3.1711 \(\int (a+b x)^3 (c+d x)^3 (e+f x)^3 \, dx\)

Optimal. Leaf size=361 \[ \frac{3 d f (a+b x)^8 \left (5 a^2 d^2 f^2-5 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )}{8 b^7}+\frac{(a+b x)^7 (-2 a d f+b c f+b d e) \left (10 a^2 d^2 f^2-10 a b d f (c f+d e)+b^2 \left (c^2 f^2+8 c d e f+d^2 e^2\right )\right )}{7 b^7}+\frac{(a+b x)^6 (b c-a d) (b e-a f) \left (5 a^2 d^2 f^2-5 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )}{2 b^7}+\frac{d^2 f^2 (a+b x)^9 (-2 a d f+b c f+b d e)}{3 b^7}+\frac{3 (a+b x)^5 (b c-a d)^2 (b e-a f)^2 (-2 a d f+b c f+b d e)}{5 b^7}+\frac{(a+b x)^4 (b c-a d)^3 (b e-a f)^3}{4 b^7}+\frac{d^3 f^3 (a+b x)^{10}}{10 b^7} \]

[Out]

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

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Rubi [A]  time = 0.729857, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ \frac{3 d f (a+b x)^8 \left (5 a^2 d^2 f^2-5 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )}{8 b^7}+\frac{(a+b x)^7 (-2 a d f+b c f+b d e) \left (10 a^2 d^2 f^2-10 a b d f (c f+d e)+b^2 \left (c^2 f^2+8 c d e f+d^2 e^2\right )\right )}{7 b^7}+\frac{(a+b x)^6 (b c-a d) (b e-a f) \left (5 a^2 d^2 f^2-5 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )}{2 b^7}+\frac{d^2 f^2 (a+b x)^9 (-2 a d f+b c f+b d e)}{3 b^7}+\frac{3 (a+b x)^5 (b c-a d)^2 (b e-a f)^2 (-2 a d f+b c f+b d e)}{5 b^7}+\frac{(a+b x)^4 (b c-a d)^3 (b e-a f)^3}{4 b^7}+\frac{d^3 f^3 (a+b x)^{10}}{10 b^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.230694, size = 653, normalized size = 1.81 \[ \frac{3}{8} b d f x^8 \left (a^2 d^2 f^2+3 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac{1}{7} x^7 \left (9 a^2 b d^2 f^2 (c f+d e)+a^3 d^3 f^3+9 a b^2 d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b^3 \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )\right )+\frac{1}{2} x^6 \left (3 a^2 b d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+a^3 d^2 f^2 (c f+d e)+a b^2 \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )+b^3 c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac{3}{5} x^5 \left (a^2 b \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )+a^3 d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+3 a b^2 c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b^3 c^2 e^2 (c f+d e)\right )+\frac{1}{4} x^4 \left (9 a^2 b c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )+a^3 \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )+9 a b^2 c^2 e^2 (c f+d e)+b^3 c^3 e^3\right )+a c e x^3 \left (a^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )+3 a b c e (c f+d e)+b^2 c^2 e^2\right )+\frac{3}{2} a^2 c^2 e^2 x^2 (a c f+a d e+b c e)+a^3 c^3 e^3 x+\frac{1}{3} b^2 d^2 f^2 x^9 (a d f+b c f+b d e)+\frac{1}{10} b^3 d^3 f^3 x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.002, size = 767, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.05834, size = 981, normalized size = 2.72 \begin{align*} \frac{1}{10} \, b^{3} d^{3} f^{3} x^{10} + a^{3} c^{3} e^{3} x + \frac{1}{3} \,{\left (b^{3} d^{3} e f^{2} +{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f^{3}\right )} x^{9} + \frac{3}{8} \,{\left (b^{3} d^{3} e^{2} f + 3 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e f^{2} +{\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{3}\right )} x^{8} + \frac{1}{7} \,{\left (b^{3} d^{3} e^{3} + 9 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e^{2} f + 9 \,{\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e f^{2} +{\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} f^{3}\right )} x^{7} + \frac{1}{2} \,{\left ({\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e^{3} + 3 \,{\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e^{2} f +{\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} e f^{2} +{\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} f^{3}\right )} x^{6} + \frac{3}{5} \,{\left ({\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e^{3} +{\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} e^{2} f + 3 \,{\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} e f^{2} +{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} f^{3}\right )} x^{5} + \frac{1}{4} \,{\left (a^{3} c^{3} f^{3} +{\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} e^{3} + 9 \,{\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} e^{2} f + 9 \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} e f^{2}\right )} x^{4} +{\left (a^{3} c^{3} e f^{2} +{\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} e^{3} + 3 \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} e^{2} f\right )} x^{3} + \frac{3}{2} \,{\left (a^{3} c^{3} e^{2} f +{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} e^{3}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.40369, size = 2136, normalized size = 5.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*f^3*d^3*b^3 + 1/3*x^9*f^2*e*d^3*b^3 + 1/3*x^9*f^3*d^2*c*b^3 + 1/3*x^9*f^3*d^3*b^2*a + 3/8*x^8*f*e^2*
d^3*b^3 + 9/8*x^8*f^2*e*d^2*c*b^3 + 3/8*x^8*f^3*d*c^2*b^3 + 9/8*x^8*f^2*e*d^3*b^2*a + 9/8*x^8*f^3*d^2*c*b^2*a
+ 3/8*x^8*f^3*d^3*b*a^2 + 1/7*x^7*e^3*d^3*b^3 + 9/7*x^7*f*e^2*d^2*c*b^3 + 9/7*x^7*f^2*e*d*c^2*b^3 + 1/7*x^7*f^
3*c^3*b^3 + 9/7*x^7*f*e^2*d^3*b^2*a + 27/7*x^7*f^2*e*d^2*c*b^2*a + 9/7*x^7*f^3*d*c^2*b^2*a + 9/7*x^7*f^2*e*d^3
*b*a^2 + 9/7*x^7*f^3*d^2*c*b*a^2 + 1/7*x^7*f^3*d^3*a^3 + 1/2*x^6*e^3*d^2*c*b^3 + 3/2*x^6*f*e^2*d*c^2*b^3 + 1/2
*x^6*f^2*e*c^3*b^3 + 1/2*x^6*e^3*d^3*b^2*a + 9/2*x^6*f*e^2*d^2*c*b^2*a + 9/2*x^6*f^2*e*d*c^2*b^2*a + 1/2*x^6*f
^3*c^3*b^2*a + 3/2*x^6*f*e^2*d^3*b*a^2 + 9/2*x^6*f^2*e*d^2*c*b*a^2 + 3/2*x^6*f^3*d*c^2*b*a^2 + 1/2*x^6*f^2*e*d
^3*a^3 + 1/2*x^6*f^3*d^2*c*a^3 + 3/5*x^5*e^3*d*c^2*b^3 + 3/5*x^5*f*e^2*c^3*b^3 + 9/5*x^5*e^3*d^2*c*b^2*a + 27/
5*x^5*f*e^2*d*c^2*b^2*a + 9/5*x^5*f^2*e*c^3*b^2*a + 3/5*x^5*e^3*d^3*b*a^2 + 27/5*x^5*f*e^2*d^2*c*b*a^2 + 27/5*
x^5*f^2*e*d*c^2*b*a^2 + 3/5*x^5*f^3*c^3*b*a^2 + 3/5*x^5*f*e^2*d^3*a^3 + 9/5*x^5*f^2*e*d^2*c*a^3 + 3/5*x^5*f^3*
d*c^2*a^3 + 1/4*x^4*e^3*c^3*b^3 + 9/4*x^4*e^3*d*c^2*b^2*a + 9/4*x^4*f*e^2*c^3*b^2*a + 9/4*x^4*e^3*d^2*c*b*a^2
+ 27/4*x^4*f*e^2*d*c^2*b*a^2 + 9/4*x^4*f^2*e*c^3*b*a^2 + 1/4*x^4*e^3*d^3*a^3 + 9/4*x^4*f*e^2*d^2*c*a^3 + 9/4*x
^4*f^2*e*d*c^2*a^3 + 1/4*x^4*f^3*c^3*a^3 + x^3*e^3*c^3*b^2*a + 3*x^3*e^3*d*c^2*b*a^2 + 3*x^3*f*e^2*c^3*b*a^2 +
x^3*e^3*d^2*c*a^3 + 3*x^3*f*e^2*d*c^2*a^3 + x^3*f^2*e*c^3*a^3 + 3/2*x^2*e^3*c^3*b*a^2 + 3/2*x^2*e^3*d*c^2*a^3
+ 3/2*x^2*f*e^2*c^3*a^3 + x*e^3*c^3*a^3

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Sympy [B]  time = 0.160139, size = 1018, normalized size = 2.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*c**3*e**3*x + b**3*d**3*f**3*x**10/10 + x**9*(a*b**2*d**3*f**3/3 + b**3*c*d**2*f**3/3 + b**3*d**3*e*f**2/
3) + x**8*(3*a**2*b*d**3*f**3/8 + 9*a*b**2*c*d**2*f**3/8 + 9*a*b**2*d**3*e*f**2/8 + 3*b**3*c**2*d*f**3/8 + 9*b
**3*c*d**2*e*f**2/8 + 3*b**3*d**3*e**2*f/8) + x**7*(a**3*d**3*f**3/7 + 9*a**2*b*c*d**2*f**3/7 + 9*a**2*b*d**3*
e*f**2/7 + 9*a*b**2*c**2*d*f**3/7 + 27*a*b**2*c*d**2*e*f**2/7 + 9*a*b**2*d**3*e**2*f/7 + b**3*c**3*f**3/7 + 9*
b**3*c**2*d*e*f**2/7 + 9*b**3*c*d**2*e**2*f/7 + b**3*d**3*e**3/7) + x**6*(a**3*c*d**2*f**3/2 + a**3*d**3*e*f**
2/2 + 3*a**2*b*c**2*d*f**3/2 + 9*a**2*b*c*d**2*e*f**2/2 + 3*a**2*b*d**3*e**2*f/2 + a*b**2*c**3*f**3/2 + 9*a*b*
*2*c**2*d*e*f**2/2 + 9*a*b**2*c*d**2*e**2*f/2 + a*b**2*d**3*e**3/2 + b**3*c**3*e*f**2/2 + 3*b**3*c**2*d*e**2*f
/2 + b**3*c*d**2*e**3/2) + x**5*(3*a**3*c**2*d*f**3/5 + 9*a**3*c*d**2*e*f**2/5 + 3*a**3*d**3*e**2*f/5 + 3*a**2
*b*c**3*f**3/5 + 27*a**2*b*c**2*d*e*f**2/5 + 27*a**2*b*c*d**2*e**2*f/5 + 3*a**2*b*d**3*e**3/5 + 9*a*b**2*c**3*
e*f**2/5 + 27*a*b**2*c**2*d*e**2*f/5 + 9*a*b**2*c*d**2*e**3/5 + 3*b**3*c**3*e**2*f/5 + 3*b**3*c**2*d*e**3/5) +
x**4*(a**3*c**3*f**3/4 + 9*a**3*c**2*d*e*f**2/4 + 9*a**3*c*d**2*e**2*f/4 + a**3*d**3*e**3/4 + 9*a**2*b*c**3*e
*f**2/4 + 27*a**2*b*c**2*d*e**2*f/4 + 9*a**2*b*c*d**2*e**3/4 + 9*a*b**2*c**3*e**2*f/4 + 9*a*b**2*c**2*d*e**3/4
+ b**3*c**3*e**3/4) + x**3*(a**3*c**3*e*f**2 + 3*a**3*c**2*d*e**2*f + a**3*c*d**2*e**3 + 3*a**2*b*c**3*e**2*f
+ 3*a**2*b*c**2*d*e**3 + a*b**2*c**3*e**3) + x**2*(3*a**3*c**3*e**2*f/2 + 3*a**3*c**2*d*e**3/2 + 3*a**2*b*c**
3*e**3/2)

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Giac [B]  time = 1.9217, size = 1311, normalized size = 3.63 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^3*d^3*f^3*x^10 + 1/3*b^3*c*d^2*f^3*x^9 + 1/3*a*b^2*d^3*f^3*x^9 + 1/3*b^3*d^3*f^2*x^9*e + 3/8*b^3*c^2*d*
f^3*x^8 + 9/8*a*b^2*c*d^2*f^3*x^8 + 3/8*a^2*b*d^3*f^3*x^8 + 9/8*b^3*c*d^2*f^2*x^8*e + 9/8*a*b^2*d^3*f^2*x^8*e
+ 1/7*b^3*c^3*f^3*x^7 + 9/7*a*b^2*c^2*d*f^3*x^7 + 9/7*a^2*b*c*d^2*f^3*x^7 + 1/7*a^3*d^3*f^3*x^7 + 3/8*b^3*d^3*
f*x^8*e^2 + 9/7*b^3*c^2*d*f^2*x^7*e + 27/7*a*b^2*c*d^2*f^2*x^7*e + 9/7*a^2*b*d^3*f^2*x^7*e + 1/2*a*b^2*c^3*f^3
*x^6 + 3/2*a^2*b*c^2*d*f^3*x^6 + 1/2*a^3*c*d^2*f^3*x^6 + 9/7*b^3*c*d^2*f*x^7*e^2 + 9/7*a*b^2*d^3*f*x^7*e^2 + 1
/2*b^3*c^3*f^2*x^6*e + 9/2*a*b^2*c^2*d*f^2*x^6*e + 9/2*a^2*b*c*d^2*f^2*x^6*e + 1/2*a^3*d^3*f^2*x^6*e + 3/5*a^2
*b*c^3*f^3*x^5 + 3/5*a^3*c^2*d*f^3*x^5 + 1/7*b^3*d^3*x^7*e^3 + 3/2*b^3*c^2*d*f*x^6*e^2 + 9/2*a*b^2*c*d^2*f*x^6
*e^2 + 3/2*a^2*b*d^3*f*x^6*e^2 + 9/5*a*b^2*c^3*f^2*x^5*e + 27/5*a^2*b*c^2*d*f^2*x^5*e + 9/5*a^3*c*d^2*f^2*x^5*
e + 1/4*a^3*c^3*f^3*x^4 + 1/2*b^3*c*d^2*x^6*e^3 + 1/2*a*b^2*d^3*x^6*e^3 + 3/5*b^3*c^3*f*x^5*e^2 + 27/5*a*b^2*c
^2*d*f*x^5*e^2 + 27/5*a^2*b*c*d^2*f*x^5*e^2 + 3/5*a^3*d^3*f*x^5*e^2 + 9/4*a^2*b*c^3*f^2*x^4*e + 9/4*a^3*c^2*d*
f^2*x^4*e + 3/5*b^3*c^2*d*x^5*e^3 + 9/5*a*b^2*c*d^2*x^5*e^3 + 3/5*a^2*b*d^3*x^5*e^3 + 9/4*a*b^2*c^3*f*x^4*e^2
+ 27/4*a^2*b*c^2*d*f*x^4*e^2 + 9/4*a^3*c*d^2*f*x^4*e^2 + a^3*c^3*f^2*x^3*e + 1/4*b^3*c^3*x^4*e^3 + 9/4*a*b^2*c
^2*d*x^4*e^3 + 9/4*a^2*b*c*d^2*x^4*e^3 + 1/4*a^3*d^3*x^4*e^3 + 3*a^2*b*c^3*f*x^3*e^2 + 3*a^3*c^2*d*f*x^3*e^2 +
a*b^2*c^3*x^3*e^3 + 3*a^2*b*c^2*d*x^3*e^3 + a^3*c*d^2*x^3*e^3 + 3/2*a^3*c^3*f*x^2*e^2 + 3/2*a^2*b*c^3*x^2*e^3
+ 3/2*a^3*c^2*d*x^2*e^3 + a^3*c^3*x*e^3