3.1514 \(\int (b+2 c x) (d+e x)^4 (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=411 \[ \frac{(d+e x)^8 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{8 e^8}+\frac{3 c^2 (d+e x)^{10} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{10 e^8}-\frac{5 c (d+e x)^9 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{9 e^8}-\frac{3 (d+e x)^7 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}+\frac{(d+e x)^6 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{6 e^8}-\frac{(d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac{7 c^3 (d+e x)^{11} (2 c d-b e)}{11 e^8}+\frac{c^4 (d+e x)^{12}}{6 e^8} \]

[Out]

-((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5)/(5*e^8) + ((c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^
2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^6)/(6*e^8) - (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*
e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^7)/(7*e^8) + ((70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^
3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^8)/(8*e^8) - (5*c*(2*c*d -
b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^9)/(9*e^8) + (3*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e
*(7*b*d - a*e))*(d + e*x)^10)/(10*e^8) - (7*c^3*(2*c*d - b*e)*(d + e*x)^11)/(11*e^8) + (c^4*(d + e*x)^12)/(6*e
^8)

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Rubi [A]  time = 0.723237, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ \frac{(d+e x)^8 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{8 e^8}+\frac{3 c^2 (d+e x)^{10} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{10 e^8}-\frac{5 c (d+e x)^9 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{9 e^8}-\frac{3 (d+e x)^7 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}+\frac{(d+e x)^6 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{6 e^8}-\frac{(d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac{7 c^3 (d+e x)^{11} (2 c d-b e)}{11 e^8}+\frac{c^4 (d+e x)^{12}}{6 e^8} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^4*(a + b*x + c*x^2)^3,x]

[Out]

-((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5)/(5*e^8) + ((c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^
2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^6)/(6*e^8) - (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*
e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^7)/(7*e^8) + ((70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^
3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^8)/(8*e^8) - (5*c*(2*c*d -
b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^9)/(9*e^8) + (3*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e
*(7*b*d - a*e))*(d + e*x)^10)/(10*e^8) - (7*c^3*(2*c*d - b*e)*(d + e*x)^11)/(11*e^8) + (c^4*(d + e*x)^12)/(6*e
^8)

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 (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{e^7}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^5}{e^7}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)^6}{e^7}+\frac{\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^7}{e^7}+\frac{5 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^8}{e^7}+\frac{3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^9}{e^7}-\frac{7 c^3 (2 c d-b e) (d+e x)^{10}}{e^7}+\frac{2 c^4 (d+e x)^{11}}{e^7}\right ) \, dx\\ &=-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 e^8}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^6}{6 e^8}-\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^7}{7 e^8}+\frac{\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^8}{8 e^8}-\frac{5 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^9}{9 e^8}+\frac{3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{10}}{10 e^8}-\frac{7 c^3 (2 c d-b e) (d+e x)^{11}}{11 e^8}+\frac{c^4 (d+e x)^{12}}{6 e^8}\\ \end{align*}

Mathematica [A]  time = 0.256042, size = 735, normalized size = 1.79 \[ \frac{1}{8} x^8 \left (6 c^2 e^2 \left (a^2 e^2+10 a b d e+9 b^2 d^2\right )+4 b^2 c e^3 (3 a e+5 b d)+4 c^3 d^2 e (9 a e+7 b d)+b^4 e^4+2 c^4 d^4\right )+\frac{1}{7} x^7 \left (b c \left (9 a^2 e^4+90 a c d^2 e^2+7 c^2 d^4\right )+3 b^3 \left (a e^4+10 c d^2 e^2\right )+12 b^2 c d e \left (4 a e^2+3 c d^2\right )+24 a c^2 d e \left (a e^2+c d^2\right )+4 b^4 d e^3\right )+\frac{1}{6} x^6 \left (3 b^2 \left (a^2 e^4+24 a c d^2 e^2+3 c^2 d^4\right )+2 a c \left (a^2 e^4+18 a c d^2 e^2+3 c^2 d^4\right )+4 b^3 \left (3 a d e^3+5 c d^3 e\right )+12 a b c d e \left (3 a e^2+5 c d^2\right )+6 b^4 d^2 e^2\right )+\frac{1}{5} x^5 \left (a b \left (a^2 e^4+54 a c d^2 e^2+15 c^2 d^4\right )+8 a^2 c d e \left (a e^2+3 c d^2\right )+b^3 \left (18 a d^2 e^2+5 c d^4\right )+12 a b^2 d e \left (a e^2+4 c d^2\right )+4 b^4 d^3 e\right )+\frac{1}{4} d x^4 \left (4 a^2 b e \left (a e^2+9 c d^2\right )+6 a^2 c d \left (2 a e^2+c d^2\right )+6 a b^2 d \left (3 a e^2+2 c d^2\right )+12 a b^3 d^2 e+b^4 d^3\right )+\frac{1}{3} a d^2 x^3 \left (8 a^2 c d e+12 a b^2 d e+3 a b \left (2 a e^2+3 c d^2\right )+3 b^3 d^2\right )+\frac{1}{2} a^2 d^3 x^2 \left (4 a b e+2 a c d+3 b^2 d\right )+a^3 b d^4 x+\frac{1}{10} c^2 e^2 x^{10} \left (2 c e (3 a e+14 b d)+9 b^2 e^2+12 c^2 d^2\right )+\frac{1}{9} c e x^9 \left (6 c^2 d e (4 a e+7 b d)+3 b c e^2 (5 a e+12 b d)+5 b^3 e^3+8 c^3 d^3\right )+\frac{1}{11} c^3 e^3 x^{11} (7 b e+8 c d)+\frac{1}{6} c^4 e^4 x^{12} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(d + e*x)^4*(a + b*x + c*x^2)^3,x]

[Out]

a^3*b*d^4*x + (a^2*d^3*(3*b^2*d + 2*a*c*d + 4*a*b*e)*x^2)/2 + (a*d^2*(3*b^3*d^2 + 12*a*b^2*d*e + 8*a^2*c*d*e +
3*a*b*(3*c*d^2 + 2*a*e^2))*x^3)/3 + (d*(b^4*d^3 + 12*a*b^3*d^2*e + 4*a^2*b*e*(9*c*d^2 + a*e^2) + 6*a^2*c*d*(c
*d^2 + 2*a*e^2) + 6*a*b^2*d*(2*c*d^2 + 3*a*e^2))*x^4)/4 + ((4*b^4*d^3*e + 8*a^2*c*d*e*(3*c*d^2 + a*e^2) + 12*a
*b^2*d*e*(4*c*d^2 + a*e^2) + b^3*(5*c*d^4 + 18*a*d^2*e^2) + a*b*(15*c^2*d^4 + 54*a*c*d^2*e^2 + a^2*e^4))*x^5)/
5 + ((6*b^4*d^2*e^2 + 12*a*b*c*d*e*(5*c*d^2 + 3*a*e^2) + 4*b^3*(5*c*d^3*e + 3*a*d*e^3) + 2*a*c*(3*c^2*d^4 + 18
*a*c*d^2*e^2 + a^2*e^4) + 3*b^2*(3*c^2*d^4 + 24*a*c*d^2*e^2 + a^2*e^4))*x^6)/6 + ((4*b^4*d*e^3 + 24*a*c^2*d*e*
(c*d^2 + a*e^2) + 12*b^2*c*d*e*(3*c*d^2 + 4*a*e^2) + 3*b^3*(10*c*d^2*e^2 + a*e^4) + b*c*(7*c^2*d^4 + 90*a*c*d^
2*e^2 + 9*a^2*e^4))*x^7)/7 + ((2*c^4*d^4 + b^4*e^4 + 4*b^2*c*e^3*(5*b*d + 3*a*e) + 4*c^3*d^2*e*(7*b*d + 9*a*e)
+ 6*c^2*e^2*(9*b^2*d^2 + 10*a*b*d*e + a^2*e^2))*x^8)/8 + (c*e*(8*c^3*d^3 + 5*b^3*e^3 + 6*c^2*d*e*(7*b*d + 4*a
*e) + 3*b*c*e^2*(12*b*d + 5*a*e))*x^9)/9 + (c^2*e^2*(12*c^2*d^2 + 9*b^2*e^2 + 2*c*e*(14*b*d + 3*a*e))*x^10)/10
+ (c^3*e^3*(8*c*d + 7*b*e)*x^11)/11 + (c^4*e^4*x^12)/6

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Maple [B]  time = 0.003, size = 1052, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a)^3,x)

[Out]

1/6*c^4*e^4*x^12+1/11*((b*e^4+8*c*d*e^3)*c^3+6*c^3*e^4*b)*x^11+1/10*((4*b*d*e^3+12*c*d^2*e^2)*c^3+3*(b*e^4+8*c
*d*e^3)*b*c^2+2*c*e^4*(a*c^2+2*b^2*c+c*(2*a*c+b^2)))*x^10+1/9*((6*b*d^2*e^2+8*c*d^3*e)*c^3+3*(4*b*d*e^3+12*c*d
^2*e^2)*b*c^2+(b*e^4+8*c*d*e^3)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+2*c*e^4*(b*(2*a*c+b^2)+4*a*b*c))*x^9+1/8*((4*b*d
^3*e+2*c*d^4)*c^3+3*(6*b*d^2*e^2+8*c*d^3*e)*b*c^2+(4*b*d*e^3+12*c*d^2*e^2)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(b*e^
4+8*c*d*e^3)*(b*(2*a*c+b^2)+4*a*b*c)+2*c*e^4*(a*(2*a*c+b^2)+2*b^2*a+c*a^2))*x^8+1/7*(b*d^4*c^3+3*(4*b*d^3*e+2*
c*d^4)*b*c^2+(6*b*d^2*e^2+8*c*d^3*e)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(4*b*d*e^3+12*c*d^2*e^2)*(b*(2*a*c+b^2)+4*a
*b*c)+(b*e^4+8*c*d*e^3)*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+6*c*e^4*a^2*b)*x^7+1/6*(3*b^2*d^4*c^2+(4*b*d^3*e+2*c*d^4
)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(6*b*d^2*e^2+8*c*d^3*e)*(b*(2*a*c+b^2)+4*a*b*c)+(4*b*d*e^3+12*c*d^2*e^2)*(a*(2
*a*c+b^2)+2*b^2*a+c*a^2)+3*(b*e^4+8*c*d*e^3)*a^2*b+2*c*e^4*a^3)*x^6+1/5*(b*d^4*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(
4*b*d^3*e+2*c*d^4)*(b*(2*a*c+b^2)+4*a*b*c)+(6*b*d^2*e^2+8*c*d^3*e)*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*(4*b*d*e^3+
12*c*d^2*e^2)*a^2*b+(b*e^4+8*c*d*e^3)*a^3)*x^5+1/4*(b*d^4*(b*(2*a*c+b^2)+4*a*b*c)+(4*b*d^3*e+2*c*d^4)*(a*(2*a*
c+b^2)+2*b^2*a+c*a^2)+3*(6*b*d^2*e^2+8*c*d^3*e)*a^2*b+(4*b*d*e^3+12*c*d^2*e^2)*a^3)*x^4+1/3*(b*d^4*(a*(2*a*c+b
^2)+2*b^2*a+c*a^2)+3*(4*b*d^3*e+2*c*d^4)*a^2*b+(6*b*d^2*e^2+8*c*d^3*e)*a^3)*x^3+1/2*(3*b^2*d^4*a^2+(4*b*d^3*e+
2*c*d^4)*a^3)*x^2+b*d^4*a^3*x

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Maxima [A]  time = 1.02793, size = 984, normalized size = 2.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

1/6*c^4*e^4*x^12 + 1/11*(8*c^4*d*e^3 + 7*b*c^3*e^4)*x^11 + 1/10*(12*c^4*d^2*e^2 + 28*b*c^3*d*e^3 + 3*(3*b^2*c^
2 + 2*a*c^3)*e^4)*x^10 + 1/9*(8*c^4*d^3*e + 42*b*c^3*d^2*e^2 + 12*(3*b^2*c^2 + 2*a*c^3)*d*e^3 + 5*(b^3*c + 3*a
*b*c^2)*e^4)*x^9 + a^3*b*d^4*x + 1/8*(2*c^4*d^4 + 28*b*c^3*d^3*e + 18*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 + 20*(b^3*
c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*x^8 + 1/7*(7*b*c^3*d^4 + 12*(3*b^2*c^2 + 2*a*c^3)*d
^3*e + 30*(b^3*c + 3*a*b*c^2)*d^2*e^2 + 4*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^3 + 3*(a*b^3 + 3*a^2*b*c)*e^4)*x^
7 + 1/6*(3*(3*b^2*c^2 + 2*a*c^3)*d^4 + 20*(b^3*c + 3*a*b*c^2)*d^3*e + 6*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^2
+ 12*(a*b^3 + 3*a^2*b*c)*d*e^3 + (3*a^2*b^2 + 2*a^3*c)*e^4)*x^6 + 1/5*(a^3*b*e^4 + 5*(b^3*c + 3*a*b*c^2)*d^4
+ 4*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e + 18*(a*b^3 + 3*a^2*b*c)*d^2*e^2 + 4*(3*a^2*b^2 + 2*a^3*c)*d*e^3)*x^5
+ 1/4*(4*a^3*b*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4 + 12*(a*b^3 + 3*a^2*b*c)*d^3*e + 6*(3*a^2*b^2 + 2*a
^3*c)*d^2*e^2)*x^4 + 1/3*(6*a^3*b*d^2*e^2 + 3*(a*b^3 + 3*a^2*b*c)*d^4 + 4*(3*a^2*b^2 + 2*a^3*c)*d^3*e)*x^3 + 1
/2*(4*a^3*b*d^3*e + (3*a^2*b^2 + 2*a^3*c)*d^4)*x^2

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Fricas [B]  time = 1.27078, size = 2045, normalized size = 4.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/6*x^12*e^4*c^4 + 8/11*x^11*e^3*d*c^4 + 7/11*x^11*e^4*c^3*b + 6/5*x^10*e^2*d^2*c^4 + 14/5*x^10*e^3*d*c^3*b +
9/10*x^10*e^4*c^2*b^2 + 3/5*x^10*e^4*c^3*a + 8/9*x^9*e*d^3*c^4 + 14/3*x^9*e^2*d^2*c^3*b + 4*x^9*e^3*d*c^2*b^2
+ 5/9*x^9*e^4*c*b^3 + 8/3*x^9*e^3*d*c^3*a + 5/3*x^9*e^4*c^2*b*a + 1/4*x^8*d^4*c^4 + 7/2*x^8*e*d^3*c^3*b + 27/4
*x^8*e^2*d^2*c^2*b^2 + 5/2*x^8*e^3*d*c*b^3 + 1/8*x^8*e^4*b^4 + 9/2*x^8*e^2*d^2*c^3*a + 15/2*x^8*e^3*d*c^2*b*a
+ 3/2*x^8*e^4*c*b^2*a + 3/4*x^8*e^4*c^2*a^2 + x^7*d^4*c^3*b + 36/7*x^7*e*d^3*c^2*b^2 + 30/7*x^7*e^2*d^2*c*b^3
+ 4/7*x^7*e^3*d*b^4 + 24/7*x^7*e*d^3*c^3*a + 90/7*x^7*e^2*d^2*c^2*b*a + 48/7*x^7*e^3*d*c*b^2*a + 3/7*x^7*e^4*b
^3*a + 24/7*x^7*e^3*d*c^2*a^2 + 9/7*x^7*e^4*c*b*a^2 + 3/2*x^6*d^4*c^2*b^2 + 10/3*x^6*e*d^3*c*b^3 + x^6*e^2*d^2
*b^4 + x^6*d^4*c^3*a + 10*x^6*e*d^3*c^2*b*a + 12*x^6*e^2*d^2*c*b^2*a + 2*x^6*e^3*d*b^3*a + 6*x^6*e^2*d^2*c^2*a
^2 + 6*x^6*e^3*d*c*b*a^2 + 1/2*x^6*e^4*b^2*a^2 + 1/3*x^6*e^4*c*a^3 + x^5*d^4*c*b^3 + 4/5*x^5*e*d^3*b^4 + 3*x^5
*d^4*c^2*b*a + 48/5*x^5*e*d^3*c*b^2*a + 18/5*x^5*e^2*d^2*b^3*a + 24/5*x^5*e*d^3*c^2*a^2 + 54/5*x^5*e^2*d^2*c*b
*a^2 + 12/5*x^5*e^3*d*b^2*a^2 + 8/5*x^5*e^3*d*c*a^3 + 1/5*x^5*e^4*b*a^3 + 1/4*x^4*d^4*b^4 + 3*x^4*d^4*c*b^2*a
+ 3*x^4*e*d^3*b^3*a + 3/2*x^4*d^4*c^2*a^2 + 9*x^4*e*d^3*c*b*a^2 + 9/2*x^4*e^2*d^2*b^2*a^2 + 3*x^4*e^2*d^2*c*a^
3 + x^4*e^3*d*b*a^3 + x^3*d^4*b^3*a + 3*x^3*d^4*c*b*a^2 + 4*x^3*e*d^3*b^2*a^2 + 8/3*x^3*e*d^3*c*a^3 + 2*x^3*e^
2*d^2*b*a^3 + 3/2*x^2*d^4*b^2*a^2 + x^2*d^4*c*a^3 + 2*x^2*e*d^3*b*a^3 + x*d^4*b*a^3

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Sympy [B]  time = 0.188385, size = 935, normalized size = 2.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**4*(c*x**2+b*x+a)**3,x)

[Out]

a**3*b*d**4*x + c**4*e**4*x**12/6 + x**11*(7*b*c**3*e**4/11 + 8*c**4*d*e**3/11) + x**10*(3*a*c**3*e**4/5 + 9*b
**2*c**2*e**4/10 + 14*b*c**3*d*e**3/5 + 6*c**4*d**2*e**2/5) + x**9*(5*a*b*c**2*e**4/3 + 8*a*c**3*d*e**3/3 + 5*
b**3*c*e**4/9 + 4*b**2*c**2*d*e**3 + 14*b*c**3*d**2*e**2/3 + 8*c**4*d**3*e/9) + x**8*(3*a**2*c**2*e**4/4 + 3*a
*b**2*c*e**4/2 + 15*a*b*c**2*d*e**3/2 + 9*a*c**3*d**2*e**2/2 + b**4*e**4/8 + 5*b**3*c*d*e**3/2 + 27*b**2*c**2*
d**2*e**2/4 + 7*b*c**3*d**3*e/2 + c**4*d**4/4) + x**7*(9*a**2*b*c*e**4/7 + 24*a**2*c**2*d*e**3/7 + 3*a*b**3*e*
*4/7 + 48*a*b**2*c*d*e**3/7 + 90*a*b*c**2*d**2*e**2/7 + 24*a*c**3*d**3*e/7 + 4*b**4*d*e**3/7 + 30*b**3*c*d**2*
e**2/7 + 36*b**2*c**2*d**3*e/7 + b*c**3*d**4) + x**6*(a**3*c*e**4/3 + a**2*b**2*e**4/2 + 6*a**2*b*c*d*e**3 + 6
*a**2*c**2*d**2*e**2 + 2*a*b**3*d*e**3 + 12*a*b**2*c*d**2*e**2 + 10*a*b*c**2*d**3*e + a*c**3*d**4 + b**4*d**2*
e**2 + 10*b**3*c*d**3*e/3 + 3*b**2*c**2*d**4/2) + x**5*(a**3*b*e**4/5 + 8*a**3*c*d*e**3/5 + 12*a**2*b**2*d*e**
3/5 + 54*a**2*b*c*d**2*e**2/5 + 24*a**2*c**2*d**3*e/5 + 18*a*b**3*d**2*e**2/5 + 48*a*b**2*c*d**3*e/5 + 3*a*b*c
**2*d**4 + 4*b**4*d**3*e/5 + b**3*c*d**4) + x**4*(a**3*b*d*e**3 + 3*a**3*c*d**2*e**2 + 9*a**2*b**2*d**2*e**2/2
+ 9*a**2*b*c*d**3*e + 3*a**2*c**2*d**4/2 + 3*a*b**3*d**3*e + 3*a*b**2*c*d**4 + b**4*d**4/4) + x**3*(2*a**3*b*
d**2*e**2 + 8*a**3*c*d**3*e/3 + 4*a**2*b**2*d**3*e + 3*a**2*b*c*d**4 + a*b**3*d**4) + x**2*(2*a**3*b*d**3*e +
a**3*c*d**4 + 3*a**2*b**2*d**4/2)

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Giac [B]  time = 1.22545, size = 1226, normalized size = 2.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

1/6*c^4*x^12*e^4 + 8/11*c^4*d*x^11*e^3 + 6/5*c^4*d^2*x^10*e^2 + 8/9*c^4*d^3*x^9*e + 1/4*c^4*d^4*x^8 + 7/11*b*c
^3*x^11*e^4 + 14/5*b*c^3*d*x^10*e^3 + 14/3*b*c^3*d^2*x^9*e^2 + 7/2*b*c^3*d^3*x^8*e + b*c^3*d^4*x^7 + 9/10*b^2*
c^2*x^10*e^4 + 3/5*a*c^3*x^10*e^4 + 4*b^2*c^2*d*x^9*e^3 + 8/3*a*c^3*d*x^9*e^3 + 27/4*b^2*c^2*d^2*x^8*e^2 + 9/2
*a*c^3*d^2*x^8*e^2 + 36/7*b^2*c^2*d^3*x^7*e + 24/7*a*c^3*d^3*x^7*e + 3/2*b^2*c^2*d^4*x^6 + a*c^3*d^4*x^6 + 5/9
*b^3*c*x^9*e^4 + 5/3*a*b*c^2*x^9*e^4 + 5/2*b^3*c*d*x^8*e^3 + 15/2*a*b*c^2*d*x^8*e^3 + 30/7*b^3*c*d^2*x^7*e^2 +
90/7*a*b*c^2*d^2*x^7*e^2 + 10/3*b^3*c*d^3*x^6*e + 10*a*b*c^2*d^3*x^6*e + b^3*c*d^4*x^5 + 3*a*b*c^2*d^4*x^5 +
1/8*b^4*x^8*e^4 + 3/2*a*b^2*c*x^8*e^4 + 3/4*a^2*c^2*x^8*e^4 + 4/7*b^4*d*x^7*e^3 + 48/7*a*b^2*c*d*x^7*e^3 + 24/
7*a^2*c^2*d*x^7*e^3 + b^4*d^2*x^6*e^2 + 12*a*b^2*c*d^2*x^6*e^2 + 6*a^2*c^2*d^2*x^6*e^2 + 4/5*b^4*d^3*x^5*e + 4
8/5*a*b^2*c*d^3*x^5*e + 24/5*a^2*c^2*d^3*x^5*e + 1/4*b^4*d^4*x^4 + 3*a*b^2*c*d^4*x^4 + 3/2*a^2*c^2*d^4*x^4 + 3
/7*a*b^3*x^7*e^4 + 9/7*a^2*b*c*x^7*e^4 + 2*a*b^3*d*x^6*e^3 + 6*a^2*b*c*d*x^6*e^3 + 18/5*a*b^3*d^2*x^5*e^2 + 54
/5*a^2*b*c*d^2*x^5*e^2 + 3*a*b^3*d^3*x^4*e + 9*a^2*b*c*d^3*x^4*e + a*b^3*d^4*x^3 + 3*a^2*b*c*d^4*x^3 + 1/2*a^2
*b^2*x^6*e^4 + 1/3*a^3*c*x^6*e^4 + 12/5*a^2*b^2*d*x^5*e^3 + 8/5*a^3*c*d*x^5*e^3 + 9/2*a^2*b^2*d^2*x^4*e^2 + 3*
a^3*c*d^2*x^4*e^2 + 4*a^2*b^2*d^3*x^3*e + 8/3*a^3*c*d^3*x^3*e + 3/2*a^2*b^2*d^4*x^2 + a^3*c*d^4*x^2 + 1/5*a^3*
b*x^5*e^4 + a^3*b*d*x^4*e^3 + 2*a^3*b*d^2*x^3*e^2 + 2*a^3*b*d^3*x^2*e + a^3*b*d^4*x