3.1821 \(\int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx\)

Optimal. Leaf size=34 \[ \frac{1}{2} x^2 \left (a e^2+c d^2\right )+a d e x+\frac{1}{3} c d e x^3 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0306304, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \frac{1}{2} x^2 \left (a e^2+c d^2\right )+a d e x+\frac{1}{3} c d e x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

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,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0000969548, size = 38, normalized size = 1.12 \[ a d e x+\frac{1}{2} a e^2 x^2+\frac{1}{2} c d^2 x^2+\frac{1}{3} c d e x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 31, normalized size = 0.9 \[ adex+{\frac{ \left ( a{e}^{2}+c{d}^{2} \right ){x}^{2}}{2}}+{\frac{cde{x}^{3}}{3}} \]

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,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.713499, size = 41, normalized size = 1.21 \[ \frac{1}{3} \, c d e x^{3} + a d e x + \frac{1}{2} \,{\left (c d^{2} + a e^{2}\right )} x^{2} \]

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,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.18286, size = 1, normalized size = 0.03 \[ \frac{1}{3} x^{3} e d c + \frac{1}{2} x^{2} d^{2} c + \frac{1}{2} x^{2} e^{2} a + x e d a \]

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,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.090911, size = 32, normalized size = 0.94 \[ a d e x + \frac{c d e x^{3}}{3} + x^{2} \left (\frac{a e^{2}}{2} + \frac{c d^{2}}{2}\right ) \]

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,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.209742, size = 42, normalized size = 1.24 \[ \frac{1}{3} \, c d x^{3} e + a d x e + \frac{1}{2} \,{\left (c d^{2} + a e^{2}\right )} x^{2} \]

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,x, algorithm="giac")

[Out]

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