### 3.1851 $$\int (d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^3 \, dx$$

Optimal. Leaf size=111 $-\frac{3 c^2 d^2 (d+e x)^8 \left (c d^2-a e^2\right )}{8 e^4}+\frac{3 c d (d+e x)^7 \left (c d^2-a e^2\right )^2}{7 e^4}-\frac{(d+e x)^6 \left (c d^2-a e^2\right )^3}{6 e^4}+\frac{c^3 d^3 (d+e x)^9}{9 e^4}$

[Out]

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

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Rubi [A]  time = 0.233257, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.057, Rules used = {626, 43} $-\frac{3 c^2 d^2 (d+e x)^8 \left (c d^2-a e^2\right )}{8 e^4}+\frac{3 c d (d+e x)^7 \left (c d^2-a e^2\right )^2}{7 e^4}-\frac{(d+e x)^6 \left (c d^2-a e^2\right )^3}{6 e^4}+\frac{c^3 d^3 (d+e x)^9}{9 e^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 626

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [B]  time = 0.0729191, size = 255, normalized size = 2.3 $\frac{1}{504} x \left (36 a^2 c d e^2 x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )+84 a^3 e^3 \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+9 a c^2 d^2 e x^2 \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )+c^3 d^3 x^3 \left (840 d^3 e^2 x^2+720 d^2 e^3 x^3+504 d^4 e x+126 d^5+315 d e^4 x^4+56 e^5 x^5\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(84*a^3*e^3*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + 36*a^2*c*d*e^2
*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 9*a*c^2*d^2*e*x^2*(56
*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + c^3*d^3*x^3*(126*d^5 +
504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5)))/504

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Maple [B]  time = 0.04, size = 801, normalized size = 7.2 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.999495, size = 409, normalized size = 3.68 \begin{align*} \frac{1}{9} \, c^{3} d^{3} e^{5} x^{9} + a^{3} d^{5} e^{3} x + \frac{1}{8} \,{\left (5 \, c^{3} d^{4} e^{4} + 3 \, a c^{2} d^{2} e^{6}\right )} x^{8} + \frac{1}{7} \,{\left (10 \, c^{3} d^{5} e^{3} + 15 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{7} + \frac{1}{6} \,{\left (10 \, c^{3} d^{6} e^{2} + 30 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{6} +{\left (c^{3} d^{7} e + 6 \, a c^{2} d^{5} e^{3} + 6 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x^{5} + \frac{1}{4} \,{\left (c^{3} d^{8} + 15 \, a c^{2} d^{6} e^{2} + 30 \, a^{2} c d^{4} e^{4} + 10 \, a^{3} d^{2} e^{6}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a c^{2} d^{7} e + 15 \, a^{2} c d^{5} e^{3} + 10 \, a^{3} d^{3} e^{5}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{6} e^{2} + 5 \, a^{3} d^{4} e^{4}\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^3*d^3*e^5*x^9 + a^3*d^5*e^3*x + 1/8*(5*c^3*d^4*e^4 + 3*a*c^2*d^2*e^6)*x^8 + 1/7*(10*c^3*d^5*e^3 + 15*a*c
^2*d^3*e^5 + 3*a^2*c*d*e^7)*x^7 + 1/6*(10*c^3*d^6*e^2 + 30*a*c^2*d^4*e^4 + 15*a^2*c*d^2*e^6 + a^3*e^8)*x^6 + (
c^3*d^7*e + 6*a*c^2*d^5*e^3 + 6*a^2*c*d^3*e^5 + a^3*d*e^7)*x^5 + 1/4*(c^3*d^8 + 15*a*c^2*d^6*e^2 + 30*a^2*c*d^
4*e^4 + 10*a^3*d^2*e^6)*x^4 + 1/3*(3*a*c^2*d^7*e + 15*a^2*c*d^5*e^3 + 10*a^3*d^3*e^5)*x^3 + 1/2*(3*a^2*c*d^6*e
^2 + 5*a^3*d^4*e^4)*x^2

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Fricas [B]  time = 1.37211, size = 694, normalized size = 6.25 \begin{align*} \frac{1}{9} x^{9} e^{5} d^{3} c^{3} + \frac{5}{8} x^{8} e^{4} d^{4} c^{3} + \frac{3}{8} x^{8} e^{6} d^{2} c^{2} a + \frac{10}{7} x^{7} e^{3} d^{5} c^{3} + \frac{15}{7} x^{7} e^{5} d^{3} c^{2} a + \frac{3}{7} x^{7} e^{7} d c a^{2} + \frac{5}{3} x^{6} e^{2} d^{6} c^{3} + 5 x^{6} e^{4} d^{4} c^{2} a + \frac{5}{2} x^{6} e^{6} d^{2} c a^{2} + \frac{1}{6} x^{6} e^{8} a^{3} + x^{5} e d^{7} c^{3} + 6 x^{5} e^{3} d^{5} c^{2} a + 6 x^{5} e^{5} d^{3} c a^{2} + x^{5} e^{7} d a^{3} + \frac{1}{4} x^{4} d^{8} c^{3} + \frac{15}{4} x^{4} e^{2} d^{6} c^{2} a + \frac{15}{2} x^{4} e^{4} d^{4} c a^{2} + \frac{5}{2} x^{4} e^{6} d^{2} a^{3} + x^{3} e d^{7} c^{2} a + 5 x^{3} e^{3} d^{5} c a^{2} + \frac{10}{3} x^{3} e^{5} d^{3} a^{3} + \frac{3}{2} x^{2} e^{2} d^{6} c a^{2} + \frac{5}{2} x^{2} e^{4} d^{4} a^{3} + x e^{3} d^{5} a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.197032, size = 335, normalized size = 3.02 \begin{align*} a^{3} d^{5} e^{3} x + \frac{c^{3} d^{3} e^{5} x^{9}}{9} + x^{8} \left (\frac{3 a c^{2} d^{2} e^{6}}{8} + \frac{5 c^{3} d^{4} e^{4}}{8}\right ) + x^{7} \left (\frac{3 a^{2} c d e^{7}}{7} + \frac{15 a c^{2} d^{3} e^{5}}{7} + \frac{10 c^{3} d^{5} e^{3}}{7}\right ) + x^{6} \left (\frac{a^{3} e^{8}}{6} + \frac{5 a^{2} c d^{2} e^{6}}{2} + 5 a c^{2} d^{4} e^{4} + \frac{5 c^{3} d^{6} e^{2}}{3}\right ) + x^{5} \left (a^{3} d e^{7} + 6 a^{2} c d^{3} e^{5} + 6 a c^{2} d^{5} e^{3} + c^{3} d^{7} e\right ) + x^{4} \left (\frac{5 a^{3} d^{2} e^{6}}{2} + \frac{15 a^{2} c d^{4} e^{4}}{2} + \frac{15 a c^{2} d^{6} e^{2}}{4} + \frac{c^{3} d^{8}}{4}\right ) + x^{3} \left (\frac{10 a^{3} d^{3} e^{5}}{3} + 5 a^{2} c d^{5} e^{3} + a c^{2} d^{7} e\right ) + x^{2} \left (\frac{5 a^{3} d^{4} e^{4}}{2} + \frac{3 a^{2} c d^{6} e^{2}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.14721, size = 419, normalized size = 3.77 \begin{align*} \frac{1}{9} \, c^{3} d^{3} x^{9} e^{5} + \frac{5}{8} \, c^{3} d^{4} x^{8} e^{4} + \frac{10}{7} \, c^{3} d^{5} x^{7} e^{3} + \frac{5}{3} \, c^{3} d^{6} x^{6} e^{2} + c^{3} d^{7} x^{5} e + \frac{1}{4} \, c^{3} d^{8} x^{4} + \frac{3}{8} \, a c^{2} d^{2} x^{8} e^{6} + \frac{15}{7} \, a c^{2} d^{3} x^{7} e^{5} + 5 \, a c^{2} d^{4} x^{6} e^{4} + 6 \, a c^{2} d^{5} x^{5} e^{3} + \frac{15}{4} \, a c^{2} d^{6} x^{4} e^{2} + a c^{2} d^{7} x^{3} e + \frac{3}{7} \, a^{2} c d x^{7} e^{7} + \frac{5}{2} \, a^{2} c d^{2} x^{6} e^{6} + 6 \, a^{2} c d^{3} x^{5} e^{5} + \frac{15}{2} \, a^{2} c d^{4} x^{4} e^{4} + 5 \, a^{2} c d^{5} x^{3} e^{3} + \frac{3}{2} \, a^{2} c d^{6} x^{2} e^{2} + \frac{1}{6} \, a^{3} x^{6} e^{8} + a^{3} d x^{5} e^{7} + \frac{5}{2} \, a^{3} d^{2} x^{4} e^{6} + \frac{10}{3} \, a^{3} d^{3} x^{3} e^{5} + \frac{5}{2} \, a^{3} d^{4} x^{2} e^{4} + a^{3} d^{5} x e^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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