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

Optimal. Leaf size=33 \[ \frac{1}{3} x^3 (b e+c d)+\frac{1}{2} b d x^2+\frac{1}{4} c e x^4 \]

[Out]

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

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Rubi [A]  time = 0.0731564, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{3} x^3 (b e+c d)+\frac{1}{2} b d x^2+\frac{1}{4} c e x^4 \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(b*x + c*x^2),x]

[Out]

(b*d*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. \[ b d \int x\, dx + \frac{c e x^{4}}{4} + x^{3} \left (\frac{b e}{3} + \frac{c d}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00822708, size = 29, normalized size = 0.88 \[ \frac{1}{12} x^2 (b (6 d+4 e x)+c x (4 d+3 e x)) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(c*x*(4*d + 3*e*x) + b*(6*d + 4*e*x)))/12

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Maple [A]  time = 0.001, size = 28, normalized size = 0.9 \[{\frac{{x}^{2}bd}{2}}+{\frac{ \left ( be+cd \right ){x}^{3}}{3}}+{\frac{ce{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+b*x),x)

[Out]

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

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Maxima [A]  time = 0.754079, size = 36, normalized size = 1.09 \[ \frac{1}{4} \, c e x^{4} + \frac{1}{2} \, b d x^{2} + \frac{1}{3} \,{\left (c d + b e\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.196854, size = 1, normalized size = 0.03 \[ \frac{1}{4} x^{4} e c + \frac{1}{3} x^{3} d c + \frac{1}{3} x^{3} e b + \frac{1}{2} x^{2} d b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(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

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Sympy [A]  time = 0.078837, size = 29, normalized size = 0.88 \[ \frac{b d x^{2}}{2} + \frac{c e x^{4}}{4} + x^{3} \left (\frac{b e}{3} + \frac{c d}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.2038, size = 42, normalized size = 1.27 \[ \frac{1}{4} \, c x^{4} e + \frac{1}{3} \, c d x^{3} + \frac{1}{3} \, b x^{3} e + \frac{1}{2} \, b d x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(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