3.1505 \(\int (b+2 c x) (d+e x)^3 (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=240 \[ \frac {4 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac {(d+e x)^6 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^6}+\frac {2 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6}-\frac {5 c^2 (d+e x)^8 (2 c d-b e)}{8 e^6}+\frac {2 c^3 (d+e x)^9}{9 e^6} \]

[Out]

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

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Rubi [A]  time = 0.32, antiderivative size = 240, 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} \[ \frac {4 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac {(d+e x)^6 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^6}+\frac {2 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6}-\frac {5 c^2 (d+e x)^8 (2 c d-b e)}{8 e^6}+\frac {2 c^3 (d+e x)^9}{9 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.12, size = 351, normalized size = 1.46 \[ \frac {1}{3} d x^3 \left (6 a^2 c d e+6 a b^2 d e+3 a b \left (a e^2+2 c d^2\right )+b^3 d^2\right )+a^2 b d^3 x+\frac {1}{6} x^6 \left (3 c^2 d e (4 a e+5 b d)+6 b c e^2 (a e+2 b d)+b^3 e^3+2 c^3 d^3\right )+\frac {1}{7} c e x^7 \left (c e (4 a e+15 b d)+4 b^2 e^2+6 c^2 d^2\right )+\frac {1}{2} a d^2 x^2 \left (3 a b e+2 a c d+2 b^2 d\right )+\frac {1}{5} x^5 \left (2 b^2 \left (a e^3+6 c d^2 e\right )+b c d \left (18 a e^2+5 c d^2\right )+2 a c e \left (a e^2+6 c d^2\right )+3 b^3 d e^2\right )+\frac {1}{4} x^4 \left (b^2 \left (6 a d e^2+4 c d^3\right )+a b e \left (a e^2+18 c d^2\right )+2 a c d \left (3 a e^2+2 c d^2\right )+3 b^3 d^2 e\right )+\frac {1}{8} c^2 e^2 x^8 (5 b e+6 c d)+\frac {2}{9} c^3 e^3 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*b*d^3*x + (a*d^2*(2*b^2*d + 2*a*c*d + 3*a*b*e)*x^2)/2 + (d*(b^3*d^2 + 6*a*b^2*d*e + 6*a^2*c*d*e + 3*a*b*(2
*c*d^2 + a*e^2))*x^3)/3 + ((3*b^3*d^2*e + a*b*e*(18*c*d^2 + a*e^2) + 2*a*c*d*(2*c*d^2 + 3*a*e^2) + b^2*(4*c*d^
3 + 6*a*d*e^2))*x^4)/4 + ((3*b^3*d*e^2 + 2*a*c*e*(6*c*d^2 + a*e^2) + b*c*d*(5*c*d^2 + 18*a*e^2) + 2*b^2*(6*c*d
^2*e + a*e^3))*x^5)/5 + ((2*c^3*d^3 + b^3*e^3 + 6*b*c*e^2*(2*b*d + a*e) + 3*c^2*d*e*(5*b*d + 4*a*e))*x^6)/6 +
(c*e*(6*c^2*d^2 + 4*b^2*e^2 + c*e*(15*b*d + 4*a*e))*x^7)/7 + (c^2*e^2*(6*c*d + 5*b*e)*x^8)/8 + (2*c^3*e^3*x^9)
/9

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fricas [A]  time = 1.07, size = 429, normalized size = 1.79 \[ \frac {2}{9} x^{9} e^{3} c^{3} + \frac {3}{4} x^{8} e^{2} d c^{3} + \frac {5}{8} x^{8} e^{3} c^{2} b + \frac {6}{7} x^{7} e d^{2} c^{3} + \frac {15}{7} x^{7} e^{2} d c^{2} b + \frac {4}{7} x^{7} e^{3} c b^{2} + \frac {4}{7} x^{7} e^{3} c^{2} a + \frac {1}{3} x^{6} d^{3} c^{3} + \frac {5}{2} x^{6} e d^{2} c^{2} b + 2 x^{6} e^{2} d c b^{2} + \frac {1}{6} x^{6} e^{3} b^{3} + 2 x^{6} e^{2} d c^{2} a + x^{6} e^{3} c b a + x^{5} d^{3} c^{2} b + \frac {12}{5} x^{5} e d^{2} c b^{2} + \frac {3}{5} x^{5} e^{2} d b^{3} + \frac {12}{5} x^{5} e d^{2} c^{2} a + \frac {18}{5} x^{5} e^{2} d c b a + \frac {2}{5} x^{5} e^{3} b^{2} a + \frac {2}{5} x^{5} e^{3} c a^{2} + x^{4} d^{3} c b^{2} + \frac {3}{4} x^{4} e d^{2} b^{3} + x^{4} d^{3} c^{2} a + \frac {9}{2} x^{4} e d^{2} c b a + \frac {3}{2} x^{4} e^{2} d b^{2} a + \frac {3}{2} x^{4} e^{2} d c a^{2} + \frac {1}{4} x^{4} e^{3} b a^{2} + \frac {1}{3} x^{3} d^{3} b^{3} + 2 x^{3} d^{3} c b a + 2 x^{3} e d^{2} b^{2} a + 2 x^{3} e d^{2} c a^{2} + x^{3} e^{2} d b a^{2} + x^{2} d^{3} b^{2} a + x^{2} d^{3} c a^{2} + \frac {3}{2} x^{2} e d^{2} b a^{2} + x d^{3} b a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*x^9*e^3*c^3 + 3/4*x^8*e^2*d*c^3 + 5/8*x^8*e^3*c^2*b + 6/7*x^7*e*d^2*c^3 + 15/7*x^7*e^2*d*c^2*b + 4/7*x^7*e
^3*c*b^2 + 4/7*x^7*e^3*c^2*a + 1/3*x^6*d^3*c^3 + 5/2*x^6*e*d^2*c^2*b + 2*x^6*e^2*d*c*b^2 + 1/6*x^6*e^3*b^3 + 2
*x^6*e^2*d*c^2*a + x^6*e^3*c*b*a + x^5*d^3*c^2*b + 12/5*x^5*e*d^2*c*b^2 + 3/5*x^5*e^2*d*b^3 + 12/5*x^5*e*d^2*c
^2*a + 18/5*x^5*e^2*d*c*b*a + 2/5*x^5*e^3*b^2*a + 2/5*x^5*e^3*c*a^2 + x^4*d^3*c*b^2 + 3/4*x^4*e*d^2*b^3 + x^4*
d^3*c^2*a + 9/2*x^4*e*d^2*c*b*a + 3/2*x^4*e^2*d*b^2*a + 3/2*x^4*e^2*d*c*a^2 + 1/4*x^4*e^3*b*a^2 + 1/3*x^3*d^3*
b^3 + 2*x^3*d^3*c*b*a + 2*x^3*e*d^2*b^2*a + 2*x^3*e*d^2*c*a^2 + x^3*e^2*d*b*a^2 + x^2*d^3*b^2*a + x^2*d^3*c*a^
2 + 3/2*x^2*e*d^2*b*a^2 + x*d^3*b*a^2

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giac [A]  time = 0.16, size = 420, normalized size = 1.75 \[ \frac {2}{9} \, c^{3} x^{9} e^{3} + \frac {3}{4} \, c^{3} d x^{8} e^{2} + \frac {6}{7} \, c^{3} d^{2} x^{7} e + \frac {1}{3} \, c^{3} d^{3} x^{6} + \frac {5}{8} \, b c^{2} x^{8} e^{3} + \frac {15}{7} \, b c^{2} d x^{7} e^{2} + \frac {5}{2} \, b c^{2} d^{2} x^{6} e + b c^{2} d^{3} x^{5} + \frac {4}{7} \, b^{2} c x^{7} e^{3} + \frac {4}{7} \, a c^{2} x^{7} e^{3} + 2 \, b^{2} c d x^{6} e^{2} + 2 \, a c^{2} d x^{6} e^{2} + \frac {12}{5} \, b^{2} c d^{2} x^{5} e + \frac {12}{5} \, a c^{2} d^{2} x^{5} e + b^{2} c d^{3} x^{4} + a c^{2} d^{3} x^{4} + \frac {1}{6} \, b^{3} x^{6} e^{3} + a b c x^{6} e^{3} + \frac {3}{5} \, b^{3} d x^{5} e^{2} + \frac {18}{5} \, a b c d x^{5} e^{2} + \frac {3}{4} \, b^{3} d^{2} x^{4} e + \frac {9}{2} \, a b c d^{2} x^{4} e + \frac {1}{3} \, b^{3} d^{3} x^{3} + 2 \, a b c d^{3} x^{3} + \frac {2}{5} \, a b^{2} x^{5} e^{3} + \frac {2}{5} \, a^{2} c x^{5} e^{3} + \frac {3}{2} \, a b^{2} d x^{4} e^{2} + \frac {3}{2} \, a^{2} c d x^{4} e^{2} + 2 \, a b^{2} d^{2} x^{3} e + 2 \, a^{2} c d^{2} x^{3} e + a b^{2} d^{3} x^{2} + a^{2} c d^{3} x^{2} + \frac {1}{4} \, a^{2} b x^{4} e^{3} + a^{2} b d x^{3} e^{2} + \frac {3}{2} \, a^{2} b d^{2} x^{2} e + a^{2} b d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*c^3*x^9*e^3 + 3/4*c^3*d*x^8*e^2 + 6/7*c^3*d^2*x^7*e + 1/3*c^3*d^3*x^6 + 5/8*b*c^2*x^8*e^3 + 15/7*b*c^2*d*x
^7*e^2 + 5/2*b*c^2*d^2*x^6*e + b*c^2*d^3*x^5 + 4/7*b^2*c*x^7*e^3 + 4/7*a*c^2*x^7*e^3 + 2*b^2*c*d*x^6*e^2 + 2*a
*c^2*d*x^6*e^2 + 12/5*b^2*c*d^2*x^5*e + 12/5*a*c^2*d^2*x^5*e + b^2*c*d^3*x^4 + a*c^2*d^3*x^4 + 1/6*b^3*x^6*e^3
+ a*b*c*x^6*e^3 + 3/5*b^3*d*x^5*e^2 + 18/5*a*b*c*d*x^5*e^2 + 3/4*b^3*d^2*x^4*e + 9/2*a*b*c*d^2*x^4*e + 1/3*b^
3*d^3*x^3 + 2*a*b*c*d^3*x^3 + 2/5*a*b^2*x^5*e^3 + 2/5*a^2*c*x^5*e^3 + 3/2*a*b^2*d*x^4*e^2 + 3/2*a^2*c*d*x^4*e^
2 + 2*a*b^2*d^2*x^3*e + 2*a^2*c*d^2*x^3*e + a*b^2*d^3*x^2 + a^2*c*d^3*x^2 + 1/4*a^2*b*x^4*e^3 + a^2*b*d*x^3*e^
2 + 3/2*a^2*b*d^2*x^2*e + a^2*b*d^3*x

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maple [A]  time = 0.04, size = 428, normalized size = 1.78 \[ \frac {2 c^{3} e^{3} x^{9}}{9}+\frac {\left (4 b \,c^{2} e^{3}+\left (b \,e^{3}+6 d \,e^{2} c \right ) c^{2}\right ) x^{8}}{8}+a^{2} b \,d^{3} x +\frac {\left (2 \left (2 a c +b^{2}\right ) c \,e^{3}+2 \left (b \,e^{3}+6 d \,e^{2} c \right ) b c +\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) c^{2}\right ) x^{7}}{7}+\frac {\left (4 a b c \,e^{3}+2 \left (3 b d \,e^{2}+6 c \,d^{2} e \right ) b c +\left (3 b \,d^{2} e +2 c \,d^{3}\right ) c^{2}+\left (b \,e^{3}+6 d \,e^{2} c \right ) \left (2 a c +b^{2}\right )\right ) x^{6}}{6}+\frac {\left (2 a^{2} c \,e^{3}+b \,c^{2} d^{3}+2 \left (b \,e^{3}+6 d \,e^{2} c \right ) a b +2 \left (3 b \,d^{2} e +2 c \,d^{3}\right ) b c +\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 b^{2} c \,d^{3}+\left (b \,e^{3}+6 d \,e^{2} c \right ) a^{2}+2 \left (3 b d \,e^{2}+6 c \,d^{2} e \right ) a b +\left (3 b \,d^{2} e +2 c \,d^{3}\right ) \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (2 a c +b^{2}\right ) b \,d^{3}+\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) a^{2}+2 \left (3 b \,d^{2} e +2 c \,d^{3}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 a \,b^{2} d^{3}+\left (3 b \,d^{2} e +2 c \,d^{3}\right ) a^{2}\right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.49, size = 343, normalized size = 1.43 \[ \frac {2}{9} \, c^{3} e^{3} x^{9} + \frac {1}{8} \, {\left (6 \, c^{3} d e^{2} + 5 \, b c^{2} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, c^{3} d^{2} e + 15 \, b c^{2} d e^{2} + 4 \, {\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{7} + a^{2} b d^{3} x + \frac {1}{6} \, {\left (2 \, c^{3} d^{3} + 15 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} + {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, b c^{2} d^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e + 3 \, {\left (b^{3} + 6 \, a b c\right )} d e^{2} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (a^{2} b e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} + 3 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e + 6 \, {\left (a b^{2} + a^{2} c\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a^{2} b d e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{3} + 6 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{2} e + 2 \, {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*c^3*e^3*x^9 + 1/8*(6*c^3*d*e^2 + 5*b*c^2*e^3)*x^8 + 1/7*(6*c^3*d^2*e + 15*b*c^2*d*e^2 + 4*(b^2*c + a*c^2)*
e^3)*x^7 + a^2*b*d^3*x + 1/6*(2*c^3*d^3 + 15*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 + (b^3 + 6*a*b*c)*e^3)*x^6
+ 1/5*(5*b*c^2*d^3 + 12*(b^2*c + a*c^2)*d^2*e + 3*(b^3 + 6*a*b*c)*d*e^2 + 2*(a*b^2 + a^2*c)*e^3)*x^5 + 1/4*(a
^2*b*e^3 + 4*(b^2*c + a*c^2)*d^3 + 3*(b^3 + 6*a*b*c)*d^2*e + 6*(a*b^2 + a^2*c)*d*e^2)*x^4 + 1/3*(3*a^2*b*d*e^2
+ (b^3 + 6*a*b*c)*d^3 + 6*(a*b^2 + a^2*c)*d^2*e)*x^3 + 1/2*(3*a^2*b*d^2*e + 2*(a*b^2 + a^2*c)*d^3)*x^2

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mupad [B]  time = 0.11, size = 349, normalized size = 1.45 \[ x^6\,\left (\frac {b^3\,e^3}{6}+2\,b^2\,c\,d\,e^2+\frac {5\,b\,c^2\,d^2\,e}{2}+a\,b\,c\,e^3+\frac {c^3\,d^3}{3}+2\,a\,c^2\,d\,e^2\right )+x^4\,\left (\frac {a^2\,b\,e^3}{4}+\frac {3\,a^2\,c\,d\,e^2}{2}+\frac {3\,a\,b^2\,d\,e^2}{2}+\frac {9\,a\,b\,c\,d^2\,e}{2}+a\,c^2\,d^3+\frac {3\,b^3\,d^2\,e}{4}+b^2\,c\,d^3\right )+x^5\,\left (\frac {2\,a^2\,c\,e^3}{5}+\frac {2\,a\,b^2\,e^3}{5}+\frac {18\,a\,b\,c\,d\,e^2}{5}+\frac {12\,a\,c^2\,d^2\,e}{5}+\frac {3\,b^3\,d\,e^2}{5}+\frac {12\,b^2\,c\,d^2\,e}{5}+b\,c^2\,d^3\right )+x^3\,\left (a^2\,b\,d\,e^2+2\,c\,a^2\,d^2\,e+2\,a\,b^2\,d^2\,e+2\,c\,a\,b\,d^3+\frac {b^3\,d^3}{3}\right )+\frac {2\,c^3\,e^3\,x^9}{9}+\frac {a\,d^2\,x^2\,\left (2\,d\,b^2+3\,a\,e\,b+2\,a\,c\,d\right )}{2}+\frac {c^2\,e^2\,x^8\,\left (5\,b\,e+6\,c\,d\right )}{8}+a^2\,b\,d^3\,x+\frac {c\,e\,x^7\,\left (4\,b^2\,e^2+15\,b\,c\,d\,e+6\,c^2\,d^2+4\,a\,c\,e^2\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^6*((b^3*e^3)/6 + (c^3*d^3)/3 + a*b*c*e^3 + 2*a*c^2*d*e^2 + (5*b*c^2*d^2*e)/2 + 2*b^2*c*d*e^2) + x^4*(a*c^2*d
^3 + (a^2*b*e^3)/4 + b^2*c*d^3 + (3*b^3*d^2*e)/4 + (3*a*b^2*d*e^2)/2 + (3*a^2*c*d*e^2)/2 + (9*a*b*c*d^2*e)/2)
+ x^5*((2*a*b^2*e^3)/5 + b*c^2*d^3 + (2*a^2*c*e^3)/5 + (3*b^3*d*e^2)/5 + (12*a*c^2*d^2*e)/5 + (12*b^2*c*d^2*e)
/5 + (18*a*b*c*d*e^2)/5) + x^3*((b^3*d^3)/3 + 2*a*b*c*d^3 + 2*a*b^2*d^2*e + a^2*b*d*e^2 + 2*a^2*c*d^2*e) + (2*
c^3*e^3*x^9)/9 + (a*d^2*x^2*(2*b^2*d + 3*a*b*e + 2*a*c*d))/2 + (c^2*e^2*x^8*(5*b*e + 6*c*d))/8 + a^2*b*d^3*x +
(c*e*x^7*(4*b^2*e^2 + 6*c^2*d^2 + 4*a*c*e^2 + 15*b*c*d*e))/7

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sympy [A]  time = 0.14, size = 430, normalized size = 1.79 \[ a^{2} b d^{3} x + \frac {2 c^{3} e^{3} x^{9}}{9} + x^{8} \left (\frac {5 b c^{2} e^{3}}{8} + \frac {3 c^{3} d e^{2}}{4}\right ) + x^{7} \left (\frac {4 a c^{2} e^{3}}{7} + \frac {4 b^{2} c e^{3}}{7} + \frac {15 b c^{2} d e^{2}}{7} + \frac {6 c^{3} d^{2} e}{7}\right ) + x^{6} \left (a b c e^{3} + 2 a c^{2} d e^{2} + \frac {b^{3} e^{3}}{6} + 2 b^{2} c d e^{2} + \frac {5 b c^{2} d^{2} e}{2} + \frac {c^{3} d^{3}}{3}\right ) + x^{5} \left (\frac {2 a^{2} c e^{3}}{5} + \frac {2 a b^{2} e^{3}}{5} + \frac {18 a b c d e^{2}}{5} + \frac {12 a c^{2} d^{2} e}{5} + \frac {3 b^{3} d e^{2}}{5} + \frac {12 b^{2} c d^{2} e}{5} + b c^{2} d^{3}\right ) + x^{4} \left (\frac {a^{2} b e^{3}}{4} + \frac {3 a^{2} c d e^{2}}{2} + \frac {3 a b^{2} d e^{2}}{2} + \frac {9 a b c d^{2} e}{2} + a c^{2} d^{3} + \frac {3 b^{3} d^{2} e}{4} + b^{2} c d^{3}\right ) + x^{3} \left (a^{2} b d e^{2} + 2 a^{2} c d^{2} e + 2 a b^{2} d^{2} e + 2 a b c d^{3} + \frac {b^{3} d^{3}}{3}\right ) + x^{2} \left (\frac {3 a^{2} b d^{2} e}{2} + a^{2} c d^{3} + a b^{2} d^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*b*d**3*x + 2*c**3*e**3*x**9/9 + x**8*(5*b*c**2*e**3/8 + 3*c**3*d*e**2/4) + x**7*(4*a*c**2*e**3/7 + 4*b**2
*c*e**3/7 + 15*b*c**2*d*e**2/7 + 6*c**3*d**2*e/7) + x**6*(a*b*c*e**3 + 2*a*c**2*d*e**2 + b**3*e**3/6 + 2*b**2*
c*d*e**2 + 5*b*c**2*d**2*e/2 + c**3*d**3/3) + x**5*(2*a**2*c*e**3/5 + 2*a*b**2*e**3/5 + 18*a*b*c*d*e**2/5 + 12
*a*c**2*d**2*e/5 + 3*b**3*d*e**2/5 + 12*b**2*c*d**2*e/5 + b*c**2*d**3) + x**4*(a**2*b*e**3/4 + 3*a**2*c*d*e**2
/2 + 3*a*b**2*d*e**2/2 + 9*a*b*c*d**2*e/2 + a*c**2*d**3 + 3*b**3*d**2*e/4 + b**2*c*d**3) + x**3*(a**2*b*d*e**2
+ 2*a**2*c*d**2*e + 2*a*b**2*d**2*e + 2*a*b*c*d**3 + b**3*d**3/3) + x**2*(3*a**2*b*d**2*e/2 + a**2*c*d**3 + a
*b**2*d**3)

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