3.1504 \(\int (b+2 c x) (d+e x)^4 (a+b x+c x^2)^2 \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.415931, antiderivative size = 240, 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} \[ \frac{c (d+e x)^8 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^6}-\frac{(d+e x)^7 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^6}+\frac{(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{(d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac{5 c^2 (d+e x)^9 (2 c d-b e)}{9 e^6}+\frac{c^3 (d+e x)^{10}}{5 e^6} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.162401, size = 433, normalized size = 1.8 \[ \frac{1}{3} x^6 \left (c e^2 \left (a^2 e^2+12 a b d e+12 b^2 d^2\right )+b^2 e^3 (a e+2 b d)+2 c^2 d^2 e (6 a e+5 b d)+c^3 d^4\right )+\frac{1}{5} x^5 \left (b \left (a^2 e^4+36 a c d^2 e^2+5 c^2 d^4\right )+8 b^2 \left (a d e^3+2 c d^3 e\right )+8 a c d e \left (a e^2+2 c d^2\right )+6 b^3 d^2 e^2\right )+\frac{1}{3} d^2 x^3 \left (8 a^2 c d e+8 a b^2 d e+6 a b \left (a e^2+c d^2\right )+b^3 d^2\right )+a^2 b d^4 x+\frac{1}{2} c e^2 x^8 \left (c e (a e+5 b d)+b^2 e^2+3 c^2 d^2\right )+\frac{1}{7} e x^7 \left (2 c^2 d e (8 a e+15 b d)+2 b c e^2 (3 a e+8 b d)+b^3 e^3+8 c^3 d^3\right )+d x^4 \left (b^2 \left (3 a d e^2+c d^3\right )+a b e \left (a e^2+6 c d^2\right )+a c d \left (3 a e^2+c d^2\right )+b^3 d^2 e\right )+a d^3 x^2 \left (2 a b e+a c d+b^2 d\right )+\frac{1}{9} c^2 e^3 x^9 (5 b e+8 c d)+\frac{1}{5} c^3 e^4 x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*b*d^4*x + a*d^3*(b^2*d + a*c*d + 2*a*b*e)*x^2 + (d^2*(b^3*d^2 + 8*a*b^2*d*e + 8*a^2*c*d*e + 6*a*b*(c*d^2 +
a*e^2))*x^3)/3 + d*(b^3*d^2*e + a*b*e*(6*c*d^2 + a*e^2) + a*c*d*(c*d^2 + 3*a*e^2) + b^2*(c*d^3 + 3*a*d*e^2))*
x^4 + ((6*b^3*d^2*e^2 + 8*a*c*d*e*(2*c*d^2 + a*e^2) + 8*b^2*(2*c*d^3*e + a*d*e^3) + b*(5*c^2*d^4 + 36*a*c*d^2*
e^2 + a^2*e^4))*x^5)/5 + ((c^3*d^4 + b^2*e^3*(2*b*d + a*e) + 2*c^2*d^2*e*(5*b*d + 6*a*e) + c*e^2*(12*b^2*d^2 +
12*a*b*d*e + a^2*e^2))*x^6)/3 + (e*(8*c^3*d^3 + b^3*e^3 + 2*b*c*e^2*(8*b*d + 3*a*e) + 2*c^2*d*e*(15*b*d + 8*a
*e))*x^7)/7 + (c*e^2*(3*c^2*d^2 + b^2*e^2 + c*e*(5*b*d + a*e))*x^8)/2 + (c^2*e^3*(8*c*d + 5*b*e)*x^9)/9 + (c^3
*e^4*x^10)/5

________________________________________________________________________________________

Maple [B]  time = 0.002, size = 554, normalized size = 2.3 \begin{align*}{\frac{{c}^{3}{e}^{4}{x}^{10}}{5}}+{\frac{ \left ( \left ( b{e}^{4}+8\,cd{e}^{3} \right ){c}^{2}+4\,{c}^{2}{e}^{4}b \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 4\,bd{e}^{3}+12\,c{d}^{2}{e}^{2} \right ){c}^{2}+2\, \left ( b{e}^{4}+8\,cd{e}^{3} \right ) bc+2\,c{e}^{4} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 6\,b{d}^{2}{e}^{2}+8\,c{d}^{3}e \right ){c}^{2}+2\, \left ( 4\,bd{e}^{3}+12\,c{d}^{2}{e}^{2} \right ) bc+ \left ( b{e}^{4}+8\,cd{e}^{3} \right ) \left ( 2\,ac+{b}^{2} \right ) +4\,c{e}^{4}ab \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,b{d}^{3}e+2\,c{d}^{4} \right ){c}^{2}+2\, \left ( 6\,b{d}^{2}{e}^{2}+8\,c{d}^{3}e \right ) bc+ \left ( 4\,bd{e}^{3}+12\,c{d}^{2}{e}^{2} \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( b{e}^{4}+8\,cd{e}^{3} \right ) ab+2\,c{e}^{4}{a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( b{d}^{4}{c}^{2}+2\, \left ( 4\,b{d}^{3}e+2\,c{d}^{4} \right ) bc+ \left ( 6\,b{d}^{2}{e}^{2}+8\,c{d}^{3}e \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 4\,bd{e}^{3}+12\,c{d}^{2}{e}^{2} \right ) ab+ \left ( b{e}^{4}+8\,cd{e}^{3} \right ){a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{2}{d}^{4}c+ \left ( 4\,b{d}^{3}e+2\,c{d}^{4} \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 6\,b{d}^{2}{e}^{2}+8\,c{d}^{3}e \right ) ab+ \left ( 4\,bd{e}^{3}+12\,c{d}^{2}{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( b{d}^{4} \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 4\,b{d}^{3}e+2\,c{d}^{4} \right ) ab+ \left ( 6\,b{d}^{2}{e}^{2}+8\,c{d}^{3}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{b}^{2}{d}^{4}a+ \left ( 4\,b{d}^{3}e+2\,c{d}^{4} \right ){a}^{2} \right ){x}^{2}}{2}}+b{d}^{4}{a}^{2}x \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)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.10827, size = 581, normalized size = 2.42 \begin{align*} \frac{1}{5} \, c^{3} e^{4} x^{10} + \frac{1}{9} \,{\left (8 \, c^{3} d e^{3} + 5 \, b c^{2} e^{4}\right )} x^{9} + \frac{1}{2} \,{\left (3 \, c^{3} d^{2} e^{2} + 5 \, b c^{2} d e^{3} +{\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{8} + a^{2} b d^{4} x + \frac{1}{7} \,{\left (8 \, c^{3} d^{3} e + 30 \, b c^{2} d^{2} e^{2} + 16 \,{\left (b^{2} c + a c^{2}\right )} d e^{3} +{\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{7} + \frac{1}{3} \,{\left (c^{3} d^{4} + 10 \, b c^{2} d^{3} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} + 2 \,{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (5 \, b c^{2} d^{4} + a^{2} b e^{4} + 16 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e + 6 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{2} + 8 \,{\left (a b^{2} + a^{2} c\right )} d e^{3}\right )} x^{5} +{\left (a^{2} b d e^{3} +{\left (b^{2} c + a c^{2}\right )} d^{4} +{\left (b^{3} + 6 \, a b c\right )} d^{3} e + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a^{2} b d^{2} e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{4} + 8 \,{\left (a b^{2} + a^{2} c\right )} d^{3} e\right )} x^{3} +{\left (2 \, a^{2} b d^{3} e +{\left (a b^{2} + a^{2} c\right )} d^{4}\right )} x^{2} \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)^2,x, algorithm="maxima")

[Out]

1/5*c^3*e^4*x^10 + 1/9*(8*c^3*d*e^3 + 5*b*c^2*e^4)*x^9 + 1/2*(3*c^3*d^2*e^2 + 5*b*c^2*d*e^3 + (b^2*c + a*c^2)*
e^4)*x^8 + a^2*b*d^4*x + 1/7*(8*c^3*d^3*e + 30*b*c^2*d^2*e^2 + 16*(b^2*c + a*c^2)*d*e^3 + (b^3 + 6*a*b*c)*e^4)
*x^7 + 1/3*(c^3*d^4 + 10*b*c^2*d^3*e + 12*(b^2*c + a*c^2)*d^2*e^2 + 2*(b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*
e^4)*x^6 + 1/5*(5*b*c^2*d^4 + a^2*b*e^4 + 16*(b^2*c + a*c^2)*d^3*e + 6*(b^3 + 6*a*b*c)*d^2*e^2 + 8*(a*b^2 + a^
2*c)*d*e^3)*x^5 + (a^2*b*d*e^3 + (b^2*c + a*c^2)*d^4 + (b^3 + 6*a*b*c)*d^3*e + 3*(a*b^2 + a^2*c)*d^2*e^2)*x^4
+ 1/3*(6*a^2*b*d^2*e^2 + (b^3 + 6*a*b*c)*d^4 + 8*(a*b^2 + a^2*c)*d^3*e)*x^3 + (2*a^2*b*d^3*e + (a*b^2 + a^2*c)
*d^4)*x^2

________________________________________________________________________________________

Fricas [B]  time = 1.68668, size = 1224, normalized size = 5.1 \begin{align*} \frac{1}{5} x^{10} e^{4} c^{3} + \frac{8}{9} x^{9} e^{3} d c^{3} + \frac{5}{9} x^{9} e^{4} c^{2} b + \frac{3}{2} x^{8} e^{2} d^{2} c^{3} + \frac{5}{2} x^{8} e^{3} d c^{2} b + \frac{1}{2} x^{8} e^{4} c b^{2} + \frac{1}{2} x^{8} e^{4} c^{2} a + \frac{8}{7} x^{7} e d^{3} c^{3} + \frac{30}{7} x^{7} e^{2} d^{2} c^{2} b + \frac{16}{7} x^{7} e^{3} d c b^{2} + \frac{1}{7} x^{7} e^{4} b^{3} + \frac{16}{7} x^{7} e^{3} d c^{2} a + \frac{6}{7} x^{7} e^{4} c b a + \frac{1}{3} x^{6} d^{4} c^{3} + \frac{10}{3} x^{6} e d^{3} c^{2} b + 4 x^{6} e^{2} d^{2} c b^{2} + \frac{2}{3} x^{6} e^{3} d b^{3} + 4 x^{6} e^{2} d^{2} c^{2} a + 4 x^{6} e^{3} d c b a + \frac{1}{3} x^{6} e^{4} b^{2} a + \frac{1}{3} x^{6} e^{4} c a^{2} + x^{5} d^{4} c^{2} b + \frac{16}{5} x^{5} e d^{3} c b^{2} + \frac{6}{5} x^{5} e^{2} d^{2} b^{3} + \frac{16}{5} x^{5} e d^{3} c^{2} a + \frac{36}{5} x^{5} e^{2} d^{2} c b a + \frac{8}{5} x^{5} e^{3} d b^{2} a + \frac{8}{5} x^{5} e^{3} d c a^{2} + \frac{1}{5} x^{5} e^{4} b a^{2} + x^{4} d^{4} c b^{2} + x^{4} e d^{3} b^{3} + x^{4} d^{4} c^{2} a + 6 x^{4} e d^{3} c b a + 3 x^{4} e^{2} d^{2} b^{2} a + 3 x^{4} e^{2} d^{2} c a^{2} + x^{4} e^{3} d b a^{2} + \frac{1}{3} x^{3} d^{4} b^{3} + 2 x^{3} d^{4} c b a + \frac{8}{3} x^{3} e d^{3} b^{2} a + \frac{8}{3} x^{3} e d^{3} c a^{2} + 2 x^{3} e^{2} d^{2} b a^{2} + x^{2} d^{4} b^{2} a + x^{2} d^{4} c a^{2} + 2 x^{2} e d^{3} b a^{2} + x d^{4} b a^{2} \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)^2,x, algorithm="fricas")

[Out]

1/5*x^10*e^4*c^3 + 8/9*x^9*e^3*d*c^3 + 5/9*x^9*e^4*c^2*b + 3/2*x^8*e^2*d^2*c^3 + 5/2*x^8*e^3*d*c^2*b + 1/2*x^8
*e^4*c*b^2 + 1/2*x^8*e^4*c^2*a + 8/7*x^7*e*d^3*c^3 + 30/7*x^7*e^2*d^2*c^2*b + 16/7*x^7*e^3*d*c*b^2 + 1/7*x^7*e
^4*b^3 + 16/7*x^7*e^3*d*c^2*a + 6/7*x^7*e^4*c*b*a + 1/3*x^6*d^4*c^3 + 10/3*x^6*e*d^3*c^2*b + 4*x^6*e^2*d^2*c*b
^2 + 2/3*x^6*e^3*d*b^3 + 4*x^6*e^2*d^2*c^2*a + 4*x^6*e^3*d*c*b*a + 1/3*x^6*e^4*b^2*a + 1/3*x^6*e^4*c*a^2 + x^5
*d^4*c^2*b + 16/5*x^5*e*d^3*c*b^2 + 6/5*x^5*e^2*d^2*b^3 + 16/5*x^5*e*d^3*c^2*a + 36/5*x^5*e^2*d^2*c*b*a + 8/5*
x^5*e^3*d*b^2*a + 8/5*x^5*e^3*d*c*a^2 + 1/5*x^5*e^4*b*a^2 + x^4*d^4*c*b^2 + x^4*e*d^3*b^3 + x^4*d^4*c^2*a + 6*
x^4*e*d^3*c*b*a + 3*x^4*e^2*d^2*b^2*a + 3*x^4*e^2*d^2*c*a^2 + x^4*e^3*d*b*a^2 + 1/3*x^3*d^4*b^3 + 2*x^3*d^4*c*
b*a + 8/3*x^3*e*d^3*b^2*a + 8/3*x^3*e*d^3*c*a^2 + 2*x^3*e^2*d^2*b*a^2 + x^2*d^4*b^2*a + x^2*d^4*c*a^2 + 2*x^2*
e*d^3*b*a^2 + x*d^4*b*a^2

________________________________________________________________________________________

Sympy [B]  time = 0.140299, size = 552, normalized size = 2.3 \begin{align*} a^{2} b d^{4} x + \frac{c^{3} e^{4} x^{10}}{5} + x^{9} \left (\frac{5 b c^{2} e^{4}}{9} + \frac{8 c^{3} d e^{3}}{9}\right ) + x^{8} \left (\frac{a c^{2} e^{4}}{2} + \frac{b^{2} c e^{4}}{2} + \frac{5 b c^{2} d e^{3}}{2} + \frac{3 c^{3} d^{2} e^{2}}{2}\right ) + x^{7} \left (\frac{6 a b c e^{4}}{7} + \frac{16 a c^{2} d e^{3}}{7} + \frac{b^{3} e^{4}}{7} + \frac{16 b^{2} c d e^{3}}{7} + \frac{30 b c^{2} d^{2} e^{2}}{7} + \frac{8 c^{3} d^{3} e}{7}\right ) + x^{6} \left (\frac{a^{2} c e^{4}}{3} + \frac{a b^{2} e^{4}}{3} + 4 a b c d e^{3} + 4 a c^{2} d^{2} e^{2} + \frac{2 b^{3} d e^{3}}{3} + 4 b^{2} c d^{2} e^{2} + \frac{10 b c^{2} d^{3} e}{3} + \frac{c^{3} d^{4}}{3}\right ) + x^{5} \left (\frac{a^{2} b e^{4}}{5} + \frac{8 a^{2} c d e^{3}}{5} + \frac{8 a b^{2} d e^{3}}{5} + \frac{36 a b c d^{2} e^{2}}{5} + \frac{16 a c^{2} d^{3} e}{5} + \frac{6 b^{3} d^{2} e^{2}}{5} + \frac{16 b^{2} c d^{3} e}{5} + b c^{2} d^{4}\right ) + x^{4} \left (a^{2} b d e^{3} + 3 a^{2} c d^{2} e^{2} + 3 a b^{2} d^{2} e^{2} + 6 a b c d^{3} e + a c^{2} d^{4} + b^{3} d^{3} e + b^{2} c d^{4}\right ) + x^{3} \left (2 a^{2} b d^{2} e^{2} + \frac{8 a^{2} c d^{3} e}{3} + \frac{8 a b^{2} d^{3} e}{3} + 2 a b c d^{4} + \frac{b^{3} d^{4}}{3}\right ) + x^{2} \left (2 a^{2} b d^{3} e + a^{2} c d^{4} + a b^{2} d^{4}\right ) \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)**2,x)

[Out]

a**2*b*d**4*x + c**3*e**4*x**10/5 + x**9*(5*b*c**2*e**4/9 + 8*c**3*d*e**3/9) + x**8*(a*c**2*e**4/2 + b**2*c*e*
*4/2 + 5*b*c**2*d*e**3/2 + 3*c**3*d**2*e**2/2) + x**7*(6*a*b*c*e**4/7 + 16*a*c**2*d*e**3/7 + b**3*e**4/7 + 16*
b**2*c*d*e**3/7 + 30*b*c**2*d**2*e**2/7 + 8*c**3*d**3*e/7) + x**6*(a**2*c*e**4/3 + a*b**2*e**4/3 + 4*a*b*c*d*e
**3 + 4*a*c**2*d**2*e**2 + 2*b**3*d*e**3/3 + 4*b**2*c*d**2*e**2 + 10*b*c**2*d**3*e/3 + c**3*d**4/3) + x**5*(a*
*2*b*e**4/5 + 8*a**2*c*d*e**3/5 + 8*a*b**2*d*e**3/5 + 36*a*b*c*d**2*e**2/5 + 16*a*c**2*d**3*e/5 + 6*b**3*d**2*
e**2/5 + 16*b**2*c*d**3*e/5 + b*c**2*d**4) + x**4*(a**2*b*d*e**3 + 3*a**2*c*d**2*e**2 + 3*a*b**2*d**2*e**2 + 6
*a*b*c*d**3*e + a*c**2*d**4 + b**3*d**3*e + b**2*c*d**4) + x**3*(2*a**2*b*d**2*e**2 + 8*a**2*c*d**3*e/3 + 8*a*
b**2*d**3*e/3 + 2*a*b*c*d**4 + b**3*d**4/3) + x**2*(2*a**2*b*d**3*e + a**2*c*d**4 + a*b**2*d**4)

________________________________________________________________________________________

Giac [B]  time = 1.18834, size = 733, normalized size = 3.05 \begin{align*} \frac{1}{5} \, c^{3} x^{10} e^{4} + \frac{8}{9} \, c^{3} d x^{9} e^{3} + \frac{3}{2} \, c^{3} d^{2} x^{8} e^{2} + \frac{8}{7} \, c^{3} d^{3} x^{7} e + \frac{1}{3} \, c^{3} d^{4} x^{6} + \frac{5}{9} \, b c^{2} x^{9} e^{4} + \frac{5}{2} \, b c^{2} d x^{8} e^{3} + \frac{30}{7} \, b c^{2} d^{2} x^{7} e^{2} + \frac{10}{3} \, b c^{2} d^{3} x^{6} e + b c^{2} d^{4} x^{5} + \frac{1}{2} \, b^{2} c x^{8} e^{4} + \frac{1}{2} \, a c^{2} x^{8} e^{4} + \frac{16}{7} \, b^{2} c d x^{7} e^{3} + \frac{16}{7} \, a c^{2} d x^{7} e^{3} + 4 \, b^{2} c d^{2} x^{6} e^{2} + 4 \, a c^{2} d^{2} x^{6} e^{2} + \frac{16}{5} \, b^{2} c d^{3} x^{5} e + \frac{16}{5} \, a c^{2} d^{3} x^{5} e + b^{2} c d^{4} x^{4} + a c^{2} d^{4} x^{4} + \frac{1}{7} \, b^{3} x^{7} e^{4} + \frac{6}{7} \, a b c x^{7} e^{4} + \frac{2}{3} \, b^{3} d x^{6} e^{3} + 4 \, a b c d x^{6} e^{3} + \frac{6}{5} \, b^{3} d^{2} x^{5} e^{2} + \frac{36}{5} \, a b c d^{2} x^{5} e^{2} + b^{3} d^{3} x^{4} e + 6 \, a b c d^{3} x^{4} e + \frac{1}{3} \, b^{3} d^{4} x^{3} + 2 \, a b c d^{4} x^{3} + \frac{1}{3} \, a b^{2} x^{6} e^{4} + \frac{1}{3} \, a^{2} c x^{6} e^{4} + \frac{8}{5} \, a b^{2} d x^{5} e^{3} + \frac{8}{5} \, a^{2} c d x^{5} e^{3} + 3 \, a b^{2} d^{2} x^{4} e^{2} + 3 \, a^{2} c d^{2} x^{4} e^{2} + \frac{8}{3} \, a b^{2} d^{3} x^{3} e + \frac{8}{3} \, a^{2} c d^{3} x^{3} e + a b^{2} d^{4} x^{2} + a^{2} c d^{4} x^{2} + \frac{1}{5} \, a^{2} b x^{5} e^{4} + a^{2} b d x^{4} e^{3} + 2 \, a^{2} b d^{2} x^{3} e^{2} + 2 \, a^{2} b d^{3} x^{2} e + a^{2} b d^{4} x \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)^2,x, algorithm="giac")

[Out]

1/5*c^3*x^10*e^4 + 8/9*c^3*d*x^9*e^3 + 3/2*c^3*d^2*x^8*e^2 + 8/7*c^3*d^3*x^7*e + 1/3*c^3*d^4*x^6 + 5/9*b*c^2*x
^9*e^4 + 5/2*b*c^2*d*x^8*e^3 + 30/7*b*c^2*d^2*x^7*e^2 + 10/3*b*c^2*d^3*x^6*e + b*c^2*d^4*x^5 + 1/2*b^2*c*x^8*e
^4 + 1/2*a*c^2*x^8*e^4 + 16/7*b^2*c*d*x^7*e^3 + 16/7*a*c^2*d*x^7*e^3 + 4*b^2*c*d^2*x^6*e^2 + 4*a*c^2*d^2*x^6*e
^2 + 16/5*b^2*c*d^3*x^5*e + 16/5*a*c^2*d^3*x^5*e + b^2*c*d^4*x^4 + a*c^2*d^4*x^4 + 1/7*b^3*x^7*e^4 + 6/7*a*b*c
*x^7*e^4 + 2/3*b^3*d*x^6*e^3 + 4*a*b*c*d*x^6*e^3 + 6/5*b^3*d^2*x^5*e^2 + 36/5*a*b*c*d^2*x^5*e^2 + b^3*d^3*x^4*
e + 6*a*b*c*d^3*x^4*e + 1/3*b^3*d^4*x^3 + 2*a*b*c*d^4*x^3 + 1/3*a*b^2*x^6*e^4 + 1/3*a^2*c*x^6*e^4 + 8/5*a*b^2*
d*x^5*e^3 + 8/5*a^2*c*d*x^5*e^3 + 3*a*b^2*d^2*x^4*e^2 + 3*a^2*c*d^2*x^4*e^2 + 8/3*a*b^2*d^3*x^3*e + 8/3*a^2*c*
d^3*x^3*e + a*b^2*d^4*x^2 + a^2*c*d^4*x^2 + 1/5*a^2*b*x^5*e^4 + a^2*b*d*x^4*e^3 + 2*a^2*b*d^2*x^3*e^2 + 2*a^2*
b*d^3*x^2*e + a^2*b*d^4*x