3.1515 \(\int (b+2 c x) (d+e x)^3 (a+b x+c x^2)^3 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.587502, 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} \[ \frac{(d+e x)^7 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{7 e^8}+\frac{c^2 (d+e x)^9 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8}-\frac{5 c (d+e x)^8 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{8 e^8}-\frac{(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^8}+\frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac{(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{4 e^8}-\frac{7 c^3 (d+e x)^{10} (2 c d-b e)}{10 e^8}+\frac{2 c^4 (d+e x)^{11}}{11 e^8} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^3*(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)^4)/(4*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)^5)/(5*e^8) - ((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)^6)/(2*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)^7)/(7*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)^8)/(8*e^8) + (c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*
b*d - a*e))*(d + e*x)^9)/(3*e^8) - (7*c^3*(2*c*d - b*e)*(d + e*x)^10)/(10*e^8) + (2*c^4*(d + e*x)^11)/(11*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)^3 \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)^3}{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)^4}{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)^5}{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)^6}{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)^7}{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)^8}{e^7}-\frac{7 c^3 (2 c d-b e) (d+e x)^9}{e^7}+\frac{2 c^4 (d+e x)^{10}}{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)^4}{4 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)^5}{5 e^8}-\frac{(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)^6}{2 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)^7}{7 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)^8}{8 e^8}+\frac{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}{3 e^8}-\frac{7 c^3 (2 c d-b e) (d+e x)^{10}}{10 e^8}+\frac{2 c^4 (d+e x)^{11}}{11 e^8}\\ \end{align*}

Mathematica [A]  time = 0.209281, size = 562, normalized size = 1.37 \[ \frac{1}{5} x^5 \left (2 a^2 c e \left (a e^2+9 c d^2\right )+b^3 \left (9 a d e^2+5 c d^3\right )+3 a b^2 e \left (a e^2+12 c d^2\right )+3 a b c d \left (9 a e^2+5 c d^2\right )+3 b^4 d^2 e\right )+\frac{1}{4} x^4 \left (a^2 b e \left (a e^2+27 c d^2\right )+6 a^2 c d \left (a e^2+c d^2\right )+3 a b^2 d \left (3 a e^2+4 c d^2\right )+9 a b^3 d^2 e+b^4 d^3\right )+a d x^3 \left (2 a^2 c d e+3 a b^2 d e+a b \left (a e^2+3 c d^2\right )+b^3 d^2\right )+\frac{1}{2} a^2 d^2 x^2 \left (3 a b e+2 a c d+3 b^2 d\right )+a^3 b d^3 x+\frac{1}{3} c^2 e x^9 \left (c e (2 a e+7 b d)+3 b^2 e^2+2 c^2 d^2\right )+\frac{1}{8} c x^8 \left (3 c^2 d e (6 a e+7 b d)+3 b c e^2 (5 a e+9 b d)+5 b^3 e^3+2 c^3 d^3\right )+\frac{1}{7} x^7 \left (3 b^2 c e \left (4 a e^2+9 c d^2\right )+b c^2 d \left (45 a e^2+7 c d^2\right )+6 a c^2 e \left (a e^2+3 c d^2\right )+15 b^3 c d e^2+b^4 e^3\right )+\frac{1}{2} x^6 \left (b^3 \left (a e^3+5 c d^2 e\right )+3 b^2 c d \left (4 a e^2+c d^2\right )+3 a b c e \left (a e^2+5 c d^2\right )+2 a c^2 d \left (3 a e^2+c d^2\right )+b^4 d e^2\right )+\frac{1}{10} c^3 e^2 x^{10} (7 b e+6 c d)+\frac{2}{11} c^4 e^3 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*b*d^3*x + (a^2*d^2*(3*b^2*d + 2*a*c*d + 3*a*b*e)*x^2)/2 + a*d*(b^3*d^2 + 3*a*b^2*d*e + 2*a^2*c*d*e + a*b*(
3*c*d^2 + a*e^2))*x^3 + ((b^4*d^3 + 9*a*b^3*d^2*e + 6*a^2*c*d*(c*d^2 + a*e^2) + a^2*b*e*(27*c*d^2 + a*e^2) + 3
*a*b^2*d*(4*c*d^2 + 3*a*e^2))*x^4)/4 + ((3*b^4*d^2*e + 2*a^2*c*e*(9*c*d^2 + a*e^2) + 3*a*b^2*e*(12*c*d^2 + a*e
^2) + 3*a*b*c*d*(5*c*d^2 + 9*a*e^2) + b^3*(5*c*d^3 + 9*a*d*e^2))*x^5)/5 + ((b^4*d*e^2 + 3*a*b*c*e*(5*c*d^2 + a
*e^2) + 2*a*c^2*d*(c*d^2 + 3*a*e^2) + 3*b^2*c*d*(c*d^2 + 4*a*e^2) + b^3*(5*c*d^2*e + a*e^3))*x^6)/2 + ((15*b^3
*c*d*e^2 + b^4*e^3 + 6*a*c^2*e*(3*c*d^2 + a*e^2) + 3*b^2*c*e*(9*c*d^2 + 4*a*e^2) + b*c^2*d*(7*c*d^2 + 45*a*e^2
))*x^7)/7 + (c*(2*c^3*d^3 + 5*b^3*e^3 + 3*b*c*e^2*(9*b*d + 5*a*e) + 3*c^2*d*e*(7*b*d + 6*a*e))*x^8)/8 + (c^2*e
*(2*c^2*d^2 + 3*b^2*e^2 + c*e*(7*b*d + 2*a*e))*x^9)/3 + (c^3*e^2*(6*c*d + 7*b*e)*x^10)/10 + (2*c^4*e^3*x^11)/1
1

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Maple [B]  time = 0., size = 830, normalized size = 2. \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)^3*(c*x^2+b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.00605, size = 763, normalized size = 1.86 \begin{align*} \frac{2}{11} \, c^{4} e^{3} x^{11} + \frac{1}{10} \,{\left (6 \, c^{4} d e^{2} + 7 \, b c^{3} e^{3}\right )} x^{10} + \frac{1}{3} \,{\left (2 \, c^{4} d^{2} e + 7 \, b c^{3} d e^{2} +{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{3}\right )} x^{9} + \frac{1}{8} \,{\left (2 \, c^{4} d^{3} + 21 \, b c^{3} d^{2} e + 9 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} + 5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} x^{8} + a^{3} b d^{3} x + \frac{1}{7} \,{\left (7 \, b c^{3} d^{3} + 9 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e + 15 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{2} +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} + 5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{2} +{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} + 3 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e + 9 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{2} +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (a^{3} b e^{3} +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} + 9 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e + 3 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{2}\right )} x^{4} +{\left (a^{3} b d e^{2} +{\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{3} b d^{2} e +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3}\right )} x^{2} \end{align*}

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)^3,x, algorithm="maxima")

[Out]

2/11*c^4*e^3*x^11 + 1/10*(6*c^4*d*e^2 + 7*b*c^3*e^3)*x^10 + 1/3*(2*c^4*d^2*e + 7*b*c^3*d*e^2 + (3*b^2*c^2 + 2*
a*c^3)*e^3)*x^9 + 1/8*(2*c^4*d^3 + 21*b*c^3*d^2*e + 9*(3*b^2*c^2 + 2*a*c^3)*d*e^2 + 5*(b^3*c + 3*a*b*c^2)*e^3)
*x^8 + a^3*b*d^3*x + 1/7*(7*b*c^3*d^3 + 9*(3*b^2*c^2 + 2*a*c^3)*d^2*e + 15*(b^3*c + 3*a*b*c^2)*d*e^2 + (b^4 +
12*a*b^2*c + 6*a^2*c^2)*e^3)*x^7 + 1/2*((3*b^2*c^2 + 2*a*c^3)*d^3 + 5*(b^3*c + 3*a*b*c^2)*d^2*e + (b^4 + 12*a*
b^2*c + 6*a^2*c^2)*d*e^2 + (a*b^3 + 3*a^2*b*c)*e^3)*x^6 + 1/5*(5*(b^3*c + 3*a*b*c^2)*d^3 + 3*(b^4 + 12*a*b^2*c
+ 6*a^2*c^2)*d^2*e + 9*(a*b^3 + 3*a^2*b*c)*d*e^2 + (3*a^2*b^2 + 2*a^3*c)*e^3)*x^5 + 1/4*(a^3*b*e^3 + (b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d^3 + 9*(a*b^3 + 3*a^2*b*c)*d^2*e + 3*(3*a^2*b^2 + 2*a^3*c)*d*e^2)*x^4 + (a^3*b*d*e^2 +
(a*b^3 + 3*a^2*b*c)*d^3 + (3*a^2*b^2 + 2*a^3*c)*d^2*e)*x^3 + 1/2*(3*a^3*b*d^2*e + (3*a^2*b^2 + 2*a^3*c)*d^3)*
x^2

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Fricas [A]  time = 1.26776, size = 1589, normalized size = 3.87 \begin{align*} \frac{2}{11} x^{11} e^{3} c^{4} + \frac{3}{5} x^{10} e^{2} d c^{4} + \frac{7}{10} x^{10} e^{3} c^{3} b + \frac{2}{3} x^{9} e d^{2} c^{4} + \frac{7}{3} x^{9} e^{2} d c^{3} b + x^{9} e^{3} c^{2} b^{2} + \frac{2}{3} x^{9} e^{3} c^{3} a + \frac{1}{4} x^{8} d^{3} c^{4} + \frac{21}{8} x^{8} e d^{2} c^{3} b + \frac{27}{8} x^{8} e^{2} d c^{2} b^{2} + \frac{5}{8} x^{8} e^{3} c b^{3} + \frac{9}{4} x^{8} e^{2} d c^{3} a + \frac{15}{8} x^{8} e^{3} c^{2} b a + x^{7} d^{3} c^{3} b + \frac{27}{7} x^{7} e d^{2} c^{2} b^{2} + \frac{15}{7} x^{7} e^{2} d c b^{3} + \frac{1}{7} x^{7} e^{3} b^{4} + \frac{18}{7} x^{7} e d^{2} c^{3} a + \frac{45}{7} x^{7} e^{2} d c^{2} b a + \frac{12}{7} x^{7} e^{3} c b^{2} a + \frac{6}{7} x^{7} e^{3} c^{2} a^{2} + \frac{3}{2} x^{6} d^{3} c^{2} b^{2} + \frac{5}{2} x^{6} e d^{2} c b^{3} + \frac{1}{2} x^{6} e^{2} d b^{4} + x^{6} d^{3} c^{3} a + \frac{15}{2} x^{6} e d^{2} c^{2} b a + 6 x^{6} e^{2} d c b^{2} a + \frac{1}{2} x^{6} e^{3} b^{3} a + 3 x^{6} e^{2} d c^{2} a^{2} + \frac{3}{2} x^{6} e^{3} c b a^{2} + x^{5} d^{3} c b^{3} + \frac{3}{5} x^{5} e d^{2} b^{4} + 3 x^{5} d^{3} c^{2} b a + \frac{36}{5} x^{5} e d^{2} c b^{2} a + \frac{9}{5} x^{5} e^{2} d b^{3} a + \frac{18}{5} x^{5} e d^{2} c^{2} a^{2} + \frac{27}{5} x^{5} e^{2} d c b a^{2} + \frac{3}{5} x^{5} e^{3} b^{2} a^{2} + \frac{2}{5} x^{5} e^{3} c a^{3} + \frac{1}{4} x^{4} d^{3} b^{4} + 3 x^{4} d^{3} c b^{2} a + \frac{9}{4} x^{4} e d^{2} b^{3} a + \frac{3}{2} x^{4} d^{3} c^{2} a^{2} + \frac{27}{4} x^{4} e d^{2} c b a^{2} + \frac{9}{4} x^{4} e^{2} d b^{2} a^{2} + \frac{3}{2} x^{4} e^{2} d c a^{3} + \frac{1}{4} x^{4} e^{3} b a^{3} + x^{3} d^{3} b^{3} a + 3 x^{3} d^{3} c b a^{2} + 3 x^{3} e d^{2} b^{2} a^{2} + 2 x^{3} e d^{2} c a^{3} + x^{3} e^{2} d b a^{3} + \frac{3}{2} x^{2} d^{3} b^{2} a^{2} + x^{2} d^{3} c a^{3} + \frac{3}{2} x^{2} e d^{2} b a^{3} + x d^{3} b a^{3} \end{align*}

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)^3,x, algorithm="fricas")

[Out]

2/11*x^11*e^3*c^4 + 3/5*x^10*e^2*d*c^4 + 7/10*x^10*e^3*c^3*b + 2/3*x^9*e*d^2*c^4 + 7/3*x^9*e^2*d*c^3*b + x^9*e
^3*c^2*b^2 + 2/3*x^9*e^3*c^3*a + 1/4*x^8*d^3*c^4 + 21/8*x^8*e*d^2*c^3*b + 27/8*x^8*e^2*d*c^2*b^2 + 5/8*x^8*e^3
*c*b^3 + 9/4*x^8*e^2*d*c^3*a + 15/8*x^8*e^3*c^2*b*a + x^7*d^3*c^3*b + 27/7*x^7*e*d^2*c^2*b^2 + 15/7*x^7*e^2*d*
c*b^3 + 1/7*x^7*e^3*b^4 + 18/7*x^7*e*d^2*c^3*a + 45/7*x^7*e^2*d*c^2*b*a + 12/7*x^7*e^3*c*b^2*a + 6/7*x^7*e^3*c
^2*a^2 + 3/2*x^6*d^3*c^2*b^2 + 5/2*x^6*e*d^2*c*b^3 + 1/2*x^6*e^2*d*b^4 + x^6*d^3*c^3*a + 15/2*x^6*e*d^2*c^2*b*
a + 6*x^6*e^2*d*c*b^2*a + 1/2*x^6*e^3*b^3*a + 3*x^6*e^2*d*c^2*a^2 + 3/2*x^6*e^3*c*b*a^2 + x^5*d^3*c*b^3 + 3/5*
x^5*e*d^2*b^4 + 3*x^5*d^3*c^2*b*a + 36/5*x^5*e*d^2*c*b^2*a + 9/5*x^5*e^2*d*b^3*a + 18/5*x^5*e*d^2*c^2*a^2 + 27
/5*x^5*e^2*d*c*b*a^2 + 3/5*x^5*e^3*b^2*a^2 + 2/5*x^5*e^3*c*a^3 + 1/4*x^4*d^3*b^4 + 3*x^4*d^3*c*b^2*a + 9/4*x^4
*e*d^2*b^3*a + 3/2*x^4*d^3*c^2*a^2 + 27/4*x^4*e*d^2*c*b*a^2 + 9/4*x^4*e^2*d*b^2*a^2 + 3/2*x^4*e^2*d*c*a^3 + 1/
4*x^4*e^3*b*a^3 + x^3*d^3*b^3*a + 3*x^3*d^3*c*b*a^2 + 3*x^3*e*d^2*b^2*a^2 + 2*x^3*e*d^2*c*a^3 + x^3*e^2*d*b*a^
3 + 3/2*x^2*d^3*b^2*a^2 + x^2*d^3*c*a^3 + 3/2*x^2*e*d^2*b*a^3 + x*d^3*b*a^3

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Sympy [A]  time = 0.15787, size = 726, normalized size = 1.77 \begin{align*} a^{3} b d^{3} x + \frac{2 c^{4} e^{3} x^{11}}{11} + x^{10} \left (\frac{7 b c^{3} e^{3}}{10} + \frac{3 c^{4} d e^{2}}{5}\right ) + x^{9} \left (\frac{2 a c^{3} e^{3}}{3} + b^{2} c^{2} e^{3} + \frac{7 b c^{3} d e^{2}}{3} + \frac{2 c^{4} d^{2} e}{3}\right ) + x^{8} \left (\frac{15 a b c^{2} e^{3}}{8} + \frac{9 a c^{3} d e^{2}}{4} + \frac{5 b^{3} c e^{3}}{8} + \frac{27 b^{2} c^{2} d e^{2}}{8} + \frac{21 b c^{3} d^{2} e}{8} + \frac{c^{4} d^{3}}{4}\right ) + x^{7} \left (\frac{6 a^{2} c^{2} e^{3}}{7} + \frac{12 a b^{2} c e^{3}}{7} + \frac{45 a b c^{2} d e^{2}}{7} + \frac{18 a c^{3} d^{2} e}{7} + \frac{b^{4} e^{3}}{7} + \frac{15 b^{3} c d e^{2}}{7} + \frac{27 b^{2} c^{2} d^{2} e}{7} + b c^{3} d^{3}\right ) + x^{6} \left (\frac{3 a^{2} b c e^{3}}{2} + 3 a^{2} c^{2} d e^{2} + \frac{a b^{3} e^{3}}{2} + 6 a b^{2} c d e^{2} + \frac{15 a b c^{2} d^{2} e}{2} + a c^{3} d^{3} + \frac{b^{4} d e^{2}}{2} + \frac{5 b^{3} c d^{2} e}{2} + \frac{3 b^{2} c^{2} d^{3}}{2}\right ) + x^{5} \left (\frac{2 a^{3} c e^{3}}{5} + \frac{3 a^{2} b^{2} e^{3}}{5} + \frac{27 a^{2} b c d e^{2}}{5} + \frac{18 a^{2} c^{2} d^{2} e}{5} + \frac{9 a b^{3} d e^{2}}{5} + \frac{36 a b^{2} c d^{2} e}{5} + 3 a b c^{2} d^{3} + \frac{3 b^{4} d^{2} e}{5} + b^{3} c d^{3}\right ) + x^{4} \left (\frac{a^{3} b e^{3}}{4} + \frac{3 a^{3} c d e^{2}}{2} + \frac{9 a^{2} b^{2} d e^{2}}{4} + \frac{27 a^{2} b c d^{2} e}{4} + \frac{3 a^{2} c^{2} d^{3}}{2} + \frac{9 a b^{3} d^{2} e}{4} + 3 a b^{2} c d^{3} + \frac{b^{4} d^{3}}{4}\right ) + x^{3} \left (a^{3} b d e^{2} + 2 a^{3} c d^{2} e + 3 a^{2} b^{2} d^{2} e + 3 a^{2} b c d^{3} + a b^{3} d^{3}\right ) + x^{2} \left (\frac{3 a^{3} b d^{2} e}{2} + a^{3} c d^{3} + \frac{3 a^{2} b^{2} d^{3}}{2}\right ) \end{align*}

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)**3,x)

[Out]

a**3*b*d**3*x + 2*c**4*e**3*x**11/11 + x**10*(7*b*c**3*e**3/10 + 3*c**4*d*e**2/5) + x**9*(2*a*c**3*e**3/3 + b*
*2*c**2*e**3 + 7*b*c**3*d*e**2/3 + 2*c**4*d**2*e/3) + x**8*(15*a*b*c**2*e**3/8 + 9*a*c**3*d*e**2/4 + 5*b**3*c*
e**3/8 + 27*b**2*c**2*d*e**2/8 + 21*b*c**3*d**2*e/8 + c**4*d**3/4) + x**7*(6*a**2*c**2*e**3/7 + 12*a*b**2*c*e*
*3/7 + 45*a*b*c**2*d*e**2/7 + 18*a*c**3*d**2*e/7 + b**4*e**3/7 + 15*b**3*c*d*e**2/7 + 27*b**2*c**2*d**2*e/7 +
b*c**3*d**3) + x**6*(3*a**2*b*c*e**3/2 + 3*a**2*c**2*d*e**2 + a*b**3*e**3/2 + 6*a*b**2*c*d*e**2 + 15*a*b*c**2*
d**2*e/2 + a*c**3*d**3 + b**4*d*e**2/2 + 5*b**3*c*d**2*e/2 + 3*b**2*c**2*d**3/2) + x**5*(2*a**3*c*e**3/5 + 3*a
**2*b**2*e**3/5 + 27*a**2*b*c*d*e**2/5 + 18*a**2*c**2*d**2*e/5 + 9*a*b**3*d*e**2/5 + 36*a*b**2*c*d**2*e/5 + 3*
a*b*c**2*d**3 + 3*b**4*d**2*e/5 + b**3*c*d**3) + x**4*(a**3*b*e**3/4 + 3*a**3*c*d*e**2/2 + 9*a**2*b**2*d*e**2/
4 + 27*a**2*b*c*d**2*e/4 + 3*a**2*c**2*d**3/2 + 9*a*b**3*d**2*e/4 + 3*a*b**2*c*d**3 + b**4*d**3/4) + x**3*(a**
3*b*d*e**2 + 2*a**3*c*d**2*e + 3*a**2*b**2*d**2*e + 3*a**2*b*c*d**3 + a*b**3*d**3) + x**2*(3*a**3*b*d**2*e/2 +
a**3*c*d**3 + 3*a**2*b**2*d**3/2)

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Giac [A]  time = 1.19703, size = 952, normalized size = 2.32 \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)^3*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

2/11*c^4*x^11*e^3 + 3/5*c^4*d*x^10*e^2 + 2/3*c^4*d^2*x^9*e + 1/4*c^4*d^3*x^8 + 7/10*b*c^3*x^10*e^3 + 7/3*b*c^3
*d*x^9*e^2 + 21/8*b*c^3*d^2*x^8*e + b*c^3*d^3*x^7 + b^2*c^2*x^9*e^3 + 2/3*a*c^3*x^9*e^3 + 27/8*b^2*c^2*d*x^8*e
^2 + 9/4*a*c^3*d*x^8*e^2 + 27/7*b^2*c^2*d^2*x^7*e + 18/7*a*c^3*d^2*x^7*e + 3/2*b^2*c^2*d^3*x^6 + a*c^3*d^3*x^6
+ 5/8*b^3*c*x^8*e^3 + 15/8*a*b*c^2*x^8*e^3 + 15/7*b^3*c*d*x^7*e^2 + 45/7*a*b*c^2*d*x^7*e^2 + 5/2*b^3*c*d^2*x^
6*e + 15/2*a*b*c^2*d^2*x^6*e + b^3*c*d^3*x^5 + 3*a*b*c^2*d^3*x^5 + 1/7*b^4*x^7*e^3 + 12/7*a*b^2*c*x^7*e^3 + 6/
7*a^2*c^2*x^7*e^3 + 1/2*b^4*d*x^6*e^2 + 6*a*b^2*c*d*x^6*e^2 + 3*a^2*c^2*d*x^6*e^2 + 3/5*b^4*d^2*x^5*e + 36/5*a
*b^2*c*d^2*x^5*e + 18/5*a^2*c^2*d^2*x^5*e + 1/4*b^4*d^3*x^4 + 3*a*b^2*c*d^3*x^4 + 3/2*a^2*c^2*d^3*x^4 + 1/2*a*
b^3*x^6*e^3 + 3/2*a^2*b*c*x^6*e^3 + 9/5*a*b^3*d*x^5*e^2 + 27/5*a^2*b*c*d*x^5*e^2 + 9/4*a*b^3*d^2*x^4*e + 27/4*
a^2*b*c*d^2*x^4*e + a*b^3*d^3*x^3 + 3*a^2*b*c*d^3*x^3 + 3/5*a^2*b^2*x^5*e^3 + 2/5*a^3*c*x^5*e^3 + 9/4*a^2*b^2*
d*x^4*e^2 + 3/2*a^3*c*d*x^4*e^2 + 3*a^2*b^2*d^2*x^3*e + 2*a^3*c*d^2*x^3*e + 3/2*a^2*b^2*d^3*x^2 + a^3*c*d^3*x^
2 + 1/4*a^3*b*x^4*e^3 + a^3*b*d*x^3*e^2 + 3/2*a^3*b*d^2*x^2*e + a^3*b*d^3*x