3.429 \(\int (a c+(b c+d) x) \, dx\)

Optimal. Leaf size=17 \[ a c x+\frac{1}{2} x^2 (b c+d) \]

[Out]

a*c*x + ((b*c + d)*x^2)/2

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Rubi [A]  time = 0.0192166, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ a c x+\frac{1}{2} x^2 (b c+d) \]

Antiderivative was successfully verified.

[In]  Int[a*c + (b*c + d)*x,x]

[Out]

a*c*x + ((b*c + d)*x^2)/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ c \int a\, dx + \left (b c + d\right ) \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a*c+(b*c+d)*x,x)

[Out]

c*Integral(a, x) + (b*c + d)*Integral(x, x)

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Mathematica [A]  time = 0.0000479975, size = 22, normalized size = 1.29 \[ a c x+\frac{1}{2} b c x^2+\frac{d x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[a*c + (b*c + d)*x,x]

[Out]

a*c*x + (b*c*x^2)/2 + (d*x^2)/2

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Maple [A]  time = 0.001, size = 16, normalized size = 0.9 \[ acx+{\frac{ \left ( bc+d \right ){x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(a*c+(b*c+d)*x,x)

[Out]

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

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Maxima [A]  time = 0.78335, size = 20, normalized size = 1.18 \[ a c x + \frac{1}{2} \,{\left (b c + d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c*x + 1/2*(b*c + d)*x^2

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Fricas [A]  time = 0.247731, size = 1, normalized size = 0.06 \[ \frac{1}{2} x^{2} c b + \frac{1}{2} x^{2} d + x c a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2*c*b + 1/2*x^2*d + x*c*a

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Sympy [A]  time = 0.069167, size = 15, normalized size = 0.88 \[ a c x + x^{2} \left (\frac{b c}{2} + \frac{d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a*c+(b*c+d)*x,x)

[Out]

a*c*x + x**2*(b*c/2 + d/2)

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GIAC/XCAS [A]  time = 0.257955, size = 20, normalized size = 1.18 \[ a c x + \frac{1}{2} \,{\left (b c + d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c*x + 1/2*(b*c + d)*x^2