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

Optimal. Leaf size=24 \[ \frac{c (a+b x)^2}{2 b}+\frac{d x^2}{2} \]

[Out]

(d*x^2)/2 + (c*(a + b*x)^2)/(2*b)

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Rubi [A]  time = 0.004717, antiderivative size = 24, 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 = {} \[ \frac{c (a+b x)^2}{2 b}+\frac{d x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[d*x + c*(a + b*x),x]

[Out]

(d*x^2)/2 + (c*(a + b*x)^2)/(2*b)

Rubi steps

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

Mathematica [A]  time = 0.0011913, size = 22, normalized size = 0.92 \[ a c x+\frac{1}{2} b c x^2+\frac{d x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[d*x + c*(a + b*x),x]

[Out]

a*c*x + (b*c*x^2)/2 + (d*x^2)/2

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Maple [A]  time = 0., size = 20, normalized size = 0.8 \begin{align*}{\frac{d{x}^{2}}{2}}+c \left ( ax+{\frac{b{x}^{2}}{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.985245, size = 27, normalized size = 1.12 \begin{align*} \frac{1}{2} \, d x^{2} + \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d*x^2 + 1/2*(b*x^2 + 2*a*x)*c

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Fricas [A]  time = 0.878348, size = 45, normalized size = 1.88 \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(d*x+c*(b*x+a),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 = 0.05406, size = 15, normalized size = 0.62 \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(d*x+c*(b*x+a),x)

[Out]

a*c*x + x**2*(b*c/2 + d/2)

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Giac [A]  time = 1.10687, size = 27, normalized size = 1.12 \begin{align*} \frac{1}{2} \, d x^{2} + \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d*x^2 + 1/2*(b*x^2 + 2*a*x)*c