3.1830 \(\int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx\)

Optimal. Leaf size=77 \[ -\frac{c d (d+e x)^4 \left (c d^2-a e^2\right )}{2 e^3}+\frac{(d+e x)^3 \left (c d^2-a e^2\right )^2}{3 e^3}+\frac{c^2 d^2 (d+e x)^5}{5 e^3} \]

[Out]

((c*d^2 - a*e^2)^2*(d + e*x)^3)/(3*e^3) - (c*d*(c*d^2 - a*e^2)*(d + e*x)^4)/(2*e
^3) + (c^2*d^2*(d + e*x)^5)/(5*e^3)

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Rubi [A]  time = 0.212987, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{c d (d+e x)^4 \left (c d^2-a e^2\right )}{2 e^3}+\frac{(d+e x)^3 \left (c d^2-a e^2\right )^2}{3 e^3}+\frac{c^2 d^2 (d+e x)^5}{5 e^3} \]

Antiderivative was successfully verified.

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

[Out]

((c*d^2 - a*e^2)^2*(d + e*x)^3)/(3*e^3) - (c*d*(c*d^2 - a*e^2)*(d + e*x)^4)/(2*e
^3) + (c^2*d^2*(d + e*x)^5)/(5*e^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 a d e \left (a e^{2} + c d^{2}\right ) \int x\, dx + \frac{c^{2} d^{2} e^{2} x^{5}}{5} + \frac{c d e x^{4} \left (a e^{2} + c d^{2}\right )}{2} + d^{2} e^{2} \int a^{2}\, dx + x^{3} \left (\frac{a^{2} e^{4}}{3} + \frac{4 a c d^{2} e^{2}}{3} + \frac{c^{2} d^{4}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*d*e*(a*e**2 + c*d**2)*Integral(x, x) + c**2*d**2*e**2*x**5/5 + c*d*e*x**4*(a
*e**2 + c*d**2)/2 + d**2*e**2*Integral(a**2, x) + x**3*(a**2*e**4/3 + 4*a*c*d**2
*e**2/3 + c**2*d**4/3)

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Mathematica [A]  time = 0.0497254, size = 87, normalized size = 1.13 \[ \frac{1}{30} x \left (10 a^2 e^2 \left (3 d^2+3 d e x+e^2 x^2\right )+5 a c d e x \left (6 d^2+8 d e x+3 e^2 x^2\right )+c^2 d^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0., size = 93, normalized size = 1.2 \[{\frac{{d}^{2}{e}^{2}{c}^{2}{x}^{5}}{5}}+{\frac{ \left ( a{e}^{2}+c{d}^{2} \right ) dec{x}^{4}}{2}}+{\frac{ \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ){x}^{3}}{3}}+aed \left ( a{e}^{2}+c{d}^{2} \right ){x}^{2}+{a}^{2}{e}^{2}{d}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.714872, size = 126, normalized size = 1.64 \[ \frac{1}{5} \, c^{2} d^{2} e^{2} x^{5} + \frac{1}{2} \,{\left (c d^{2} + a e^{2}\right )} c d e x^{4} + a^{2} d^{2} e^{2} x + \frac{1}{3} \,{\left (c d^{2} + a e^{2}\right )}^{2} x^{3} + \frac{1}{3} \,{\left (2 \, c d e x^{3} + 3 \,{\left (c d^{2} + a e^{2}\right )} x^{2}\right )} a d e \]

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.216502, size = 1, normalized size = 0.01 \[ \frac{1}{5} x^{5} e^{2} d^{2} c^{2} + \frac{1}{2} x^{4} e d^{3} c^{2} + \frac{1}{2} x^{4} e^{3} d c a + \frac{1}{3} x^{3} d^{4} c^{2} + \frac{4}{3} x^{3} e^{2} d^{2} c a + \frac{1}{3} x^{3} e^{4} a^{2} + x^{2} e d^{3} c a + x^{2} e^{3} d a^{2} + x e^{2} d^{2} a^{2} \]

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)^2,x, algorithm="fricas")

[Out]

1/5*x^5*e^2*d^2*c^2 + 1/2*x^4*e*d^3*c^2 + 1/2*x^4*e^3*d*c*a + 1/3*x^3*d^4*c^2 +
4/3*x^3*e^2*d^2*c*a + 1/3*x^3*e^4*a^2 + x^2*e*d^3*c*a + x^2*e^3*d*a^2 + x*e^2*d^
2*a^2

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Sympy [A]  time = 0.163502, size = 104, normalized size = 1.35 \[ a^{2} d^{2} e^{2} x + \frac{c^{2} d^{2} e^{2} x^{5}}{5} + x^{4} \left (\frac{a c d e^{3}}{2} + \frac{c^{2} d^{3} e}{2}\right ) + x^{3} \left (\frac{a^{2} e^{4}}{3} + \frac{4 a c d^{2} e^{2}}{3} + \frac{c^{2} d^{4}}{3}\right ) + x^{2} \left (a^{2} d e^{3} + a c d^{3} e\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*d**2*e**2*x + c**2*d**2*e**2*x**5/5 + x**4*(a*c*d*e**3/2 + c**2*d**3*e/2) +
x**3*(a**2*e**4/3 + 4*a*c*d**2*e**2/3 + c**2*d**4/3) + x**2*(a**2*d*e**3 + a*c*
d**3*e)

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GIAC/XCAS [A]  time = 0.207043, size = 136, normalized size = 1.77 \[ \frac{1}{5} \, c^{2} d^{2} x^{5} e^{2} + \frac{1}{2} \, c^{2} d^{3} x^{4} e + \frac{1}{3} \, c^{2} d^{4} x^{3} + \frac{1}{2} \, a c d x^{4} e^{3} + \frac{4}{3} \, a c d^{2} x^{3} e^{2} + a c d^{3} x^{2} e + \frac{1}{3} \, a^{2} x^{3} e^{4} + a^{2} d x^{2} e^{3} + a^{2} d^{2} x e^{2} \]

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)^2,x, algorithm="giac")

[Out]

1/5*c^2*d^2*x^5*e^2 + 1/2*c^2*d^3*x^4*e + 1/3*c^2*d^4*x^3 + 1/2*a*c*d*x^4*e^3 +
4/3*a*c*d^2*x^3*e^2 + a*c*d^3*x^2*e + 1/3*a^2*x^3*e^4 + a^2*d*x^2*e^3 + a^2*d^2*
x*e^2