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

Optimal. Leaf size=305 \[ \frac {1}{4} A b^3 d^3 x^4+\frac {1}{3} c e x^9 \left (A c e (b e+c d)+B \left (b^2 e^2+3 b c d e+c^2 d^2\right )\right )+\frac {1}{2} b d x^6 \left (b^2 e (A e+B d)+b c d (3 A e+B d)+A c^2 d^2\right )+\frac {1}{5} b^2 d^2 x^5 (3 A b e+3 A c d+b B d)+\frac {1}{7} x^7 \left (b^3 e^2 (A e+3 B d)+9 b^2 c d e (A e+B d)+3 b c^2 d^2 (3 A e+B d)+A c^3 d^3\right )+\frac {1}{8} x^8 \left (3 A c e \left (b^2 e^2+3 b c d e+c^2 d^2\right )+B \left (b^3 e^3+9 b^2 c d e^2+9 b c^2 d^2 e+c^3 d^3\right )\right )+\frac {1}{10} c^2 e^2 x^{10} (A c e+3 B (b e+c d))+\frac {1}{11} B c^3 e^3 x^{11} \]

________________________________________________________________________________________

Rubi [A]  time = 0.48, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} \frac {1}{3} c e x^9 \left (A c e (b e+c d)+B \left (b^2 e^2+3 b c d e+c^2 d^2\right )\right )+\frac {1}{8} x^8 \left (3 A c e \left (b^2 e^2+3 b c d e+c^2 d^2\right )+B \left (9 b^2 c d e^2+b^3 e^3+9 b c^2 d^2 e+c^3 d^3\right )\right )+\frac {1}{7} x^7 \left (9 b^2 c d e (A e+B d)+b^3 e^2 (A e+3 B d)+3 b c^2 d^2 (3 A e+B d)+A c^3 d^3\right )+\frac {1}{2} b d x^6 \left (b^2 e (A e+B d)+b c d (3 A e+B d)+A c^2 d^2\right )+\frac {1}{5} b^2 d^2 x^5 (3 A b e+3 A c d+b B d)+\frac {1}{4} A b^3 d^3 x^4+\frac {1}{10} c^2 e^2 x^{10} (A c e+3 B (b e+c d))+\frac {1}{11} B c^3 e^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 d^3 x^3+b^2 d^2 (b B d+3 A c d+3 A b e) x^4+3 b d \left (A c^2 d^2+b^2 e (B d+A e)+b c d (B d+3 A e)\right ) x^5+\left (A c^3 d^3+9 b^2 c d e (B d+A e)+b^3 e^2 (3 B d+A e)+3 b c^2 d^2 (B d+3 A e)\right ) x^6+\left (3 A c e \left (c^2 d^2+3 b c d e+b^2 e^2\right )+B \left (c^3 d^3+9 b c^2 d^2 e+9 b^2 c d e^2+b^3 e^3\right )\right ) x^7+3 c e \left (A c e (c d+b e)+B \left (c^2 d^2+3 b c d e+b^2 e^2\right )\right ) x^8+c^2 e^2 (A c e+3 B (c d+b e)) x^9+B c^3 e^3 x^{10}\right ) \, dx\\ &=\frac {1}{4} A b^3 d^3 x^4+\frac {1}{5} b^2 d^2 (b B d+3 A c d+3 A b e) x^5+\frac {1}{2} b d \left (A c^2 d^2+b^2 e (B d+A e)+b c d (B d+3 A e)\right ) x^6+\frac {1}{7} \left (A c^3 d^3+9 b^2 c d e (B d+A e)+b^3 e^2 (3 B d+A e)+3 b c^2 d^2 (B d+3 A e)\right ) x^7+\frac {1}{8} \left (3 A c e \left (c^2 d^2+3 b c d e+b^2 e^2\right )+B \left (c^3 d^3+9 b c^2 d^2 e+9 b^2 c d e^2+b^3 e^3\right )\right ) x^8+\frac {1}{3} c e \left (A c e (c d+b e)+B \left (c^2 d^2+3 b c d e+b^2 e^2\right )\right ) x^9+\frac {1}{10} c^2 e^2 (A c e+3 B (c d+b e)) x^{10}+\frac {1}{11} B c^3 e^3 x^{11}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 305, normalized size = 1.00 \begin {gather*} \frac {1}{4} A b^3 d^3 x^4+\frac {1}{3} c e x^9 \left (A c e (b e+c d)+B \left (b^2 e^2+3 b c d e+c^2 d^2\right )\right )+\frac {1}{2} b d x^6 \left (b^2 e (A e+B d)+b c d (3 A e+B d)+A c^2 d^2\right )+\frac {1}{5} b^2 d^2 x^5 (3 A b e+3 A c d+b B d)+\frac {1}{7} x^7 \left (b^3 e^2 (A e+3 B d)+9 b^2 c d e (A e+B d)+3 b c^2 d^2 (3 A e+B d)+A c^3 d^3\right )+\frac {1}{8} x^8 \left (3 A c e \left (b^2 e^2+3 b c d e+c^2 d^2\right )+B \left (b^3 e^3+9 b^2 c d e^2+9 b c^2 d^2 e+c^3 d^3\right )\right )+\frac {1}{10} c^2 e^2 x^{10} (A c e+3 B (b e+c d))+\frac {1}{11} B c^3 e^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.38, size = 416, normalized size = 1.36 \begin {gather*} \frac {1}{11} x^{11} e^{3} c^{3} B + \frac {3}{10} x^{10} e^{2} d c^{3} B + \frac {3}{10} x^{10} e^{3} c^{2} b B + \frac {1}{10} x^{10} e^{3} c^{3} A + \frac {1}{3} x^{9} e d^{2} c^{3} B + x^{9} e^{2} d c^{2} b B + \frac {1}{3} x^{9} e^{3} c b^{2} B + \frac {1}{3} x^{9} e^{2} d c^{3} A + \frac {1}{3} x^{9} e^{3} c^{2} b A + \frac {1}{8} x^{8} d^{3} c^{3} B + \frac {9}{8} x^{8} e d^{2} c^{2} b B + \frac {9}{8} x^{8} e^{2} d c b^{2} B + \frac {1}{8} x^{8} e^{3} b^{3} B + \frac {3}{8} x^{8} e d^{2} c^{3} A + \frac {9}{8} x^{8} e^{2} d c^{2} b A + \frac {3}{8} x^{8} e^{3} c b^{2} A + \frac {3}{7} x^{7} d^{3} c^{2} b B + \frac {9}{7} x^{7} e d^{2} c b^{2} B + \frac {3}{7} x^{7} e^{2} d b^{3} B + \frac {1}{7} x^{7} d^{3} c^{3} A + \frac {9}{7} x^{7} e d^{2} c^{2} b A + \frac {9}{7} x^{7} e^{2} d c b^{2} A + \frac {1}{7} x^{7} e^{3} b^{3} A + \frac {1}{2} x^{6} d^{3} c b^{2} B + \frac {1}{2} x^{6} e d^{2} b^{3} B + \frac {1}{2} x^{6} d^{3} c^{2} b A + \frac {3}{2} x^{6} e d^{2} c b^{2} A + \frac {1}{2} x^{6} e^{2} d b^{3} A + \frac {1}{5} x^{5} d^{3} b^{3} B + \frac {3}{5} x^{5} d^{3} c b^{2} A + \frac {3}{5} x^{5} e d^{2} b^{3} A + \frac {1}{4} x^{4} d^{3} b^{3} 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)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.20, size = 408, normalized size = 1.34 \begin {gather*} \frac {1}{11} \, B c^{3} x^{11} e^{3} + \frac {3}{10} \, B c^{3} d x^{10} e^{2} + \frac {1}{3} \, B c^{3} d^{2} x^{9} e + \frac {1}{8} \, B c^{3} d^{3} x^{8} + \frac {3}{10} \, B b c^{2} x^{10} e^{3} + \frac {1}{10} \, A c^{3} x^{10} e^{3} + B b c^{2} d x^{9} e^{2} + \frac {1}{3} \, A c^{3} d x^{9} e^{2} + \frac {9}{8} \, B b c^{2} d^{2} x^{8} e + \frac {3}{8} \, A c^{3} d^{2} x^{8} e + \frac {3}{7} \, B b c^{2} d^{3} x^{7} + \frac {1}{7} \, A c^{3} d^{3} x^{7} + \frac {1}{3} \, B b^{2} c x^{9} e^{3} + \frac {1}{3} \, A b c^{2} x^{9} e^{3} + \frac {9}{8} \, B b^{2} c d x^{8} e^{2} + \frac {9}{8} \, A b c^{2} d x^{8} e^{2} + \frac {9}{7} \, B b^{2} c d^{2} x^{7} e + \frac {9}{7} \, A b c^{2} d^{2} x^{7} e + \frac {1}{2} \, B b^{2} c d^{3} x^{6} + \frac {1}{2} \, A b c^{2} d^{3} x^{6} + \frac {1}{8} \, B b^{3} x^{8} e^{3} + \frac {3}{8} \, A b^{2} c x^{8} e^{3} + \frac {3}{7} \, B b^{3} d x^{7} e^{2} + \frac {9}{7} \, A b^{2} c d x^{7} e^{2} + \frac {1}{2} \, B b^{3} d^{2} x^{6} e + \frac {3}{2} \, A b^{2} c d^{2} x^{6} e + \frac {1}{5} \, B b^{3} d^{3} x^{5} + \frac {3}{5} \, A b^{2} c d^{3} x^{5} + \frac {1}{7} \, A b^{3} x^{7} e^{3} + \frac {1}{2} \, A b^{3} d x^{6} e^{2} + \frac {3}{5} \, A b^{3} d^{2} x^{5} e + \frac {1}{4} \, A b^{3} d^{3} x^{4} \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)^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 342, normalized size = 1.12 \begin {gather*} \frac {B \,c^{3} e^{3} x^{11}}{11}+\frac {A \,b^{3} d^{3} x^{4}}{4}+\frac {\left (3 B b \,c^{2} e^{3}+\left (A \,e^{3}+3 B d \,e^{2}\right ) c^{3}\right ) x^{10}}{10}+\frac {\left (3 B \,b^{2} c \,e^{3}+3 \left (A \,e^{3}+3 B d \,e^{2}\right ) b \,c^{2}+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) c^{3}\right ) x^{9}}{9}+\frac {\left (B \,b^{3} e^{3}+3 \left (A \,e^{3}+3 B d \,e^{2}\right ) b^{2} c +3 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b \,c^{2}+\left (3 A \,d^{2} e +B \,d^{3}\right ) c^{3}\right ) x^{8}}{8}+\frac {\left (A \,c^{3} d^{3}+\left (A \,e^{3}+3 B d \,e^{2}\right ) b^{3}+3 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b^{2} c +3 \left (3 A \,d^{2} e +B \,d^{3}\right ) b \,c^{2}\right ) x^{7}}{7}+\frac {\left (3 A b \,c^{2} d^{3}+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b^{3}+3 \left (3 A \,d^{2} e +B \,d^{3}\right ) b^{2} c \right ) x^{6}}{6}+\frac {\left (3 A \,b^{2} c \,d^{3}+\left (3 A \,d^{2} e +B \,d^{3}\right ) b^{3}\right ) x^{5}}{5} \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)^3,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.59, size = 329, normalized size = 1.08 \begin {gather*} \frac {1}{11} \, B c^{3} e^{3} x^{11} + \frac {1}{4} \, A b^{3} d^{3} x^{4} + \frac {1}{10} \, {\left (3 \, B c^{3} d e^{2} + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{3}\right )} x^{10} + \frac {1}{3} \, {\left (B c^{3} d^{2} e + {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{2} + {\left (B b^{2} c + A b c^{2}\right )} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{3} + 3 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e + 9 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{2} + {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (A b^{3} e^{3} + {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} + 9 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e + 3 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{2}\right )} x^{7} + \frac {1}{2} \, {\left (A b^{3} d e^{2} + {\left (B b^{2} c + A b c^{2}\right )} d^{3} + {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e\right )} x^{6} + \frac {1}{5} \, {\left (3 \, A b^{3} d^{2} e + {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3}\right )} x^{5} \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)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.42, size = 340, normalized size = 1.11 \begin {gather*} x^6\,\left (\frac {B\,b^3\,d^2\,e}{2}+\frac {A\,b^3\,d\,e^2}{2}+\frac {B\,b^2\,c\,d^3}{2}+\frac {3\,A\,b^2\,c\,d^2\,e}{2}+\frac {A\,b\,c^2\,d^3}{2}\right )+x^9\,\left (\frac {B\,b^2\,c\,e^3}{3}+B\,b\,c^2\,d\,e^2+\frac {A\,b\,c^2\,e^3}{3}+\frac {B\,c^3\,d^2\,e}{3}+\frac {A\,c^3\,d\,e^2}{3}\right )+x^7\,\left (\frac {3\,B\,b^3\,d\,e^2}{7}+\frac {A\,b^3\,e^3}{7}+\frac {9\,B\,b^2\,c\,d^2\,e}{7}+\frac {9\,A\,b^2\,c\,d\,e^2}{7}+\frac {3\,B\,b\,c^2\,d^3}{7}+\frac {9\,A\,b\,c^2\,d^2\,e}{7}+\frac {A\,c^3\,d^3}{7}\right )+x^8\,\left (\frac {B\,b^3\,e^3}{8}+\frac {9\,B\,b^2\,c\,d\,e^2}{8}+\frac {3\,A\,b^2\,c\,e^3}{8}+\frac {9\,B\,b\,c^2\,d^2\,e}{8}+\frac {9\,A\,b\,c^2\,d\,e^2}{8}+\frac {B\,c^3\,d^3}{8}+\frac {3\,A\,c^3\,d^2\,e}{8}\right )+\frac {b^2\,d^2\,x^5\,\left (3\,A\,b\,e+3\,A\,c\,d+B\,b\,d\right )}{5}+\frac {c^2\,e^2\,x^{10}\,\left (A\,c\,e+3\,B\,b\,e+3\,B\,c\,d\right )}{10}+\frac {A\,b^3\,d^3\,x^4}{4}+\frac {B\,c^3\,e^3\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 430, normalized size = 1.41 \begin {gather*} \frac {A b^{3} d^{3} x^{4}}{4} + \frac {B c^{3} e^{3} x^{11}}{11} + x^{10} \left (\frac {A c^{3} e^{3}}{10} + \frac {3 B b c^{2} e^{3}}{10} + \frac {3 B c^{3} d e^{2}}{10}\right ) + x^{9} \left (\frac {A b c^{2} e^{3}}{3} + \frac {A c^{3} d e^{2}}{3} + \frac {B b^{2} c e^{3}}{3} + B b c^{2} d e^{2} + \frac {B c^{3} d^{2} e}{3}\right ) + x^{8} \left (\frac {3 A b^{2} c e^{3}}{8} + \frac {9 A b c^{2} d e^{2}}{8} + \frac {3 A c^{3} d^{2} e}{8} + \frac {B b^{3} e^{3}}{8} + \frac {9 B b^{2} c d e^{2}}{8} + \frac {9 B b c^{2} d^{2} e}{8} + \frac {B c^{3} d^{3}}{8}\right ) + x^{7} \left (\frac {A b^{3} e^{3}}{7} + \frac {9 A b^{2} c d e^{2}}{7} + \frac {9 A b c^{2} d^{2} e}{7} + \frac {A c^{3} d^{3}}{7} + \frac {3 B b^{3} d e^{2}}{7} + \frac {9 B b^{2} c d^{2} e}{7} + \frac {3 B b c^{2} d^{3}}{7}\right ) + x^{6} \left (\frac {A b^{3} d e^{2}}{2} + \frac {3 A b^{2} c d^{2} e}{2} + \frac {A b c^{2} d^{3}}{2} + \frac {B b^{3} d^{2} e}{2} + \frac {B b^{2} c d^{3}}{2}\right ) + x^{5} \left (\frac {3 A b^{3} d^{2} e}{5} + \frac {3 A b^{2} c d^{3}}{5} + \frac {B b^{3} d^{3}}{5}\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)**3,x)

[Out]

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

________________________________________________________________________________________