### 3.2132 $$\int (d+e x)^3 (a+b x+c x^2)^3 \, dx$$

Optimal. Leaf size=272 $\frac{3 c (d+e x)^8 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{8 e^7}-\frac{(d+e x)^7 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^7}+\frac{(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}-\frac{3 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )^3}{4 e^7}-\frac{c^2 (d+e x)^9 (2 c d-b e)}{3 e^7}+\frac{c^3 (d+e x)^{10}}{10 e^7}$

[Out]

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

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Rubi [A]  time = 0.367624, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {698} $\frac{3 c (d+e x)^8 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{8 e^7}-\frac{(d+e x)^7 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^7}+\frac{(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}-\frac{3 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )^3}{4 e^7}-\frac{c^2 (d+e x)^9 (2 c d-b e)}{3 e^7}+\frac{c^3 (d+e x)^{10}}{10 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.136436, size = 372, normalized size = 1.37 $\frac{1}{4} x^4 \left (a^2 e \left (a e^2+9 c d^2\right )+9 a b^2 d^2 e+3 a b d \left (3 a e^2+2 c d^2\right )+b^3 d^3\right )+\frac{3}{2} a^2 d^2 x^2 (a e+b d)+a^3 d^3 x+\frac{3}{8} c e x^8 \left (c e (a e+3 b d)+b^2 e^2+c^2 d^2\right )+\frac{1}{7} x^7 \left (9 c^2 d e (a e+b d)+3 b c e^2 (2 a e+3 b d)+b^3 e^3+c^3 d^3\right )+\frac{1}{2} x^6 \left (b^2 \left (a e^3+3 c d^2 e\right )+b c d \left (6 a e^2+c d^2\right )+a c e \left (a e^2+3 c d^2\right )+b^3 d e^2\right )+\frac{3}{5} x^5 \left (b^2 \left (3 a d e^2+c d^3\right )+a b e \left (a e^2+6 c d^2\right )+a c d \left (3 a e^2+c d^2\right )+b^3 d^2 e\right )+a d x^3 \left (3 a b d e+a \left (a e^2+c d^2\right )+b^2 d^2\right )+\frac{1}{3} c^2 e^2 x^9 (b e+c d)+\frac{1}{10} c^3 e^3 x^{10}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.039, size = 495, normalized size = 1.8 \begin{align*}{\frac{{c}^{3}{e}^{3}{x}^{10}}{10}}+{\frac{ \left ( 3\,{e}^{3}b{c}^{2}+3\,d{e}^{2}{c}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,{d}^{2}e{c}^{3}+9\,d{e}^{2}b{c}^{2}+{e}^{3} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{8}}{8}}+{\frac{ \left ({c}^{3}{d}^{3}+9\,b{c}^{2}{d}^{2}e+3\,d{e}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +{e}^{3} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{d}^{3}b{c}^{2}+3\,{d}^{2}e \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +3\,d{e}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +{e}^{3} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({d}^{3} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +3\,{d}^{2}e \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +3\,d{e}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +3\,{e}^{3}b{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({d}^{3} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +3\,{d}^{2}e \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +9\,d{e}^{2}{a}^{2}b+{a}^{3}{e}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{3} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +9\,{d}^{2}eb{a}^{2}+3\,d{e}^{2}{a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{3}+3\,{d}^{3}b{a}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{3}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0093, size = 495, normalized size = 1.82 \begin{align*} \frac{1}{10} \, c^{3} e^{3} x^{10} + \frac{1}{3} \,{\left (c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{9} + \frac{3}{8} \,{\left (c^{3} d^{2} e + 3 \, b c^{2} d e^{2} +{\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} +{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x^{7} + a^{3} d^{3} x + \frac{1}{2} \,{\left (b c^{2} d^{3} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e +{\left (b^{3} + 6 \, a b c\right )} d e^{2} +{\left (a b^{2} + a^{2} c\right )} e^{3}\right )} x^{6} + \frac{3}{5} \,{\left (a^{2} b e^{3} +{\left (b^{2} c + a c^{2}\right )} d^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{2} e + 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (9 \, a^{2} b d e^{2} + a^{3} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{3} + 9 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e\right )} x^{4} +{\left (3 \, a^{2} b d^{2} e + a^{3} d e^{2} +{\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{3} + \frac{3}{2} \,{\left (a^{2} b d^{3} + a^{3} d^{2} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.74845, size = 1075, normalized size = 3.95 \begin{align*} \frac{1}{10} x^{10} e^{3} c^{3} + \frac{1}{3} x^{9} e^{2} d c^{3} + \frac{1}{3} x^{9} e^{3} c^{2} b + \frac{3}{8} x^{8} e d^{2} c^{3} + \frac{9}{8} x^{8} e^{2} d c^{2} b + \frac{3}{8} x^{8} e^{3} c b^{2} + \frac{3}{8} x^{8} e^{3} c^{2} a + \frac{1}{7} x^{7} d^{3} c^{3} + \frac{9}{7} x^{7} e d^{2} c^{2} b + \frac{9}{7} x^{7} e^{2} d c b^{2} + \frac{1}{7} x^{7} e^{3} b^{3} + \frac{9}{7} x^{7} e^{2} d c^{2} a + \frac{6}{7} x^{7} e^{3} c b a + \frac{1}{2} x^{6} d^{3} c^{2} b + \frac{3}{2} x^{6} e d^{2} c b^{2} + \frac{1}{2} x^{6} e^{2} d b^{3} + \frac{3}{2} x^{6} e d^{2} c^{2} a + 3 x^{6} e^{2} d c b a + \frac{1}{2} x^{6} e^{3} b^{2} a + \frac{1}{2} x^{6} e^{3} c a^{2} + \frac{3}{5} x^{5} d^{3} c b^{2} + \frac{3}{5} x^{5} e d^{2} b^{3} + \frac{3}{5} x^{5} d^{3} c^{2} a + \frac{18}{5} x^{5} e d^{2} c b a + \frac{9}{5} x^{5} e^{2} d b^{2} a + \frac{9}{5} x^{5} e^{2} d c a^{2} + \frac{3}{5} x^{5} e^{3} b a^{2} + \frac{1}{4} x^{4} d^{3} b^{3} + \frac{3}{2} x^{4} d^{3} c b a + \frac{9}{4} x^{4} e d^{2} b^{2} a + \frac{9}{4} x^{4} e d^{2} c a^{2} + \frac{9}{4} x^{4} e^{2} d b a^{2} + \frac{1}{4} x^{4} e^{3} a^{3} + x^{3} d^{3} b^{2} a + x^{3} d^{3} c a^{2} + 3 x^{3} e d^{2} b a^{2} + x^{3} e^{2} d a^{3} + \frac{3}{2} x^{2} d^{3} b a^{2} + \frac{3}{2} x^{2} e d^{2} a^{3} + x d^{3} a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.141338, size = 484, normalized size = 1.78 \begin{align*} a^{3} d^{3} x + \frac{c^{3} e^{3} x^{10}}{10} + x^{9} \left (\frac{b c^{2} e^{3}}{3} + \frac{c^{3} d e^{2}}{3}\right ) + x^{8} \left (\frac{3 a c^{2} e^{3}}{8} + \frac{3 b^{2} c e^{3}}{8} + \frac{9 b c^{2} d e^{2}}{8} + \frac{3 c^{3} d^{2} e}{8}\right ) + x^{7} \left (\frac{6 a b c e^{3}}{7} + \frac{9 a c^{2} d e^{2}}{7} + \frac{b^{3} e^{3}}{7} + \frac{9 b^{2} c d e^{2}}{7} + \frac{9 b c^{2} d^{2} e}{7} + \frac{c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac{a^{2} c e^{3}}{2} + \frac{a b^{2} e^{3}}{2} + 3 a b c d e^{2} + \frac{3 a c^{2} d^{2} e}{2} + \frac{b^{3} d e^{2}}{2} + \frac{3 b^{2} c d^{2} e}{2} + \frac{b c^{2} d^{3}}{2}\right ) + x^{5} \left (\frac{3 a^{2} b e^{3}}{5} + \frac{9 a^{2} c d e^{2}}{5} + \frac{9 a b^{2} d e^{2}}{5} + \frac{18 a b c d^{2} e}{5} + \frac{3 a c^{2} d^{3}}{5} + \frac{3 b^{3} d^{2} e}{5} + \frac{3 b^{2} c d^{3}}{5}\right ) + x^{4} \left (\frac{a^{3} e^{3}}{4} + \frac{9 a^{2} b d e^{2}}{4} + \frac{9 a^{2} c d^{2} e}{4} + \frac{9 a b^{2} d^{2} e}{4} + \frac{3 a b c d^{3}}{2} + \frac{b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{3} d e^{2} + 3 a^{2} b d^{2} e + a^{2} c d^{3} + a b^{2} d^{3}\right ) + x^{2} \left (\frac{3 a^{3} d^{2} e}{2} + \frac{3 a^{2} b d^{3}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13292, size = 633, normalized size = 2.33 \begin{align*} \frac{1}{10} \, c^{3} x^{10} e^{3} + \frac{1}{3} \, c^{3} d x^{9} e^{2} + \frac{3}{8} \, c^{3} d^{2} x^{8} e + \frac{1}{7} \, c^{3} d^{3} x^{7} + \frac{1}{3} \, b c^{2} x^{9} e^{3} + \frac{9}{8} \, b c^{2} d x^{8} e^{2} + \frac{9}{7} \, b c^{2} d^{2} x^{7} e + \frac{1}{2} \, b c^{2} d^{3} x^{6} + \frac{3}{8} \, b^{2} c x^{8} e^{3} + \frac{3}{8} \, a c^{2} x^{8} e^{3} + \frac{9}{7} \, b^{2} c d x^{7} e^{2} + \frac{9}{7} \, a c^{2} d x^{7} e^{2} + \frac{3}{2} \, b^{2} c d^{2} x^{6} e + \frac{3}{2} \, a c^{2} d^{2} x^{6} e + \frac{3}{5} \, b^{2} c d^{3} x^{5} + \frac{3}{5} \, a c^{2} d^{3} x^{5} + \frac{1}{7} \, b^{3} x^{7} e^{3} + \frac{6}{7} \, a b c x^{7} e^{3} + \frac{1}{2} \, b^{3} d x^{6} e^{2} + 3 \, a b c d x^{6} e^{2} + \frac{3}{5} \, b^{3} d^{2} x^{5} e + \frac{18}{5} \, a b c d^{2} x^{5} e + \frac{1}{4} \, b^{3} d^{3} x^{4} + \frac{3}{2} \, a b c d^{3} x^{4} + \frac{1}{2} \, a b^{2} x^{6} e^{3} + \frac{1}{2} \, a^{2} c x^{6} e^{3} + \frac{9}{5} \, a b^{2} d x^{5} e^{2} + \frac{9}{5} \, a^{2} c d x^{5} e^{2} + \frac{9}{4} \, a b^{2} d^{2} x^{4} e + \frac{9}{4} \, a^{2} c d^{2} x^{4} e + a b^{2} d^{3} x^{3} + a^{2} c d^{3} x^{3} + \frac{3}{5} \, a^{2} b x^{5} e^{3} + \frac{9}{4} \, a^{2} b d x^{4} e^{2} + 3 \, a^{2} b d^{2} x^{3} e + \frac{3}{2} \, a^{2} b d^{3} x^{2} + \frac{1}{4} \, a^{3} x^{4} e^{3} + a^{3} d x^{3} e^{2} + \frac{3}{2} \, a^{3} d^{2} x^{2} e + a^{3} d^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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