3.1516 \(\int (b+2 c x) (d+e x)^2 (a+b x+c x^2)^3 \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.454805, 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} \[ \frac{(d+e x)^6 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{6 e^8}+\frac{3 c^2 (d+e x)^8 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{8 e^8}-\frac{5 c (d+e x)^7 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}-\frac{3 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^8}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{4 e^8}-\frac{(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8}-\frac{7 c^3 (d+e x)^9 (2 c d-b e)}{9 e^8}+\frac{c^4 (d+e x)^{10}}{5 e^8} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.155048, size = 413, normalized size = 1. \[ \frac{1}{5} x^5 \left (12 a^2 c^2 d e+b^3 \left (3 a e^2+5 c d^2\right )+24 a b^2 c d e+3 a b c \left (3 a e^2+5 c d^2\right )+2 b^4 d e\right )+\frac{1}{4} x^4 \left (18 a^2 b c d e+2 a^2 c \left (a e^2+3 c d^2\right )+3 a b^2 \left (a e^2+4 c d^2\right )+6 a b^3 d e+b^4 d^2\right )+\frac{1}{3} a x^3 \left (4 a^2 c d e+6 a b^2 d e+a b \left (a e^2+9 c d^2\right )+3 b^3 d^2\right )+\frac{1}{2} a^2 d x^2 \left (2 a b e+2 a c d+3 b^2 d\right )+a^3 b d^2 x+\frac{1}{8} c^2 x^8 \left (2 c e (3 a e+7 b d)+9 b^2 e^2+2 c^2 d^2\right )+\frac{1}{7} c x^7 \left (b c \left (15 a e^2+7 c d^2\right )+12 a c^2 d e+18 b^2 c d e+5 b^3 e^2\right )+\frac{1}{6} x^6 \left (3 b^2 c \left (4 a e^2+3 c d^2\right )+30 a b c^2 d e+6 a c^2 \left (a e^2+c d^2\right )+10 b^3 c d e+b^4 e^2\right )+\frac{1}{9} c^3 e x^9 (7 b e+4 c d)+\frac{1}{5} c^4 e^2 x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*b*d^2*x + (a^2*d*(3*b^2*d + 2*a*c*d + 2*a*b*e)*x^2)/2 + (a*(3*b^3*d^2 + 6*a*b^2*d*e + 4*a^2*c*d*e + a*b*(9
*c*d^2 + a*e^2))*x^3)/3 + ((b^4*d^2 + 6*a*b^3*d*e + 18*a^2*b*c*d*e + 2*a^2*c*(3*c*d^2 + a*e^2) + 3*a*b^2*(4*c*
d^2 + a*e^2))*x^4)/4 + ((2*b^4*d*e + 24*a*b^2*c*d*e + 12*a^2*c^2*d*e + b^3*(5*c*d^2 + 3*a*e^2) + 3*a*b*c*(5*c*
d^2 + 3*a*e^2))*x^5)/5 + ((10*b^3*c*d*e + 30*a*b*c^2*d*e + b^4*e^2 + 6*a*c^2*(c*d^2 + a*e^2) + 3*b^2*c*(3*c*d^
2 + 4*a*e^2))*x^6)/6 + (c*(18*b^2*c*d*e + 12*a*c^2*d*e + 5*b^3*e^2 + b*c*(7*c*d^2 + 15*a*e^2))*x^7)/7 + (c^2*(
2*c^2*d^2 + 9*b^2*e^2 + 2*c*e*(7*b*d + 3*a*e))*x^8)/8 + (c^3*e*(4*c*d + 7*b*e)*x^9)/9 + (c^4*e^2*x^10)/5

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 608, normalized size = 1.5 \begin{align*}{\frac{{c}^{4}{e}^{2}{x}^{10}}{5}}+{\frac{ \left ( \left ( b{e}^{2}+4\,cde \right ){c}^{3}+6\,{c}^{3}{e}^{2}b \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 2\,bde+2\,c{d}^{2} \right ){c}^{3}+3\, \left ( b{e}^{2}+4\,cde \right ) b{c}^{2}+2\,c{e}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( b{d}^{2}{c}^{3}+3\, \left ( 2\,bde+2\,c{d}^{2} \right ) b{c}^{2}+ \left ( b{e}^{2}+4\,cde \right ) \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,c{e}^{2} \left ( b \left ( 2\,ac+{b}^{2} \right ) +4\,abc \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{b}^{2}{d}^{2}{c}^{2}+ \left ( 2\,bde+2\,c{d}^{2} \right ) \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( b{e}^{2}+4\,cde \right ) \left ( b \left ( 2\,ac+{b}^{2} \right ) +4\,abc \right ) +2\,c{e}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( b{d}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( 2\,bde+2\,c{d}^{2} \right ) \left ( b \left ( 2\,ac+{b}^{2} \right ) +4\,abc \right ) + \left ( b{e}^{2}+4\,cde \right ) \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +6\,c{e}^{2}{a}^{2}b \right ){x}^{5}}{5}}+{\frac{ \left ( b{d}^{2} \left ( b \left ( 2\,ac+{b}^{2} \right ) +4\,abc \right ) + \left ( 2\,bde+2\,c{d}^{2} \right ) \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\, \left ( b{e}^{2}+4\,cde \right ){a}^{2}b+2\,c{e}^{2}{a}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( b{d}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\, \left ( 2\,bde+2\,c{d}^{2} \right ){a}^{2}b+ \left ( b{e}^{2}+4\,cde \right ){a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{b}^{2}{d}^{2}{a}^{2}+ \left ( 2\,bde+2\,c{d}^{2} \right ){a}^{3} \right ){x}^{2}}{2}}+b{d}^{2}{a}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.01613, size = 563, normalized size = 1.37 \begin{align*} \frac{1}{5} \, c^{4} e^{2} x^{10} + \frac{1}{9} \,{\left (4 \, c^{4} d e + 7 \, b c^{3} e^{2}\right )} x^{9} + \frac{1}{8} \,{\left (2 \, c^{4} d^{2} + 14 \, b c^{3} d e + 3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left (7 \, b c^{3} d^{2} + 6 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e + 5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{2}\right )} x^{7} + a^{3} b d^{2} x + \frac{1}{6} \,{\left (3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} + 10 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d e +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} + 2 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e + 3 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} + 6 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d e +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (a^{3} b e^{2} + 3 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} + 2 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a^{3} b d e +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.29892, size = 1111, normalized size = 2.7 \begin{align*} \frac{1}{5} x^{10} e^{2} c^{4} + \frac{4}{9} x^{9} e d c^{4} + \frac{7}{9} x^{9} e^{2} c^{3} b + \frac{1}{4} x^{8} d^{2} c^{4} + \frac{7}{4} x^{8} e d c^{3} b + \frac{9}{8} x^{8} e^{2} c^{2} b^{2} + \frac{3}{4} x^{8} e^{2} c^{3} a + x^{7} d^{2} c^{3} b + \frac{18}{7} x^{7} e d c^{2} b^{2} + \frac{5}{7} x^{7} e^{2} c b^{3} + \frac{12}{7} x^{7} e d c^{3} a + \frac{15}{7} x^{7} e^{2} c^{2} b a + \frac{3}{2} x^{6} d^{2} c^{2} b^{2} + \frac{5}{3} x^{6} e d c b^{3} + \frac{1}{6} x^{6} e^{2} b^{4} + x^{6} d^{2} c^{3} a + 5 x^{6} e d c^{2} b a + 2 x^{6} e^{2} c b^{2} a + x^{6} e^{2} c^{2} a^{2} + x^{5} d^{2} c b^{3} + \frac{2}{5} x^{5} e d b^{4} + 3 x^{5} d^{2} c^{2} b a + \frac{24}{5} x^{5} e d c b^{2} a + \frac{3}{5} x^{5} e^{2} b^{3} a + \frac{12}{5} x^{5} e d c^{2} a^{2} + \frac{9}{5} x^{5} e^{2} c b a^{2} + \frac{1}{4} x^{4} d^{2} b^{4} + 3 x^{4} d^{2} c b^{2} a + \frac{3}{2} x^{4} e d b^{3} a + \frac{3}{2} x^{4} d^{2} c^{2} a^{2} + \frac{9}{2} x^{4} e d c b a^{2} + \frac{3}{4} x^{4} e^{2} b^{2} a^{2} + \frac{1}{2} x^{4} e^{2} c a^{3} + x^{3} d^{2} b^{3} a + 3 x^{3} d^{2} c b a^{2} + 2 x^{3} e d b^{2} a^{2} + \frac{4}{3} x^{3} e d c a^{3} + \frac{1}{3} x^{3} e^{2} b a^{3} + \frac{3}{2} x^{2} d^{2} b^{2} a^{2} + x^{2} d^{2} c a^{3} + x^{2} e d b a^{3} + x d^{2} b a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^10*e^2*c^4 + 4/9*x^9*e*d*c^4 + 7/9*x^9*e^2*c^3*b + 1/4*x^8*d^2*c^4 + 7/4*x^8*e*d*c^3*b + 9/8*x^8*e^2*c^2
*b^2 + 3/4*x^8*e^2*c^3*a + x^7*d^2*c^3*b + 18/7*x^7*e*d*c^2*b^2 + 5/7*x^7*e^2*c*b^3 + 12/7*x^7*e*d*c^3*a + 15/
7*x^7*e^2*c^2*b*a + 3/2*x^6*d^2*c^2*b^2 + 5/3*x^6*e*d*c*b^3 + 1/6*x^6*e^2*b^4 + x^6*d^2*c^3*a + 5*x^6*e*d*c^2*
b*a + 2*x^6*e^2*c*b^2*a + x^6*e^2*c^2*a^2 + x^5*d^2*c*b^3 + 2/5*x^5*e*d*b^4 + 3*x^5*d^2*c^2*b*a + 24/5*x^5*e*d
*c*b^2*a + 3/5*x^5*e^2*b^3*a + 12/5*x^5*e*d*c^2*a^2 + 9/5*x^5*e^2*c*b*a^2 + 1/4*x^4*d^2*b^4 + 3*x^4*d^2*c*b^2*
a + 3/2*x^4*e*d*b^3*a + 3/2*x^4*d^2*c^2*a^2 + 9/2*x^4*e*d*c*b*a^2 + 3/4*x^4*e^2*b^2*a^2 + 1/2*x^4*e^2*c*a^3 +
x^3*d^2*b^3*a + 3*x^3*d^2*c*b*a^2 + 2*x^3*e*d*b^2*a^2 + 4/3*x^3*e*d*c*a^3 + 1/3*x^3*e^2*b*a^3 + 3/2*x^2*d^2*b^
2*a^2 + x^2*d^2*c*a^3 + x^2*e*d*b*a^3 + x*d^2*b*a^3

________________________________________________________________________________________

Sympy [A]  time = 0.133632, size = 503, normalized size = 1.22 \begin{align*} a^{3} b d^{2} x + \frac{c^{4} e^{2} x^{10}}{5} + x^{9} \left (\frac{7 b c^{3} e^{2}}{9} + \frac{4 c^{4} d e}{9}\right ) + x^{8} \left (\frac{3 a c^{3} e^{2}}{4} + \frac{9 b^{2} c^{2} e^{2}}{8} + \frac{7 b c^{3} d e}{4} + \frac{c^{4} d^{2}}{4}\right ) + x^{7} \left (\frac{15 a b c^{2} e^{2}}{7} + \frac{12 a c^{3} d e}{7} + \frac{5 b^{3} c e^{2}}{7} + \frac{18 b^{2} c^{2} d e}{7} + b c^{3} d^{2}\right ) + x^{6} \left (a^{2} c^{2} e^{2} + 2 a b^{2} c e^{2} + 5 a b c^{2} d e + a c^{3} d^{2} + \frac{b^{4} e^{2}}{6} + \frac{5 b^{3} c d e}{3} + \frac{3 b^{2} c^{2} d^{2}}{2}\right ) + x^{5} \left (\frac{9 a^{2} b c e^{2}}{5} + \frac{12 a^{2} c^{2} d e}{5} + \frac{3 a b^{3} e^{2}}{5} + \frac{24 a b^{2} c d e}{5} + 3 a b c^{2} d^{2} + \frac{2 b^{4} d e}{5} + b^{3} c d^{2}\right ) + x^{4} \left (\frac{a^{3} c e^{2}}{2} + \frac{3 a^{2} b^{2} e^{2}}{4} + \frac{9 a^{2} b c d e}{2} + \frac{3 a^{2} c^{2} d^{2}}{2} + \frac{3 a b^{3} d e}{2} + 3 a b^{2} c d^{2} + \frac{b^{4} d^{2}}{4}\right ) + x^{3} \left (\frac{a^{3} b e^{2}}{3} + \frac{4 a^{3} c d e}{3} + 2 a^{2} b^{2} d e + 3 a^{2} b c d^{2} + a b^{3} d^{2}\right ) + x^{2} \left (a^{3} b d e + a^{3} c d^{2} + \frac{3 a^{2} b^{2} d^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.14753, size = 678, normalized size = 1.65 \begin{align*} \frac{1}{5} \, c^{4} x^{10} e^{2} + \frac{4}{9} \, c^{4} d x^{9} e + \frac{1}{4} \, c^{4} d^{2} x^{8} + \frac{7}{9} \, b c^{3} x^{9} e^{2} + \frac{7}{4} \, b c^{3} d x^{8} e + b c^{3} d^{2} x^{7} + \frac{9}{8} \, b^{2} c^{2} x^{8} e^{2} + \frac{3}{4} \, a c^{3} x^{8} e^{2} + \frac{18}{7} \, b^{2} c^{2} d x^{7} e + \frac{12}{7} \, a c^{3} d x^{7} e + \frac{3}{2} \, b^{2} c^{2} d^{2} x^{6} + a c^{3} d^{2} x^{6} + \frac{5}{7} \, b^{3} c x^{7} e^{2} + \frac{15}{7} \, a b c^{2} x^{7} e^{2} + \frac{5}{3} \, b^{3} c d x^{6} e + 5 \, a b c^{2} d x^{6} e + b^{3} c d^{2} x^{5} + 3 \, a b c^{2} d^{2} x^{5} + \frac{1}{6} \, b^{4} x^{6} e^{2} + 2 \, a b^{2} c x^{6} e^{2} + a^{2} c^{2} x^{6} e^{2} + \frac{2}{5} \, b^{4} d x^{5} e + \frac{24}{5} \, a b^{2} c d x^{5} e + \frac{12}{5} \, a^{2} c^{2} d x^{5} e + \frac{1}{4} \, b^{4} d^{2} x^{4} + 3 \, a b^{2} c d^{2} x^{4} + \frac{3}{2} \, a^{2} c^{2} d^{2} x^{4} + \frac{3}{5} \, a b^{3} x^{5} e^{2} + \frac{9}{5} \, a^{2} b c x^{5} e^{2} + \frac{3}{2} \, a b^{3} d x^{4} e + \frac{9}{2} \, a^{2} b c d x^{4} e + a b^{3} d^{2} x^{3} + 3 \, a^{2} b c d^{2} x^{3} + \frac{3}{4} \, a^{2} b^{2} x^{4} e^{2} + \frac{1}{2} \, a^{3} c x^{4} e^{2} + 2 \, a^{2} b^{2} d x^{3} e + \frac{4}{3} \, a^{3} c d x^{3} e + \frac{3}{2} \, a^{2} b^{2} d^{2} x^{2} + a^{3} c d^{2} x^{2} + \frac{1}{3} \, a^{3} b x^{3} e^{2} + a^{3} b d x^{2} e + a^{3} b d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^4*x^10*e^2 + 4/9*c^4*d*x^9*e + 1/4*c^4*d^2*x^8 + 7/9*b*c^3*x^9*e^2 + 7/4*b*c^3*d*x^8*e + b*c^3*d^2*x^7 +
9/8*b^2*c^2*x^8*e^2 + 3/4*a*c^3*x^8*e^2 + 18/7*b^2*c^2*d*x^7*e + 12/7*a*c^3*d*x^7*e + 3/2*b^2*c^2*d^2*x^6 + a
*c^3*d^2*x^6 + 5/7*b^3*c*x^7*e^2 + 15/7*a*b*c^2*x^7*e^2 + 5/3*b^3*c*d*x^6*e + 5*a*b*c^2*d*x^6*e + b^3*c*d^2*x^
5 + 3*a*b*c^2*d^2*x^5 + 1/6*b^4*x^6*e^2 + 2*a*b^2*c*x^6*e^2 + a^2*c^2*x^6*e^2 + 2/5*b^4*d*x^5*e + 24/5*a*b^2*c
*d*x^5*e + 12/5*a^2*c^2*d*x^5*e + 1/4*b^4*d^2*x^4 + 3*a*b^2*c*d^2*x^4 + 3/2*a^2*c^2*d^2*x^4 + 3/5*a*b^3*x^5*e^
2 + 9/5*a^2*b*c*x^5*e^2 + 3/2*a*b^3*d*x^4*e + 9/2*a^2*b*c*d*x^4*e + a*b^3*d^2*x^3 + 3*a^2*b*c*d^2*x^3 + 3/4*a^
2*b^2*x^4*e^2 + 1/2*a^3*c*x^4*e^2 + 2*a^2*b^2*d*x^3*e + 4/3*a^3*c*d*x^3*e + 3/2*a^2*b^2*d^2*x^2 + a^3*c*d^2*x^
2 + 1/3*a^3*b*x^3*e^2 + a^3*b*d*x^2*e + a^3*b*d^2*x