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 = 0.0058426, 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., Rules used = {} \[ a c x+\frac{1}{2} x^2 (b c+d) \]

Antiderivative was successfully verified.

[In]

Int[a*c + (b*c + d)*x,x]

[Out]

a*c*x + ((b*c + d)*x^2)/2

Rubi steps

\begin{align*} \int (a c+(b c+d) x) \, dx &=a c x+\frac{1}{2} (b c+d) x^2\\ \end{align*}

Mathematica [A]  time = 0.0000349, 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 = 0.002, size = 16, normalized size = 0.9 \begin{align*} acx+{\frac{ \left ( bc+d \right ){x}^{2}}{2}} \end{align*}

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 = 0.968542, size = 20, normalized size = 1.18 \begin{align*} a c x + \frac{1}{2} \,{\left (b c + d\right )} x^{2} \end{align*}

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 = 0.901043, size = 45, normalized size = 2.65 \begin{align*} \frac{1}{2} x^{2} c b + \frac{1}{2} x^{2} d + x c a \end{align*}

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 = 0.052922, size = 15, normalized size = 0.88 \begin{align*} a c x + x^{2} \left (\frac{b c}{2} + \frac{d}{2}\right ) \end{align*}

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 [A]  time = 1.13595, size = 20, normalized size = 1.18 \begin{align*} a c x + \frac{1}{2} \,{\left (b c + d\right )} x^{2} \end{align*}

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