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]

-1/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^5/e^8+1/6*(a*e^2-b*d*e+c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-
a*e+7*b*d))*(e*x+d)^6/e^8-3/7*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^
7/e^8+1/8*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20*c^3*d^2*e*(-3*a*e+7*b*d)+6*c^2*e^2*(a^2*e^2-10*a*b
*d*e+15*b^2*d^2))*(e*x+d)^8/e^8-5/9*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^9/e^8+3/10*c
^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^10/e^8-7/11*c^3*(-b*e+2*c*d)*(e*x+d)^11/e^8+1/6*c^4*(e*x+
d)^12/e^8

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Rubi [A]  time = 0.72, antiderivative size = 411, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.24, size = 735, normalized size = 1.79 \[ a^3 b d^4 x+\frac {1}{2} a^2 d^3 x^2 \left (4 a b e+2 a c d+3 b^2 d\right )+\frac {1}{8} x^8 \left (6 c^2 e^2 \left (a^2 e^2+10 a b d e+9 b^2 d^2\right )+4 b^2 c e^3 (3 a e+5 b d)+4 c^3 d^2 e (9 a e+7 b d)+b^4 e^4+2 c^4 d^4\right )+\frac {1}{3} a d^2 x^3 \left (8 a^2 c d e+12 a b^2 d e+3 a b \left (2 a e^2+3 c d^2\right )+3 b^3 d^2\right )+\frac {1}{7} x^7 \left (b c \left (9 a^2 e^4+90 a c d^2 e^2+7 c^2 d^4\right )+3 b^3 \left (a e^4+10 c d^2 e^2\right )+12 b^2 c d e \left (4 a e^2+3 c d^2\right )+24 a c^2 d e \left (a e^2+c d^2\right )+4 b^4 d e^3\right )+\frac {1}{5} x^5 \left (a b \left (a^2 e^4+54 a c d^2 e^2+15 c^2 d^4\right )+8 a^2 c d e \left (a e^2+3 c d^2\right )+b^3 \left (18 a d^2 e^2+5 c d^4\right )+12 a b^2 d e \left (a e^2+4 c d^2\right )+4 b^4 d^3 e\right )+\frac {1}{6} x^6 \left (3 b^2 \left (a^2 e^4+24 a c d^2 e^2+3 c^2 d^4\right )+2 a c \left (a^2 e^4+18 a c d^2 e^2+3 c^2 d^4\right )+4 b^3 \left (3 a d e^3+5 c d^3 e\right )+12 a b c d e \left (3 a e^2+5 c d^2\right )+6 b^4 d^2 e^2\right )+\frac {1}{4} d x^4 \left (4 a^2 b e \left (a e^2+9 c d^2\right )+6 a^2 c d \left (2 a e^2+c d^2\right )+12 a b^3 d^2 e+6 a b^2 d \left (3 a e^2+2 c d^2\right )+b^4 d^3\right )+\frac {1}{9} c e x^9 \left (6 c^2 d e (4 a e+7 b d)+3 b c e^2 (5 a e+12 b d)+5 b^3 e^3+8 c^3 d^3\right )+\frac {1}{10} c^2 e^2 x^{10} \left (2 c e (3 a e+14 b d)+9 b^2 e^2+12 c^2 d^2\right )+\frac {1}{11} c^3 e^3 x^{11} (7 b e+8 c d)+\frac {1}{6} c^4 e^4 x^{12} \]

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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fricas [B]  time = 0.72, size = 936, normalized size = 2.28 \[ \frac {1}{6} x^{12} e^{4} c^{4} + \frac {8}{11} x^{11} e^{3} d c^{4} + \frac {7}{11} x^{11} e^{4} c^{3} b + \frac {6}{5} x^{10} e^{2} d^{2} c^{4} + \frac {14}{5} x^{10} e^{3} d c^{3} b + \frac {9}{10} x^{10} e^{4} c^{2} b^{2} + \frac {3}{5} x^{10} e^{4} c^{3} a + \frac {8}{9} x^{9} e d^{3} c^{4} + \frac {14}{3} x^{9} e^{2} d^{2} c^{3} b + 4 x^{9} e^{3} d c^{2} b^{2} + \frac {5}{9} x^{9} e^{4} c b^{3} + \frac {8}{3} x^{9} e^{3} d c^{3} a + \frac {5}{3} x^{9} e^{4} c^{2} b a + \frac {1}{4} x^{8} d^{4} c^{4} + \frac {7}{2} x^{8} e d^{3} c^{3} b + \frac {27}{4} x^{8} e^{2} d^{2} c^{2} b^{2} + \frac {5}{2} x^{8} e^{3} d c b^{3} + \frac {1}{8} x^{8} e^{4} b^{4} + \frac {9}{2} x^{8} e^{2} d^{2} c^{3} a + \frac {15}{2} x^{8} e^{3} d c^{2} b a + \frac {3}{2} x^{8} e^{4} c b^{2} a + \frac {3}{4} x^{8} e^{4} c^{2} a^{2} + x^{7} d^{4} c^{3} b + \frac {36}{7} x^{7} e d^{3} c^{2} b^{2} + \frac {30}{7} x^{7} e^{2} d^{2} c b^{3} + \frac {4}{7} x^{7} e^{3} d b^{4} + \frac {24}{7} x^{7} e d^{3} c^{3} a + \frac {90}{7} x^{7} e^{2} d^{2} c^{2} b a + \frac {48}{7} x^{7} e^{3} d c b^{2} a + \frac {3}{7} x^{7} e^{4} b^{3} a + \frac {24}{7} x^{7} e^{3} d c^{2} a^{2} + \frac {9}{7} x^{7} e^{4} c b a^{2} + \frac {3}{2} x^{6} d^{4} c^{2} b^{2} + \frac {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} + \frac {1}{2} x^{6} e^{4} b^{2} a^{2} + \frac {1}{3} x^{6} e^{4} c a^{3} + x^{5} d^{4} c b^{3} + \frac {4}{5} x^{5} e d^{3} b^{4} + 3 x^{5} d^{4} c^{2} b a + \frac {48}{5} x^{5} e d^{3} c b^{2} a + \frac {18}{5} x^{5} e^{2} d^{2} b^{3} a + \frac {24}{5} x^{5} e d^{3} c^{2} a^{2} + \frac {54}{5} x^{5} e^{2} d^{2} c b a^{2} + \frac {12}{5} x^{5} e^{3} d b^{2} a^{2} + \frac {8}{5} x^{5} e^{3} d c a^{3} + \frac {1}{5} x^{5} e^{4} b a^{3} + \frac {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 + \frac {3}{2} x^{4} d^{4} c^{2} a^{2} + 9 x^{4} e d^{3} c b a^{2} + \frac {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} + \frac {8}{3} x^{3} e d^{3} c a^{3} + 2 x^{3} e^{2} d^{2} b a^{3} + \frac {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} \]

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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giac [B]  time = 0.17, size = 908, normalized size = 2.21 \[ \frac {1}{6} \, c^{4} x^{12} e^{4} + \frac {8}{11} \, c^{4} d x^{11} e^{3} + \frac {6}{5} \, c^{4} d^{2} x^{10} e^{2} + \frac {8}{9} \, c^{4} d^{3} x^{9} e + \frac {1}{4} \, c^{4} d^{4} x^{8} + \frac {7}{11} \, b c^{3} x^{11} e^{4} + \frac {14}{5} \, b c^{3} d x^{10} e^{3} + \frac {14}{3} \, b c^{3} d^{2} x^{9} e^{2} + \frac {7}{2} \, b c^{3} d^{3} x^{8} e + b c^{3} d^{4} x^{7} + \frac {9}{10} \, b^{2} c^{2} x^{10} e^{4} + \frac {3}{5} \, a c^{3} x^{10} e^{4} + 4 \, b^{2} c^{2} d x^{9} e^{3} + \frac {8}{3} \, a c^{3} d x^{9} e^{3} + \frac {27}{4} \, b^{2} c^{2} d^{2} x^{8} e^{2} + \frac {9}{2} \, a c^{3} d^{2} x^{8} e^{2} + \frac {36}{7} \, b^{2} c^{2} d^{3} x^{7} e + \frac {24}{7} \, a c^{3} d^{3} x^{7} e + \frac {3}{2} \, b^{2} c^{2} d^{4} x^{6} + a c^{3} d^{4} x^{6} + \frac {5}{9} \, b^{3} c x^{9} e^{4} + \frac {5}{3} \, a b c^{2} x^{9} e^{4} + \frac {5}{2} \, b^{3} c d x^{8} e^{3} + \frac {15}{2} \, a b c^{2} d x^{8} e^{3} + \frac {30}{7} \, b^{3} c d^{2} x^{7} e^{2} + \frac {90}{7} \, a b c^{2} d^{2} x^{7} e^{2} + \frac {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} + \frac {1}{8} \, b^{4} x^{8} e^{4} + \frac {3}{2} \, a b^{2} c x^{8} e^{4} + \frac {3}{4} \, a^{2} c^{2} x^{8} e^{4} + \frac {4}{7} \, b^{4} d x^{7} e^{3} + \frac {48}{7} \, a b^{2} c d x^{7} e^{3} + \frac {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} + \frac {4}{5} \, b^{4} d^{3} x^{5} e + \frac {48}{5} \, a b^{2} c d^{3} x^{5} e + \frac {24}{5} \, a^{2} c^{2} d^{3} x^{5} e + \frac {1}{4} \, b^{4} d^{4} x^{4} + 3 \, a b^{2} c d^{4} x^{4} + \frac {3}{2} \, a^{2} c^{2} d^{4} x^{4} + \frac {3}{7} \, a b^{3} x^{7} e^{4} + \frac {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} + \frac {18}{5} \, a b^{3} d^{2} x^{5} e^{2} + \frac {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} + \frac {1}{2} \, a^{2} b^{2} x^{6} e^{4} + \frac {1}{3} \, a^{3} c x^{6} e^{4} + \frac {12}{5} \, a^{2} b^{2} d x^{5} e^{3} + \frac {8}{5} \, a^{3} c d x^{5} e^{3} + \frac {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 + \frac {8}{3} \, a^{3} c d^{3} x^{3} e + \frac {3}{2} \, a^{2} b^{2} d^{4} x^{2} + a^{3} c d^{4} x^{2} + \frac {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 \]

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

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maple [B]  time = 0.04, size = 1052, normalized size = 2.56 \[ \frac {c^{4} e^{4} x^{12}}{6}+\frac {\left (6 b \,c^{3} e^{4}+\left (b \,e^{4}+8 c d \,e^{3}\right ) c^{3}\right ) x^{11}}{11}+\frac {\left (2 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) c \,e^{4}+3 \left (b \,e^{4}+8 c d \,e^{3}\right ) b \,c^{2}+\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) c^{3}\right ) x^{10}}{10}+a^{3} b \,d^{4} x +\frac {\left (2 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) c \,e^{4}+3 \left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) b \,c^{2}+\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) c^{3}+\left (b \,e^{4}+8 c d \,e^{3}\right ) \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right )\right ) x^{9}}{9}+\frac {\left (2 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) c \,e^{4}+3 \left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) b \,c^{2}+\left (4 b \,d^{3} e +2 c \,d^{4}\right ) c^{3}+\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right )+\left (b \,e^{4}+8 c d \,e^{3}\right ) \left (4 a b c +\left (2 a c +b^{2}\right ) b \right )\right ) x^{8}}{8}+\frac {\left (6 a^{2} b c \,e^{4}+b \,c^{3} d^{4}+3 \left (4 b \,d^{3} e +2 c \,d^{4}\right ) b \,c^{2}+\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right )+\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) \left (4 a b c +\left (2 a c +b^{2}\right ) b \right )+\left (b \,e^{4}+8 c d \,e^{3}\right ) \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right )\right ) x^{7}}{7}+\frac {\left (2 a^{3} c \,e^{4}+3 b^{2} c^{2} d^{4}+3 \left (b \,e^{4}+8 c d \,e^{3}\right ) a^{2} b +\left (4 b \,d^{3} e +2 c \,d^{4}\right ) \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right )+\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) \left (4 a b c +\left (2 a c +b^{2}\right ) b \right )+\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right )\right ) x^{6}}{6}+\frac {\left (\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b \,d^{4}+\left (b \,e^{4}+8 c d \,e^{3}\right ) a^{3}+3 \left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) a^{2} b +\left (4 b \,d^{3} e +2 c \,d^{4}\right ) \left (4 a b c +\left (2 a c +b^{2}\right ) b \right )+\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right )\right ) x^{5}}{5}+\frac {\left (\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b \,d^{4}+\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) a^{3}+3 \left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) a^{2} b +\left (4 b \,d^{3} e +2 c \,d^{4}\right ) \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right )\right ) x^{4}}{4}+\frac {\left (\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b \,d^{4}+\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) a^{3}+3 \left (4 b \,d^{3} e +2 c \,d^{4}\right ) a^{2} b \right ) x^{3}}{3}+\frac {\left (3 a^{2} b^{2} d^{4}+\left (4 b \,d^{3} e +2 c \,d^{4}\right ) a^{3}\right ) x^{2}}{2} \]

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+(2*a*c+b^2)*c))*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+(2*a*c+b^2)*c)+2*c*e^4*(4*a*b*c+(2*a*c+b^2)*b))*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+(2*a*c+b^2)*c)+(b*e^
4+8*c*d*e^3)*(4*a*b*c+(2*a*c+b^2)*b)+2*c*e^4*(a*(2*a*c+b^2)+2*a*b^2+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+(2*a*c+b^2)*c)+(4*b*d*e^3+12*c*d^2*e^2)*(4*a*b*c+(2*a*c+b^
2)*b)+(b*e^4+8*c*d*e^3)*(a*(2*a*c+b^2)+2*a*b^2+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+(2*a*c+b^2)*c)+(6*b*d^2*e^2+8*c*d^3*e)*(4*a*b*c+(2*a*c+b^2)*b)+(4*b*d*e^3+12*c*d^2*e^2)*(a*(2
*a*c+b^2)+2*a*b^2+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+(2*a*c+b^2)*c)+(
4*b*d^3*e+2*c*d^4)*(4*a*b*c+(2*a*c+b^2)*b)+(6*b*d^2*e^2+8*c*d^3*e)*(a*(2*a*c+b^2)+2*a*b^2+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*(4*a*b*c+(2*a*c+b^2)*b)+(4*b*d^3*e+2*c*d^4)*(a*(2*a*
c+b^2)+2*a*b^2+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*a*b^2+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 = 0.55, size = 729, normalized size = 1.77 \[ \frac {1}{6} \, c^{4} e^{4} x^{12} + \frac {1}{11} \, {\left (8 \, c^{4} d e^{3} + 7 \, b c^{3} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (12 \, c^{4} d^{2} e^{2} + 28 \, b c^{3} d e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (8 \, c^{4} d^{3} e + 42 \, b c^{3} d^{2} e^{2} + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{3} + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{4}\right )} x^{9} + a^{3} b d^{4} x + \frac {1}{8} \, {\left (2 \, c^{4} d^{4} + 28 \, b c^{3} d^{3} e + 18 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} + 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (7 \, b c^{3} d^{4} + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e + 30 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{2} + 4 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{3} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} + 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e + 6 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{2} + 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{3} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (a^{3} b e^{4} + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} + 4 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e + 18 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{2} + 4 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a^{3} b d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} + 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e + 6 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{3} b d^{2} e^{2} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{4} + 4 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} e + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{4}\right )} x^{2} \]

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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mupad [B]  time = 0.24, size = 768, normalized size = 1.87 \[ x^3\,\left (2\,a^3\,b\,d^2\,e^2+\frac {8\,c\,a^3\,d^3\,e}{3}+4\,a^2\,b^2\,d^3\,e+3\,c\,a^2\,b\,d^4+a\,b^3\,d^4\right )+x^5\,\left (\frac {a^3\,b\,e^4}{5}+\frac {8\,a^3\,c\,d\,e^3}{5}+\frac {12\,a^2\,b^2\,d\,e^3}{5}+\frac {54\,a^2\,b\,c\,d^2\,e^2}{5}+\frac {24\,a^2\,c^2\,d^3\,e}{5}+\frac {18\,a\,b^3\,d^2\,e^2}{5}+\frac {48\,a\,b^2\,c\,d^3\,e}{5}+3\,a\,b\,c^2\,d^4+\frac {4\,b^4\,d^3\,e}{5}+b^3\,c\,d^4\right )+x^7\,\left (\frac {9\,a^2\,b\,c\,e^4}{7}+\frac {24\,a^2\,c^2\,d\,e^3}{7}+\frac {3\,a\,b^3\,e^4}{7}+\frac {48\,a\,b^2\,c\,d\,e^3}{7}+\frac {90\,a\,b\,c^2\,d^2\,e^2}{7}+\frac {24\,a\,c^3\,d^3\,e}{7}+\frac {4\,b^4\,d\,e^3}{7}+\frac {30\,b^3\,c\,d^2\,e^2}{7}+\frac {36\,b^2\,c^2\,d^3\,e}{7}+b\,c^3\,d^4\right )+x^6\,\left (\frac {a^3\,c\,e^4}{3}+\frac {a^2\,b^2\,e^4}{2}+6\,a^2\,b\,c\,d\,e^3+6\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+12\,a\,b^2\,c\,d^2\,e^2+10\,a\,b\,c^2\,d^3\,e+a\,c^3\,d^4+b^4\,d^2\,e^2+\frac {10\,b^3\,c\,d^3\,e}{3}+\frac {3\,b^2\,c^2\,d^4}{2}\right )+x^8\,\left (\frac {3\,a^2\,c^2\,e^4}{4}+\frac {3\,a\,b^2\,c\,e^4}{2}+\frac {15\,a\,b\,c^2\,d\,e^3}{2}+\frac {9\,a\,c^3\,d^2\,e^2}{2}+\frac {b^4\,e^4}{8}+\frac {5\,b^3\,c\,d\,e^3}{2}+\frac {27\,b^2\,c^2\,d^2\,e^2}{4}+\frac {7\,b\,c^3\,d^3\,e}{2}+\frac {c^4\,d^4}{4}\right )+x^9\,\left (\frac {5\,b^3\,c\,e^4}{9}+4\,b^2\,c^2\,d\,e^3+\frac {14\,b\,c^3\,d^2\,e^2}{3}+\frac {5\,a\,b\,c^2\,e^4}{3}+\frac {8\,c^4\,d^3\,e}{9}+\frac {8\,a\,c^3\,d\,e^3}{3}\right )+x^4\,\left (a^3\,b\,d\,e^3+3\,a^3\,c\,d^2\,e^2+\frac {9\,a^2\,b^2\,d^2\,e^2}{2}+9\,a^2\,b\,c\,d^3\,e+\frac {3\,a^2\,c^2\,d^4}{2}+3\,a\,b^3\,d^3\,e+3\,a\,b^2\,c\,d^4+\frac {b^4\,d^4}{4}\right )+\frac {c^4\,e^4\,x^{12}}{6}+\frac {c^3\,e^3\,x^{11}\,\left (7\,b\,e+8\,c\,d\right )}{11}+\frac {a^2\,d^3\,x^2\,\left (3\,d\,b^2+4\,a\,e\,b+2\,a\,c\,d\right )}{2}+\frac {c^2\,e^2\,x^{10}\,\left (9\,b^2\,e^2+28\,b\,c\,d\,e+12\,c^2\,d^2+6\,a\,c\,e^2\right )}{10}+a^3\,b\,d^4\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*(a*b^3*d^4 + 4*a^2*b^2*d^3*e + 2*a^3*b*d^2*e^2 + 3*a^2*b*c*d^4 + (8*a^3*c*d^3*e)/3) + x^5*((a^3*b*e^4)/5 +
b^3*c*d^4 + (4*b^4*d^3*e)/5 + (18*a*b^3*d^2*e^2)/5 + (12*a^2*b^2*d*e^3)/5 + (24*a^2*c^2*d^3*e)/5 + 3*a*b*c^2*
d^4 + (8*a^3*c*d*e^3)/5 + (48*a*b^2*c*d^3*e)/5 + (54*a^2*b*c*d^2*e^2)/5) + x^7*((3*a*b^3*e^4)/7 + b*c^3*d^4 +
(4*b^4*d*e^3)/7 + (24*a^2*c^2*d*e^3)/7 + (36*b^2*c^2*d^3*e)/7 + (30*b^3*c*d^2*e^2)/7 + (9*a^2*b*c*e^4)/7 + (24
*a*c^3*d^3*e)/7 + (48*a*b^2*c*d*e^3)/7 + (90*a*b*c^2*d^2*e^2)/7) + x^6*(a*c^3*d^4 + (a^3*c*e^4)/3 + (a^2*b^2*e
^4)/2 + (3*b^2*c^2*d^4)/2 + b^4*d^2*e^2 + 6*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + (10*b^3*c*d^3*e)/3 + 10*a*b*c^2*
d^3*e + 6*a^2*b*c*d*e^3 + 12*a*b^2*c*d^2*e^2) + x^8*((b^4*e^4)/8 + (c^4*d^4)/4 + (3*a^2*c^2*e^4)/4 + (9*a*c^3*
d^2*e^2)/2 + (27*b^2*c^2*d^2*e^2)/4 + (3*a*b^2*c*e^4)/2 + (7*b*c^3*d^3*e)/2 + (5*b^3*c*d*e^3)/2 + (15*a*b*c^2*
d*e^3)/2) + x^9*((5*b^3*c*e^4)/9 + (8*c^4*d^3*e)/9 + (14*b*c^3*d^2*e^2)/3 + 4*b^2*c^2*d*e^3 + (5*a*b*c^2*e^4)/
3 + (8*a*c^3*d*e^3)/3) + x^4*((b^4*d^4)/4 + (3*a^2*c^2*d^4)/2 + 3*a^3*c*d^2*e^2 + (9*a^2*b^2*d^2*e^2)/2 + 3*a*
b^2*c*d^4 + 3*a*b^3*d^3*e + a^3*b*d*e^3 + 9*a^2*b*c*d^3*e) + (c^4*e^4*x^12)/6 + (c^3*e^3*x^11*(7*b*e + 8*c*d))
/11 + (a^2*d^3*x^2*(3*b^2*d + 4*a*b*e + 2*a*c*d))/2 + (c^2*e^2*x^10*(9*b^2*e^2 + 12*c^2*d^2 + 6*a*c*e^2 + 28*b
*c*d*e))/10 + a^3*b*d^4*x

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sympy [B]  time = 0.21, size = 935, normalized size = 2.27 \[ a^{3} b d^{4} x + \frac {c^{4} e^{4} x^{12}}{6} + x^{11} \left (\frac {7 b c^{3} e^{4}}{11} + \frac {8 c^{4} d e^{3}}{11}\right ) + x^{10} \left (\frac {3 a c^{3} e^{4}}{5} + \frac {9 b^{2} c^{2} e^{4}}{10} + \frac {14 b c^{3} d e^{3}}{5} + \frac {6 c^{4} d^{2} e^{2}}{5}\right ) + x^{9} \left (\frac {5 a b c^{2} e^{4}}{3} + \frac {8 a c^{3} d e^{3}}{3} + \frac {5 b^{3} c e^{4}}{9} + 4 b^{2} c^{2} d e^{3} + \frac {14 b c^{3} d^{2} e^{2}}{3} + \frac {8 c^{4} d^{3} e}{9}\right ) + x^{8} \left (\frac {3 a^{2} c^{2} e^{4}}{4} + \frac {3 a b^{2} c e^{4}}{2} + \frac {15 a b c^{2} d e^{3}}{2} + \frac {9 a c^{3} d^{2} e^{2}}{2} + \frac {b^{4} e^{4}}{8} + \frac {5 b^{3} c d e^{3}}{2} + \frac {27 b^{2} c^{2} d^{2} e^{2}}{4} + \frac {7 b c^{3} d^{3} e}{2} + \frac {c^{4} d^{4}}{4}\right ) + x^{7} \left (\frac {9 a^{2} b c e^{4}}{7} + \frac {24 a^{2} c^{2} d e^{3}}{7} + \frac {3 a b^{3} e^{4}}{7} + \frac {48 a b^{2} c d e^{3}}{7} + \frac {90 a b c^{2} d^{2} e^{2}}{7} + \frac {24 a c^{3} d^{3} e}{7} + \frac {4 b^{4} d e^{3}}{7} + \frac {30 b^{3} c d^{2} e^{2}}{7} + \frac {36 b^{2} c^{2} d^{3} e}{7} + b c^{3} d^{4}\right ) + x^{6} \left (\frac {a^{3} c e^{4}}{3} + \frac {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} + \frac {10 b^{3} c d^{3} e}{3} + \frac {3 b^{2} c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac {a^{3} b e^{4}}{5} + \frac {8 a^{3} c d e^{3}}{5} + \frac {12 a^{2} b^{2} d e^{3}}{5} + \frac {54 a^{2} b c d^{2} e^{2}}{5} + \frac {24 a^{2} c^{2} d^{3} e}{5} + \frac {18 a b^{3} d^{2} e^{2}}{5} + \frac {48 a b^{2} c d^{3} e}{5} + 3 a b c^{2} d^{4} + \frac {4 b^{4} d^{3} e}{5} + b^{3} c d^{4}\right ) + x^{4} \left (a^{3} b d e^{3} + 3 a^{3} c d^{2} e^{2} + \frac {9 a^{2} b^{2} d^{2} e^{2}}{2} + 9 a^{2} b c d^{3} e + \frac {3 a^{2} c^{2} d^{4}}{2} + 3 a b^{3} d^{3} e + 3 a b^{2} c d^{4} + \frac {b^{4} d^{4}}{4}\right ) + x^{3} \left (2 a^{3} b d^{2} e^{2} + \frac {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}\right ) + x^{2} \left (2 a^{3} b d^{3} e + a^{3} c d^{4} + \frac {3 a^{2} b^{2} d^{4}}{2}\right ) \]

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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