3.1840 \(\int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx\)

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 = 0.527003, 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 \[ -\frac{3 c^2 d^2 (d+e x)^8 \left (c d^2-a e^2\right )}{8 e^4}+\frac{3 c d (d+e x)^7 \left (c d^2-a e^2\right )^2}{7 e^4}-\frac{(d+e x)^6 \left (c d^2-a e^2\right )^3}{6 e^4}+\frac{c^3 d^3 (d+e x)^9}{9 e^4} \]

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)

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Rubi in Sympy [A]  time = 67.0257, size = 100, normalized size = 0.9 \[ \frac{c^{3} d^{3} \left (d + e x\right )^{9}}{9 e^{4}} + \frac{3 c^{2} d^{2} \left (d + e x\right )^{8} \left (a e^{2} - c d^{2}\right )}{8 e^{4}} + \frac{3 c d \left (d + e x\right )^{7} \left (a e^{2} - c d^{2}\right )^{2}}{7 e^{4}} + \frac{\left (d + e x\right )^{6} \left (a e^{2} - c d^{2}\right )^{3}}{6 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

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Mathematica [B]  time = 0.145617, size = 255, normalized size = 2.3 \[ \frac{1}{504} x \left (84 a^3 e^3 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+36 a^2 c d e^2 x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+9 a c^2 d^2 e x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+c^3 d^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )\right ) \]

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 = 0.001, size = 801, normalized size = 7.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2*(a*e*d+(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*e^3*d^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*e^3*d^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*e^2*d^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*e^3*d^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*e^2*d^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*e*d*(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2)+2*(a*e^2+c*d^2)^2*a*e*
d+d^3*e^3*c*a^2))*x^5+1/4*(d^2*(4*a*e^2*d^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*e*d*(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2)+2*(a*e^2
+c*d^2)^2*a*e*d+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*e*d*
(2*a*c*d^2*e^2+(a*e^2+c*d^2)^2)+2*(a*e^2+c*d^2)^2*a*e*d+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 [A]  time = 0.730226, size = 409, normalized size = 3.68 \[ \frac{1}{9} \, c^{3} d^{3} e^{5} x^{9} + a^{3} d^{5} e^{3} x + \frac{1}{8} \,{\left (5 \, c^{3} d^{4} e^{4} + 3 \, a c^{2} d^{2} e^{6}\right )} x^{8} + \frac{1}{7} \,{\left (10 \, c^{3} d^{5} e^{3} + 15 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{7} + \frac{1}{6} \,{\left (10 \, c^{3} d^{6} e^{2} + 30 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{6} +{\left (c^{3} d^{7} e + 6 \, a c^{2} d^{5} e^{3} + 6 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x^{5} + \frac{1}{4} \,{\left (c^{3} d^{8} + 15 \, a c^{2} d^{6} e^{2} + 30 \, a^{2} c d^{4} e^{4} + 10 \, a^{3} d^{2} e^{6}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a c^{2} d^{7} e + 15 \, a^{2} c d^{5} e^{3} + 10 \, a^{3} d^{3} e^{5}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{6} e^{2} + 5 \, a^{3} d^{4} e^{4}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3*(e*x + d)^2,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 + 3
0*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 [A]  time = 0.184627, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} e^{5} d^{3} c^{3} + \frac{5}{8} x^{8} e^{4} d^{4} c^{3} + \frac{3}{8} x^{8} e^{6} d^{2} c^{2} a + \frac{10}{7} x^{7} e^{3} d^{5} c^{3} + \frac{15}{7} x^{7} e^{5} d^{3} c^{2} a + \frac{3}{7} x^{7} e^{7} d c a^{2} + \frac{5}{3} x^{6} e^{2} d^{6} c^{3} + 5 x^{6} e^{4} d^{4} c^{2} a + \frac{5}{2} x^{6} e^{6} d^{2} c a^{2} + \frac{1}{6} x^{6} e^{8} a^{3} + x^{5} e d^{7} c^{3} + 6 x^{5} e^{3} d^{5} c^{2} a + 6 x^{5} e^{5} d^{3} c a^{2} + x^{5} e^{7} d a^{3} + \frac{1}{4} x^{4} d^{8} c^{3} + \frac{15}{4} x^{4} e^{2} d^{6} c^{2} a + \frac{15}{2} x^{4} e^{4} d^{4} c a^{2} + \frac{5}{2} x^{4} e^{6} d^{2} a^{3} + x^{3} e d^{7} c^{2} a + 5 x^{3} e^{3} d^{5} c a^{2} + \frac{10}{3} x^{3} e^{5} d^{3} a^{3} + \frac{3}{2} x^{2} e^{2} d^{6} c a^{2} + \frac{5}{2} x^{2} e^{4} d^{4} a^{3} + x e^{3} d^{5} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3*(e*x + d)^2,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 [A]  time = 0.334889, size = 335, normalized size = 3.02 \[ a^{3} d^{5} e^{3} x + \frac{c^{3} d^{3} e^{5} x^{9}}{9} + x^{8} \left (\frac{3 a c^{2} d^{2} e^{6}}{8} + \frac{5 c^{3} d^{4} e^{4}}{8}\right ) + x^{7} \left (\frac{3 a^{2} c d e^{7}}{7} + \frac{15 a c^{2} d^{3} e^{5}}{7} + \frac{10 c^{3} d^{5} e^{3}}{7}\right ) + x^{6} \left (\frac{a^{3} e^{8}}{6} + \frac{5 a^{2} c d^{2} e^{6}}{2} + 5 a c^{2} d^{4} e^{4} + \frac{5 c^{3} d^{6} e^{2}}{3}\right ) + x^{5} \left (a^{3} d e^{7} + 6 a^{2} c d^{3} e^{5} + 6 a c^{2} d^{5} e^{3} + c^{3} d^{7} e\right ) + x^{4} \left (\frac{5 a^{3} d^{2} e^{6}}{2} + \frac{15 a^{2} c d^{4} e^{4}}{2} + \frac{15 a c^{2} d^{6} e^{2}}{4} + \frac{c^{3} d^{8}}{4}\right ) + x^{3} \left (\frac{10 a^{3} d^{3} e^{5}}{3} + 5 a^{2} c d^{5} e^{3} + a c^{2} d^{7} e\right ) + x^{2} \left (\frac{5 a^{3} d^{4} e^{4}}{2} + \frac{3 a^{2} c d^{6} e^{2}}{2}\right ) \]

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/XCAS [A]  time = 0.211274, size = 419, normalized size = 3.77 \[ \frac{1}{9} \, c^{3} d^{3} x^{9} e^{5} + \frac{5}{8} \, c^{3} d^{4} x^{8} e^{4} + \frac{10}{7} \, c^{3} d^{5} x^{7} e^{3} + \frac{5}{3} \, c^{3} d^{6} x^{6} e^{2} + c^{3} d^{7} x^{5} e + \frac{1}{4} \, c^{3} d^{8} x^{4} + \frac{3}{8} \, a c^{2} d^{2} x^{8} e^{6} + \frac{15}{7} \, a c^{2} d^{3} x^{7} e^{5} + 5 \, a c^{2} d^{4} x^{6} e^{4} + 6 \, a c^{2} d^{5} x^{5} e^{3} + \frac{15}{4} \, a c^{2} d^{6} x^{4} e^{2} + a c^{2} d^{7} x^{3} e + \frac{3}{7} \, a^{2} c d x^{7} e^{7} + \frac{5}{2} \, a^{2} c d^{2} x^{6} e^{6} + 6 \, a^{2} c d^{3} x^{5} e^{5} + \frac{15}{2} \, a^{2} c d^{4} x^{4} e^{4} + 5 \, a^{2} c d^{5} x^{3} e^{3} + \frac{3}{2} \, a^{2} c d^{6} x^{2} e^{2} + \frac{1}{6} \, a^{3} x^{6} e^{8} + a^{3} d x^{5} e^{7} + \frac{5}{2} \, a^{3} d^{2} x^{4} e^{6} + \frac{10}{3} \, a^{3} d^{3} x^{3} e^{5} + \frac{5}{2} \, a^{3} d^{4} x^{2} e^{4} + a^{3} d^{5} x e^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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