3.2099 \(\int (d+e x) \left (a+b x+c x^2\right ) \, dx\)

Optimal. Leaf size=42 \[ \frac{1}{2} x^2 (a e+b d)+a d x+\frac{1}{3} x^3 (b e+c d)+\frac{1}{4} c e x^4 \]

[Out]

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

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Rubi [A]  time = 0.0759425, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{1}{2} x^2 (a e+b d)+a d x+\frac{1}{3} x^3 (b e+c d)+\frac{1}{4} c e x^4 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0252163, size = 42, normalized size = 1. \[ \frac{1}{2} x^2 (a e+b d)+a d x+\frac{1}{3} x^3 (b e+c d)+\frac{1}{4} c e x^4 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 37, normalized size = 0.9 \[ adx+{\frac{ \left ( ae+bd \right ){x}^{2}}{2}}+{\frac{ \left ( be+cd \right ){x}^{3}}{3}}+{\frac{ce{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+b*x+a),x)

[Out]

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

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Maxima [A]  time = 0.812731, size = 49, normalized size = 1.17 \[ \frac{1}{4} \, c e x^{4} + \frac{1}{3} \,{\left (c d + b e\right )} x^{3} + a d x + \frac{1}{2} \,{\left (b d + a e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.181663, size = 1, normalized size = 0.02 \[ \frac{1}{4} x^{4} e c + \frac{1}{3} x^{3} d c + \frac{1}{3} x^{3} e b + \frac{1}{2} x^{2} d b + \frac{1}{2} x^{2} e a + x d a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.091038, size = 39, normalized size = 0.93 \[ a d x + \frac{c e x^{4}}{4} + x^{3} \left (\frac{b e}{3} + \frac{c d}{3}\right ) + x^{2} \left (\frac{a e}{2} + \frac{b d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.201956, size = 58, normalized size = 1.38 \[ \frac{1}{4} \, c x^{4} e + \frac{1}{3} \, c d x^{3} + \frac{1}{3} \, b x^{3} e + \frac{1}{2} \, b d x^{2} + \frac{1}{2} \, a x^{2} e + a d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d),x, algorithm="giac")

[Out]

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