (\frac {A\,a^3\,e}{2}+\frac {B\,a^3\,d}{2}+\frac {3\,A\,a^2\,b\,d}{2}\right )+x^8\,\left (\frac {A\,c^3\,e}{8}+\frac {B\,c^3\,d}{8}+\frac {3\,B\,b\,c^2\,e}{8}\right )+x^4\,\left (\frac {A\,b^3\,d}{4}+\frac {3\,A\,a\,b^2\,e}{4}+\frac {3\,B\,a\,b^2\,d}{4}+\frac {3\,A\,a^2\,c\,e}{4}+\frac {3\,B\,a^2\,b\,e}{4}+\frac {3\,B\,a^2\,c\,d}{4}+\frac {3\,A\,a\,b\,c\,d}{2}\right )+x^6\,\left (\frac {B\,b^3\,e}{6}+\frac {A\,a\,c^2\,e}{2}+\frac {A\,b\,c^2\,d}{2}+\frac {B\,a\,c^2\,d}{2}+\frac {A\,b^2\,c\,e}{2}+\frac {B\,b^2\,c\,d}{2}+B\,a\,b\,c\,e\right )+x^3\,\left (\frac {B\,a^3\,e}{3}+A\,a\,b^2\,d+A\,a^2\,b\,e+A\,a^2\,c\,d+B\,a^2\,b\,d\right )+A\,a^3\,d\,x+\frac {B\,c^3\,e\,x^9}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.13, size = 435, normalized size = 1.40 \[ A a^{3} d x + \frac {B c^{3} e x^{9}}{9} + x^{8} \left (\frac {A c^{3} e}{8} + \frac {3 B b c^{2} e}{8} + \frac {B c^{3} d}{8}\right ) + x^{7} \left (\frac {3 A b c^{2} e}{7} + \frac {A c^{3} d}{7} + \frac {3 B a c^{2} e}{7} + \frac {3 B b^{2} c e}{7} + \frac {3 B b c^{2} d}{7}\right ) + x^{6} \left (\frac {A a c^{2} e}{2} + \frac {A b^{2} c e}{2} + \frac {A b c^{2} d}{2} + B a b c e + \frac {B a c^{2} d}{2} + \frac {B b^{3} e}{6} + \frac {B b^{2} c d}{2}\right ) + x^{5} \left (\frac {6 A a b c e}{5} + \frac {3 A a c^{2} d}{5} + \frac {A b^{3} e}{5} + \frac {3 A b^{2} c d}{5} + \frac {3 B a^{2} c e}{5} + \frac {3 B a b^{2} e}{5} + \frac {6 B a b c d}{5} + \frac {B b^{3} d}{5}\right ) + x^{4} \left (\frac {3 A a^{2} c e}{4} + \frac {3 A a b^{2} e}{4} + \frac {3 A a b c d}{2} + \frac {A b^{3} d}{4} + \frac {3 B a^{2} b e}{4} + \frac {3 B a^{2} c d}{4} + \frac {3 B a b^{2} d}{4}\right ) + x^{3} \left (A a^{2} b e + A a^{2} c d + A a b^{2} d + \frac {B a^{3} e}{3} + B a^{2} b d\right ) + x^{2} \left (\frac {A a^{3} e}{2} + \frac {3 A a^{2} b d}{2} + \frac {B a^{3} d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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